A244349 - OEIS

%I #8 Jun 26 2014 09:03:55

%S 232272,254294,326943,686574,1021708,1251812,1254205,1347543,1562959,

%T 2236112,2607137,2682138,2810880,2884861,3041717,3049727,3049728,

%U 3159941,3159942,3314220

%N Positive integers n such that all the differences prime(n+j) - prime(n+i) with 0 <= i < j <= 9 are practical numbers.

%C Conjecture: For any integer m > 0, there are infinitely many positive integers n such that all the differences prime(n+j) - prime(n+i) with 0 <= i < j <= m are practical numbers.

%H Zhi-Wei Sun, <a href="/A244349/b244349.txt">Table of n, a(n) for n = 1..300</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.

%e a(1) = 232272 with prime(232272+i) (i=0..9) having values 3230477, 3230483, 3230489, 3230501, 3230519, 3230531, 3230537, 3230543, 3230567, 3230573 respectively. Note that the set {prime(232272+j) - prime(232272+i): 0 <= i < j <= 9} coincides with {6*k: k = 1, ..., 16} whose elements are all practical numbers.

%t f[n_]:=FactorInteger[n]

%t Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])

%t 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 pr[n_]:=n>0&&(n<3||Mod[n,2]+Con[n]==0)

%t m=0;Do[Do[If[pr[Prime[n+j]-Prime[n+i]]==False,Goto[aa]],{j,1,9},{i,0,j-1}];m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,3314220}]

%Y Cf. A000040, A005153, A244254, A244264, A244266, A244294, A244308.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Jun 26 2014