

A082919


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


17



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
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OFFSET

1,1


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 n and n+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 first 10000 terms, from all 80000 numbers a(n)+k, k=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


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 and Zak Seidov, 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.


EXAMPLE

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.


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


CROSSREFS

Cf. A001358, A082130, A082131, A056809, A070552, A092207, A092125, A092126, A092127, A092128, A092129, A092209.
Sequence in context: A231862 A088846 A092208 * A217222 A252144 A201802
Adjacent sequences: A082916 A082917 A082918 * A082920 A082921 A082922


KEYWORD

nonn


AUTHOR

Hugo Pfoertner, Apr 22 2003


STATUS

approved



