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

1,2

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.

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.)

Example

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.

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]

Crossrefs

Cf. A001318, A005153, A085787, A208244.

Sequence in context: A086598 A211261 A344174 * A074746 A133188 A261036

Adjacent sequences: A336428 A336429 A336430 * A336432 A336433 A336434

Keyword

nonn

Author

Zhi-Wei Sun, Oct 04 2020

Status

approved