A244349 Positive integers n such that all the differences prime(n+j) - prime(n+i) with 0 <= i < j <= 9 are practical numbers.

232272, 254294, 326943, 686574, 1021708, 1251812, 1254205, 1347543, 1562959, 2236112, 2607137, 2682138, 2810880, 2884861, 3041717, 3049727, 3049728, 3159941, 3159942, 3314220

Offset

1,1

Comments

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Example

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.

Mathematica

f[n_]:=FactorInteger[n]
Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
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}]

Crossrefs

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

Sequence in context: A357826 A050998 A237462 * A128484 A257638 A116463

Adjacent sequences: A244346 A244347 A244348 * A244350 A244351 A244352

Keyword

nonn

Author

Zhi-Wei Sun, Jun 26 2014

Status

approved