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Number of ways to write n as the sum of a practical number (A005153) and a triangular number (A000217).
+20
21
1, 2, 1, 2, 2, 1, 3, 2, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 3, 1, 3, 3, 3, 2, 2, 2, 3, 2, 3, 4, 3, 2, 4, 3, 2, 3, 3, 3, 3, 4, 2, 4, 3, 2, 3, 4, 2, 4, 3, 1, 4, 3, 2, 3, 2, 4, 6, 2, 2, 4, 4, 1, 5, 4, 2, 4, 4, 3, 4, 4, 2, 4, 3, 2, 5, 3, 2, 4, 4, 2, 5, 4, 2, 6, 4, 3, 5, 3, 1, 6, 3, 3, 5, 5, 3, 5, 3, 3, 5, 4
OFFSET
1,2
COMMENTS
Conjecture: a(n)>0 for all n>0.
The author has verified this for n up to 10^8, and also guessed the following refinement: If n>6 is not among 20, 104, 272, 464, 1664, then n can be written as p+q with p an even practical number and q a positive triangular number.
Somu and Tran (2024) proved the conjecture that a(n)>0 for n>0. - Duc Van Khanh Tran, Apr 24 2024
LINKS
Giuseppe Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Sai Teja Somu and Duc Van Khanh Tran, On Sums of Practical Numbers and Polygonal Numbers, arXiv:2403.13533 [math.NT], 2024.
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(15)=1 since 15=12+3 with 12 a practical number and 3 a triangular number.
MATHEMATICA
f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[n-k(k+1)/2]==True, 1, 0], {k, 0, (Sqrt[8n+1]-1)/2}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Zhi-Wei Sun, Jan 11 2013
STATUS
approved
Number of ways to write 2n-1 = p+q, where p is a prime, and both q and q+2 are practical numbers (A005153).
+20
20
0, 0, 1, 2, 3, 2, 2, 2, 2, 3, 4, 4, 3, 2, 3, 3, 5, 6, 3, 3, 4, 4, 5, 7, 4, 3, 4, 2, 5, 7, 4, 4, 5, 4, 5, 7, 4, 5, 8, 2, 5, 7, 5, 5, 6, 6, 4, 7, 4, 5, 9, 3, 5, 9, 4, 6, 6, 5, 5, 7, 3, 3, 7, 3, 6, 8, 5, 4, 8, 4, 5, 8, 4, 4, 5, 3, 5, 8, 6, 3, 6, 4, 5, 12, 5, 5, 5, 3, 6, 8, 5, 4, 8, 4, 4, 8, 4, 6, 9, 5
OFFSET
1,4
COMMENTS
Conjecture: a(n)>0 for all n=3,4,...
The author has verified this for n up to 2*10^8. It is known that there are infinitely many practical numbers q with q+2 also practical.
Zhi-Wei Sun also made the following similar conjectures:
(1) Each odd number n>5 can be written as p+q with p and p+6 both prime and q practical. Also, any odd number n>3 not equal to 55 can be written as p+q with p and p+2 both prime and q practical.
(2) Each integer n>10 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+6 both practical.
Also, any integer n>=6360 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+2 both practical.
LINKS
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(14)=2 since 2*14-1=27=11+16=23+4, where 11 and 23 are primes, 16,16+2,4,4+2 are practical numbers.
MATHEMATICA
f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[2k]==True&&pr[2k+2]==True&&PrimeQ[2n-1-2k]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 11 2013
STATUS
approved
Characteristic function of practical numbers, A005153: If n is in A005153, a(n) = 1, otherwise a(n) = 0.
+20
7
1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1
OFFSET
1
COMMENTS
Characteristic function of practical numbers, A005153.
FORMULA
a(n) = [A030057(n) > n], where [ ] is the Iverson bracket.
PROG
(PARI) A322860 = is_A005153 \\ Please do not duplicate code by pasting copies from one place to others, it will be near to impossible to track down all copies if something has to be corrected or improved. - M. F. Hasler, Jun 19 2023
CROSSREFS
Cf. A005153, A030057, A225559 (partial sums).
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 31 2018
STATUS
approved
Primes of the form p - 1, where p is a practical number (A005153).
