cp's OEIS Frontend

A233654 |{prime p < n: n - p = sigma(k) for some k > 0}|, where sigma(k) is the sum of all (positive) divisors of k.

%I A233654 #10 Dec 14 2013 14:43:48
%S A233654 0,0,1,1,1,3,1,3,3,3,3,2,2,4,4,3,5,4,4,6,4,3,5,3,6,5,5,1,4,4,7,5,6,4,
%T A233654 8,3,8,5,5,2,9,5,10,8,8,4,10,3,11,6,10,2,8,4,11,5,8,3,12,5,13,7,7,3,
%U A233654 13,3,14,7,7,5,12,3,14,9,11,6,12,2,16,7,11,5,12,3,18,8,12,2,11,3,19,6,11,4,13,4,17,8,10,6
%N A233654 |{prime p < n: n - p = sigma(k) for some k > 0}|, where sigma(k) is the sum of all (positive) divisors of k.
%C A233654 Conjecture: (i) Let n > 1 be an integer. Then we have a(2*n) > 0. Also, 2*n + 1 can be written as p + sigma(k), where p is a Sophie Germain prime and k is a positive integer.
%C A233654 (ii) Each odd number greater than one can be written as sigma(k^2) + phi(m), where k and m are positive integers, and phi(.) is Euler's totient function.
%C A233654 That a(2*n+1) > 0 for n > 1 is a consequence of Goldbach's conjecture, for, if 2*n = p + q with p and q both prime, then 2*n + 1 = p + sigma(q) = q + sigma(p).
%H A233654 Zhi-Wei Sun, <a href="/A233654/b233654.txt">Table of n, a(n) for n = 1..10000</a>
%e A233654 a(3) = 1 since 3 = 2 + 1 = 2 + sigma(1) with 2 prime.
%e A233654 a(7) = 1 since 7 = 3 + 4 = 3 + sigma(3) with 3 prime.
%e A233654 a(10) = 3 since 10 = 2 + 8 = 2 + sigma(7) with 2 prime, 10 = 3 + 7 = 3 + sigma(4) with 3 prime, and 10 = 7 + 3 = 7 + sigma(2) with 7 prime.
%e A233654 a(13) = 2 since 13 = 5 + 8 = 5 + sigma(7) with 5 prime, and 13 = 7 + 6 = 7 + sigma(5) with 7 prime.
%e A233654 a(28) = 1 since 28 = 13 + 15 = 13 + sigma(8) with 13 prime.
%e A233654 a(36) = 3 since 36 = 5 + 31 = 5 + sigma(16) = 5 + sigma(25) with 5 prime, 36 = 23 + 13 = 23 + sigma(9) with 23 prime, and 36 = 29 + 7 = 29 + sigma(4) with 29 prime.
%e A233654 a(148) = 1 since 148 = 109 + 39 = 109 + sigma(18) with 109 prime.
%t A233654 f[n_]:=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
%t A233654 S[n_]:=Union[Table[f[j],{j,1,n}]]
%t A233654 PQ[n_]:=n>0&&PrimeQ[n]
%t A233654 a[n_]:=Sum[If[PQ[n-Part[S[n],i]],1,0],{i,1,Length[S[n]]}]
%t A233654 Table[a[n],{n,1,100}]
%Y A233654 Cf. A000010, A000040, A000203, A002372, A002375, A005384, A232270, A233542, A233544, A233547, A233549.
%K A233654 nonn
%O A233654 1,6
%A A233654 _Zhi-Wei Sun_, Dec 14 2013