cp's OEIS Frontend

A235061 Number of ways to write n = k*(k+1)/2 + m with k > 0 and m > 0 such that prime(k*(k+1)/2) + phi(m) is prime, where phi(.) is Euler's totient function.

%I A235061 #18 Apr 06 2014 04:07:07
%S A235061 0,1,1,0,0,1,1,0,1,1,1,1,2,2,1,3,2,2,1,1,1,2,3,3,2,1,1,3,3,3,3,2,3,2,
%T A235061 3,2,3,4,2,4,4,2,3,1,4,4,1,4,3,3,3,4,5,4,3,1,3,3,5,4,4,5,1,5,3,5,5,4,
%U A235061 2,2,5,4,5,1,1,6,5,6,6,4,5,5,8,5,2,1,4,6,4,6,7,3,3,6,4,7,5,2,7,6
%N A235061 Number of ways to write n = k*(k+1)/2 + m with k > 0 and m > 0 such that prime(k*(k+1)/2) + phi(m) is prime, where phi(.) is Euler's totient function.
%C A235061 Conjecture: (i) a(n) > 0 for all n > 8.
%C A235061 (ii) Any integer n > 100 can be written as k^2 + m with k > 0 and m > 0 such that phi(k^2) + prime(m) is prime.
%C A235061 (iii) Any integer n > 187 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m) is a triangular number. Also, each integer n > 45 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a triangular number.
%C A235061 (iv)  Any integer n > 293 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m) is a square. Also, each integer n > 83 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a square.
%H A235061 Zhi-Wei Sun, <a href="/A235061/b235061.txt">Table of n, a(n) for n = 1..10000</a>
%H A235061 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A235061 a(10) = 1 since 10 = 2*(2+1)/2 + 7 = 3 + 7 with prime(3) + phi(7) = 5 + 6 = 11 prime.
%e A235061 a(20) = 1 since 20 = 3*(3+1)/2 + 14 = 6 + 14 with prime(6) + phi(14) = 13 + 6 = 19 prime.
%e A235061 a(86) = 1 since 86 = 12*(12+1)/2 + 8 = 78 + 8 with prime(78) + phi(8) = 397 + 4 = 401 prime.
%t A235061 a[n_]:=Sum[If[PrimeQ[Prime[k(k+1)/2]+EulerPhi[n-k(k+1)/2]],1,0],{k,1,(Sqrt[8n-7]-1)/2}];Table[a[n],{n,1,100}]
%Y A235061 Cf. A000010, A000040, A000217, A234615.
%K A235061 nonn
%O A235061 1,13
%A A235061 _Zhi-Wei Sun_, Jan 03 2014