+20
4
3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 179, 191, 197, 199, 223, 227, 233, 239, 251, 263, 269, 271, 293, 307, 311, 347, 359, 367, 379, 383, 389, 419, 431, 439, 449, 461, 463, 467, 479, 499, 503, 509
OFFSET
1,1
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
a(5)=17 as 18 is a practical number, 18-1=17 and it is the 5th such prime.
MATHEMATICA
PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
Select[Table[Prime[n]+1, {n, 1, 200}], PracticalQ]-1 (* using T. D. Noe's program A005153 *)
PROG
(PARI) isPractical(n)={
if(n%2, return(n==1));
my(f=factor(n), P=1);
for(i=1, #f[, 1]-1,
P*=sigma(f[i, 1]^f[i, 2]);
if(f[i+1, 1]>P+1, return(0))
);
n>0
};
select(p->isPractical(p+1), primes(300)) \\ Charles R Greathouse IV, May 03 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, May 02 2013
STATUS
approved
The number of practical numbers <= n where the practical numbers are A005153.
+20
4
1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22
OFFSET
1,2
COMMENTS
a(n) is analogous to A000720.
LINKS
EXAMPLE
a(13)=6 as there are 6 practical numbers <= 13, namely 1, 2, 4, 6, 8 and 12.
MAPLE
isprac:= proc(n) local L, i, P;
L:= sort(ifactors(n)[2], (a, b) -> a[1]<b[1]);
if L[1][1] <> 2 then return false fi;
P:= 2^(L[1][2]+1)-1;
for i from 2 to nops(L) do
if L[i][1] > P+1 then return false fi;
P:= P*(L[i][1]^(L[i][2]+1)-1)/(L[i][1]-1);
od;
true
end proc:
isprac(1):= true:
N:= 100: # to get a(1)..a(N)
P:= select(isprac, [1, seq(i, i=2..N, 2)]):
V:= Vector(N):
for n from 2 to nops(P) do V[P[n-1] .. P[n]-1]:= n-1 od:
V[P[-1]..N]:= n:
convert(V, list); # Robert Israel, May 29 2019
MATHEMATICA
PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; t={}; n3=1; n4=0; While[n3<100, (If[PracticalQ[n3], n4++]; AppendTo[t, n4]; n3++)]; t (* using T. D. Noe's program A005153 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, May 10 2013
STATUS
approved
Zumkeller numbers (A083207) that are not practical numbers (A005153).
+20
4
70, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 350, 354, 366, 372, 402, 426, 438, 444, 474, 490, 492, 498, 516, 534, 550, 564, 572, 582, 606, 618, 636, 642, 650, 654, 678, 708, 732, 762, 770, 786, 804, 822, 834, 836, 852, 876, 894, 906, 910, 940, 942, 945, 948, 978, 996
OFFSET
1,1
COMMENTS
Different from A007621: A007621 contains no odd numbers, while every odd term in A083207 is here. The numbers 738, 748, 774, 846, ... are in A007621 and are not here.
But the subsequence of even terms (A005843 intersect this sequence) is a subsequence of A007621:
- A005843 intersect this sequence = (A005843 intersect A083207) \ A005153;
- A083207 is a subsequence of A023196, and every perfect number is practical;
- So, (A005843 intersect A083207) \ A005153 is a subsequence of A173490, and A005153 is a supersequence of A007620.
LINKS
EXAMPLE
70 is a term since 70 is a Zumkeller number but not a practical number: 1+5+7+10+14+35 = 2+70, so 70 is a Zumkeller number; but 4 cannot be written as a sum of distinct divisors of 70, so 70 is not practical.
PROG
(PARI) isA353061(n) = is(n) && !is_A005153(n) \\ See A083207 for is(n) and A005153 for is_A005153(n)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Apr 20 2022
STATUS
approved
Largest primitive practical number p that divides the n-th practical number - A005153(n) such that the radical of the quotient A005153(n)/p is a divisor of p.
