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A082919
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Numbers n such that n, n+2, n+4, n+6, n+8, n+10, n+12 and n+14 are semiprimes.
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19
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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
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1,1
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COMMENTS
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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.
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
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REFERENCES
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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
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CROSSREFS
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Cf. A001358, A082130, A082131, A056809, A070552, A092207, A092125, A092126, A092127, A092128, A092129, A092209, A217222 (consecutive semiprimes).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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