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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 22 05:35 EDT 2019. Contains 326172 sequences. (Running on oeis4.)