+20
4
1, 2, 2, 6, 2, 6, 2, 6, 20, 6, 28, 30, 2, 6, 20, 42, 6, 6, 28, 30, 2, 66, 6, 78, 20, 42, 88, 30, 6, 20, 104, 6, 28, 30, 42, 2, 66, 140, 6, 30, 78, 20, 6, 42, 88, 30, 6, 28, 66, 20, 204, 104, 210, 6, 220, 28, 228, 78, 30, 42, 2, 260, 66, 30, 272, 276, 140, 6, 42, 30, 304, 306, 308, 78, 20, 6, 330, 42, 340, 342, 348, 88, 30, 364, 368, 42, 380, 6, 390, 28, 66
OFFSET
1,2
COMMENTS
The relationship between a practical number and the largest primitive practical number that divides it such that the radical of the quotient is a divisor of same primitive provides an equivalence relation. Practical numbers that have the same progenitive primitive. This property arises from the characteristic of practical numbers and, in particular, primitives that says a practical number multiplied by power combinations of any of its divisors is also practical. This sequence identifies the primitive progenitor of each practical number A005153(n).
Note that this sequence and A378202 are similar but the first difference is at a(63) as explained in the example.
LINKS
EXAMPLE
a(63) = 66. A005153(63) = 264 and the largest primitive practical number that divides the practical number 264 is 88. However the radical of the quotient 264/88 is 3 and 3 is not a divisor of 88. The next greatest primitive divisor of 264 is 66 and the radical of the quotient 264/66 is 2 and 2 is a divisor of 66.
a(131) = 306. A005153(131) = 612 and it is divisible by two primitive practical numbers 204 and 306 with their quotient a divisor of their primitive in both cases but 306 is chosen as the larger primitive.
MATHEMATICA
plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; pplst=Last/@ReadList["https://oeis.org/A267124/b267124.txt", {Number, Number}]; Rad[n_] := Times @@ First /@ FactorInteger[n]; getpplst[n_] := Module[{}, Select[pplst, #<=n &]]; lst1={}; Do[lst=getpplst[plst[[n]]]; lnh=Length@lst; m=0; While[Mod[j=plst[[n]], k=lst[[lnh-m]]]!=0||Mod[k, Rad[j/k]]!=0, m++]; AppendTo[lst1, {j, k}], {n, 1, 100}]; Last/@lst1
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Dec 20 2024
STATUS
approved
Primes p such that no practical number (A005153) exists between p and its successor.
+20
3
43, 67, 101, 137, 163, 181, 229, 241, 281, 313, 353, 421, 433, 487, 563, 601, 617, 641, 653, 673, 769, 821, 823, 853, 883, 907, 937, 941, 1009, 1061, 1093, 1277, 1303, 1361, 1423, 1429, 1433, 1447, 1489, 1549, 1571, 1579, 1601, 1607, 1609, 1613, 1657, 1667, 1697, 1741, 1747
OFFSET
1,1
COMMENTS
According to Margenstern and proved by Weingartner (see links) the density of practical numbers is greater than the density of primes. Margenstern calculated that the density of practical numbers was approx 1.2767 (1.3411/1.059) times greater than the density of primes in the interval 1 to 10^12. This sequence shows that the set of places where no practical number exists between successive primes has a degree of regularity and appears to be infinite.
LINKS
Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory 37 (1): 1-36, 1991.
A. Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics 66 (2): 743-758, 2015.
Wikipedia, Practical number
EXAMPLE
a(6) = 181, the next prime is 191. In the integer interval [181, 191] there are no practical numbers. It is the 6th such occurrence.
MATHEMATICA
PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod = prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; count[n_Integer] := Module[{t=0, m}, Do[If[PracticalQ[m], t++], {m, Prime[n], Prime[n + 1] - 1}]; t]; lst = {}; Do[If[count[n]==0, AppendTo[lst, Prime[n]]], {n, 1, 1000}]; lst
CROSSREFS
Cf. A005153.
KEYWORD
nonn
AUTHOR
Frank M Jackson, May 29 2016
STATUS
approved
a(n) = 0 whenever n is a practical number (A005153) otherwise least powers of 2 that when multiplied by n becomes practical.
