cp's OEIS Frontend

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

%I A336431 #25 Oct 06 2020 12:22:39
%S A336431 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,
%T A336431 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,
%U A336431 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
%N A336431 Number of ordered ways to write n as the sum of a practical number (A005153) and a generalized heptagonal number (A085787).
%C A336431 a(n) > 0 for all n = 1..5*10^7.
%C A336431 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.
%C A336431 Conjecture 2: Any positive integer can be written as the sum of a practical number and a generalized pentagonal number.
%C A336431 Both conjectures are motivated by A208244.
%H A336431 Zhi-Wei Sun, <a href="/A336431/b336431.txt">Table of n, a(n) for n = 1..10000</a>
%H A336431 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv</a>, arXiv:1211.1588 [math.NT], 2012-2017.)
%e A336431 Set p_7(x) = x*(5*x-3)/2.
%e A336431 a(14) = 1, and 14 = 1 + p_7(-2) with 1 practical.
%e A336431 a(80) = 1, and 80 = 80 + p_7(0) with 80 practical.
%e A336431 a(425) = 1, and 425 = 160 + p_7(-10) with 160 practical.
%e A336431 a(1340) = 1, and 1340 = 800 + p_7(15) with 800 practical.
%e A336431 a(2273) = 1, and 2273 = 544 + p_7(-26) with 544 practical.
%e A336431 a(5591) = 1, and 5591 = 2752 + p_7(34) with 2752 practical.
%t A336431 f[n_]:=f[n]=FactorInteger[n];
%t A336431 Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
%t A336431 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}];
%t A336431 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
%t A336431 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]
%Y A336431 Cf. A001318, A005153, A085787, A208244.
%K A336431 nonn
%O A336431 1,2
%A A336431 _Zhi-Wei Sun_, Oct 04 2020