cp's OEIS Frontend

A335641 Number of ordered ways to write 2n+1 as p + x*(9x+7) with p prime and x an integer.

%I A335641 #18 Jan 08 2025 11:22:07
%S A335641 1,2,2,1,1,2,1,1,3,2,2,2,2,3,2,3,2,1,3,2,2,3,2,1,1,4,2,2,3,3,5,2,3,4,
%T A335641 2,4,3,2,3,3,4,1,2,4,3,2,2,3,2,3,4,4,3,4,3,4,2,2,5,4,4,3,3,5,4,5,2,1,
%U A335641 6,1,3,2,3,4,3,5,2,4,4,3,5,2,3,4,1,5,4,3,4,4,4,3,3,5,4,3,6,4,6,5
%N A335641 Number of ordered ways to write 2n+1 as p + x*(9x+7) with p prime and x an integer.
%C A335641 Conjecture 1: a(n) > 0 for all n > 0. Also, a(n) = 1 only for n = 1, 4, 5, 7, 8, 18, 24, 25, 42, 68, 70, 85, 117, 118, 196, 238, 287, 497, 628, 677, 732.
%C A335641 We have verified a(n) > 0 for all n = 1..2*10^8.
%C A335641 Conjecture 2: Let f(x) be any of the polynomials x*(3x+1), x*(5x+1), 2x*(3x+1), 2x*(3x+2). Then, each odd integer greater than one can be written as p + f(x) with p prime and x an integer.
%H A335641 Zhi-Wei Sun, <a href="/A335641/b335641.txt">Table of n, a(n) for n = 1..10000</a>
%H A335641 Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. and Number Theory 1(2009), no. 1, 65-76. See also <a href="https://arxiv.org/abs/0803.3737">arXiv:0803.3737</a> [math.NT], 2008.
%e A335641 a(68) = 1, and 2*68+1 = 137 + 0*(9*0+7) with 137 prime.
%e A335641 a(117) = 1, and 2*117+1 = 233 + (-1)*(9*(-1)+7) with 233 prime.
%e A335641 a(238) = 1, and 2*238+1 = 461 + 1*(9*1+7) with 461 prime.
%e A335641 a(287) = 1, and 2*287+1 = 293 + (-6)*(9*(-6)+7) with 293 prime.
%e A335641 a(732) = 1, and 2*732+1 = 673 + 9*(9*9+7) with 673 prime.
%t A335641 tab={};Do[r=0;Do[If[PrimeQ[2n+1-x*(9*x+7)],r=r+1],{x,-Floor[(Sqrt[36(2n+1)+49]+7)/18],(Sqrt[36(2n+1)+49]-7)/18}];
%t A335641 tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A335641 Cf. A000040, A132399, A144590, A256071.
%K A335641 nonn
%O A335641 1,2
%A A335641 _Zhi-Wei Sun_, Oct 03 2020