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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 04 2020
STATUS
approved