A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487, 802199, 858179, 991289, 1471727, 1474607, 1963829, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 4738907, 6156137, 7813559, 8090759

Offset: 1

Author

Hugo Pfoertner, Apr 22 2003

Keywords

• nonn

Comments

Start of a cluster of 8 consecutive odd semiprimes. Semiprimes in arithmetic progression. All terms are odd, see also A056809.

Note that there cannot exist 9 consecutive odd semiprimes. Out of any 9 consecutive odd numbers, one of them will be divisible by 9. The only multiple of 9 which is a semiprime is 9 itself and it is easy to see that's not part of a solution. - Jack Brennen, Jan 04 2006

For the first 500 terms, a(n) is roughly 40000*n^1.6, so the sequence appears to be infinite. Note that (a(n)+4)/3 and (a(n)+10)/3 are twin primes. - Don Reble, Jan 05 2006

All terms == 11 (mod 18). - Zak Seidov, Sep 27 2012

There is at least one even semiprime between k and k+14 for 1812 of the first 10000 terms. - Donovan Johnson, Oct 01 2012

All terms == {29,47,83} (mod 90). - Zak Seidov, Sep 13 2014

Among the first 10000 terms, from all 80000 numbers a(n)+m, m=0,2,4,6,8,10,12,14, the only square is a(4637) + 2 = 23538003241 = 153421^2 (153421 is prime, of course). - Zak Seidov, Dec 22 2014

Examples

			a(1)=8129 because 8129=11*739, 8131=47*173, 8133=3*2711, 8135=5*1627, 8137=79*103, 8139=3*2713, 8141=7*1163, 8143=17*479 are semiprimes.
		

References

• Author of this sequence is Jack Brennen, who provided the terms up to 991289 in a posting to the seqfan mailing list on April 5, 2003.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms a(1001) to a(2000) from Zak Seidov)
• Eric Weisstein's World of Mathematics, Semiprime.

Crossrefs

Cf. A001358, A082130, A082131, A056809, A070552, A092207, A092125, A092126, A092127, A092128, A092129, A092209, A217222 (consecutive semiprimes).

Programs

Mathematica

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[3*10^6], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == PrimeFactorExponentsAdded[ # + 12] == PrimeFactorExponentsAdded[ # + 14] == 2 &] (* Robert G. Wilson v and Zak Seidov, Feb 24 2004 *)