+20
3
0, 0, 1, 0, 2, 0, 2, 0, 1, 1, 3, 0, 3, 1, 1, 0, 4, 0, 4, 0, 1, 2, 4, 0, 2, 2, 1, 0, 4, 0, 4, 0, 1, 3, 2, 0, 5, 3, 1, 0, 5, 0, 5, 1, 1, 3, 5, 0, 2, 1, 2, 1, 5, 0, 2, 0, 2, 3, 5, 0, 5, 3, 1, 0, 2, 0, 6, 2, 2, 1, 6, 0, 6, 4, 1, 2, 2, 0, 6, 0, 1, 4, 6, 0, 2, 4, 2, 0, 6, 0, 2, 2, 3, 4, 2, 0, 6, 1, 1, 0
OFFSET
1,5
COMMENTS
A conjecture by Zhi-Wei Sun states that any rational number can be expressed as the sum of distinct unit fractions whose denominators are practical numbers. To prove this conjecture, David Eppstein (see link) used the fact that every natural number when repeatedly multiplied by 2 will eventually become practical.
LINKS
EXAMPLE
a(11) = 3 because 11 * 2^3 = 88 is a practical number and 3 is the least power of 2 which when multiplied by 11 becomes practical.
MATHEMATICA
practicalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 ||(n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod * p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Table[(m = n; k = 0; While[! practicalQ[m], m = 2 * m; k++]; k), {n, 100}]
CROSSREFS
Cf. A005153.
KEYWORD
nonn
AUTHOR
Frank M Jackson, Dec 04 2016
STATUS
approved
The practical component of n: the largest divisor of n which is a practical number (A005153).
+20
3
1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 2, 1, 24, 1, 2, 1, 28, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 56, 1, 2, 1, 60, 1, 2, 1, 64, 1, 66, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 78, 1, 80, 1
OFFSET
1,2
COMMENTS
From Andreas Weingartner, Jun 30 2021: (Start)
Let r_m be the natural density of the set of integers n with a(n) = m. Then r_m is positive if and only if m is practical. In that case, r_m = (1/m)*P_m, where P_m is the product of (1-1/p) over primes p <= sigma(m) + 1 (see Cor. 1 of Weingartner 2015). The first few values of (m, r_m) are (1, 1/2), (2, 1/6), (4, 2/35), (6, 32/1001), (8, 24/1001), (12, 36864/2800733), ...
As y grows, the natural density of integers n, which satisfy a(n) > y, is asymptotic to c*exp(-gamma)/log(y), where c = 1.33607... is the constant factor in the asymptotic for the count of practical numbers (A005153) and gamma = 0.577215... is Euler's constant (see Eq. (3) of Weingartner (2015)). For example, about 1% of integers n satisfy a(n) > exp(75), because c*exp(-gamma)/75 = 0.010... (End)
LINKS
Paul Pollack and Lola Thompson, Practical pretenders, Publicationes Mathematicae Debrecen, Vol. 82, No. 3-4 (2013), pp. 651-717, arXiv preprint, arXiv:1201.3168 [math.NT], 2012.
Andreas Weingartner, Integers with large practical component, Publicationes Mathematicae Debrecen, Vol. 87, No. 3-4 (2015), pp. 439-447, arXiv preprint, arXiv:1411.6974v2 [math.NT], 2014-2015.
FORMULA
If n = Product_{i=1..r} p_i^e_i, then define n_0 = 1, n_j = Product_{i=1..j} p_i^e_i. a(n) = n_j where j is the first index for which p_{j+1} > sigma(n_j) + 1, or j = r if no such index exists.
A number n is practical if and only if a(n) = n.
a(n) = 1 if and only if n is odd.
A000203(a(n)) = A225561(n).
EXAMPLE
a(22) = 2 since the divisors of 22 are {1, 2, 11, 22}, of them {1, 2} are practical, and 2 being the largest.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[n_] := If[(ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}, n, Times @@ (Power @@@ fct[[1 ;; ind[[1, 1]] - 1]])]; Array[a, 100]
PROG
(PARI) \\ using is_A005153
a(n) = fordiv(n, d, if(is_A005153(n/d), return(n/d))); \\ Michel Marcus, Jul 03 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 27 2019
STATUS
approved

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