A336431 Number of ordered ways to write n as the sum of a practical number (A005153) and a generalized heptagonal number (A085787).

1, 2, 1, 1, 2, 2, 1, 3, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 4, 3, 2, 2, 1, 3, 3, 1, 1, 3, 3, 2, 4, 2, 3, 2, 3, 4, 3, 2, 2, 3, 2, 3, 4, 1, 2, 3, 3, 3, 3, 3, 1, 3, 1, 4, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 4, 5, 1, 2, 3, 3, 3, 4, 4, 2, 4, 1, 4, 4, 1, 2, 6, 3, 3, 5, 2, 4, 5, 2, 4, 5, 1, 3, 4, 2, 2, 5, 3, 2, 6

Offset: 1

Author

Zhi-Wei Sun, Oct 04 2020

Keywords

• nonn

Comments

a(n) > 0 for all n = 1..5*10^7.

Conjecture 1: a(n) > 0 for all n > 0. Also, a(n) = 1 only for n = 1, 3, 4, 7, 10, 11, 14, 18, 23, 26, 27, 44, 51, 53, 65, 68, 77, 80, 92, 125, 143, 170, 179, 182, 185, 191, 206, 296, 362, 383, 425, 437, 674, 1340, 1622, 2273, 2558, 3167, 5591.

Conjecture 2: Any positive integer can be written as the sum of a practical number and a generalized pentagonal number.

Both conjectures are motivated by A208244.

Examples

			Set p_7(x) = x*(5*x-3)/2.
a(14) = 1, and 14 = 1 + p_7(-2) with 1 practical.
a(80) = 1, and 80 = 80 + p_7(0) with 80 practical.
a(425) = 1, and 425 = 160 + p_7(-10) with 160 practical.
a(1340) = 1, and 1340 = 800 + p_7(15) with 800 practical.
a(2273) = 1, and 2273 = 544 + p_7(-26) with 544 practical.
a(5591) = 1, and 5591 = 2752 + p_7(34) with 2752 practical.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A001318, A005153, A085787, A208244.

Programs

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);
tab={};Do[r=0;Do[If[pr[n-x*(5*x-3)/2],r=r+1],{x,-Floor[(Sqrt[40n+9]-3)/10],(Sqrt[40n+9]+3)/10}];tab=Append[tab,r],{n,1,100}];Print[tab]