A082919 - OEIS

%I #48 Feb 16 2025 08:32:49

%S 8129,9983,99443,132077,190937,237449,401429,441677,452639,604487,

%T 802199,858179,991289,1471727,1474607,1963829,1999937,2376893,2714987,

%U 3111977,3302039,3869237,4622087,4738907,6156137,7813559,8090759

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

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

%C 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

%C 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

%C All terms == 11 (mod 18). - _Zak Seidov_, Sep 27 2012

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

%C All terms == {29,47,83} (mod 90). - _Zak Seidov_, Sep 13 2014

%C 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

%D 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.

%H Donovan Johnson, <a href="/A082919/b082919.txt">Table of n, a(n) for n = 1..10000</a> (terms a(1001) to a(2000) from Zak Seidov)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.

%e 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.

%t 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 *)

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

%K nonn,changed

%O 1,1

%A _Hugo Pfoertner_, Apr 22 2003