A215232 Least semiprime m such that the next semiprime is m + A215231(n).

4, 6, 10, 15, 26, 95, 597, 1418, 2681, 6559, 16053, 17965, 32777, 35103, 35981, 340894, 1069541, 1589662, 3586843, 5835191, 139139887, 251306317, 285074689, 327023206, 751411951, 981270902, 2655397631, 5238280946, 6498130361, 8512915573, 16328958619

Offset: 1

Author

T. D. Noe, Aug 07 2012

Keywords

• nonn
• hard
• nice

Comments

The semiprime m + A215231(n) is in A217851.

Matomäki & Teräväinen prove that there is almost always (in the sense of natural density) a semiprime in (x, x + log(x)^2.1]. Under RH the exponent can be chosen as 2 + e for any e > 0. - Charles R Greathouse IV, Oct 03 2022

Links

• Donovan Johnson, Table of n, a(n) for n = 1..38 (terms < 10^13)
• Kaisa Matomäki and Joni Teräväinen, Almost primes in almost all short intervals II, arXiv:2207.05038 [math.NT].

Crossrefs

Cf. A001358 (semiprimes), A131109, A215231, A217851.

Cf. A002386 (increasing gaps between primes).

Programs

Mathematica

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nextSemiprime[n_] := Module[{m = n + 1}, While[! SemiPrimeQ[m], m++]; m]; t = {{0, 0}}; s1 = nextSemiprime[1]; While[s1 < 10^7, s2 = nextSemiprime[s1]; d = s2 - s1; If[d > t[[-1, 1]], AppendTo[t, {d, s1}]; Print[{d, s1}]]; s1 = s2]; t = Rest[t]; Transpose[t][[2]]

PARI

r=0;s=2;for(n=3,1e7,if(bigomega(n)==2,if(n-s>r,r=n-s;print1(s", "));s=n)) \\ Charles R Greathouse IV, Sep 07 2012

Extensions

a(27)-a(31) from Donovan Johnson, Aug 07 2012