A108605 Semiprimes with prime sum of factors: twice the lesser of the twin prime pairs.

6, 10, 22, 34, 58, 82, 118, 142, 202, 214, 274, 298, 358, 382, 394, 454, 478, 538, 562, 622, 694, 838, 862, 922, 1042, 1138, 1198, 1234, 1282, 1318, 1618, 1642, 1654, 1714, 1762, 2038, 2062, 2098, 2122, 2182, 2302, 2458, 2554, 2578, 2602, 2638, 2854, 2902

Offset

1,1

Comments

All terms are even. (Cf. formula.)

The definition implies that the sum of factors is the sum over the prime factors with multiplicity, as in A001414. - R. J. Mathar, Nov 28 2008

The sum of factors of a semiprime pq is p+q, which can only be prime if {p, q} = {2, odd prime}. Requiring the sum to be prime then implies that the semiprime is twice the lesser of a twin prime pair. - M. F. Hasler, Apr 07 2015

Subsequence of A288814, each term being of the form A288814(p), where p is the greatest of a pair of twin primes. - David James Sycamore, Aug 29 2017

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n)=2*p, with p and 2+p twin primes: a(n)=2*A001359(n).

Example

58=2*29 and 2+29 is prime.

Mathematica

Select[Range[2, 3000, 2], !IntegerQ[Sqrt[ # ]]&&Plus@@(Transpose[FactorInteger[ # ]])[[2]]==2&&PrimeQ[Plus@@(Transpose[FactorInteger[ # ]])[[1]]]&]
Select[Range[2, 3000, 2], PrimeOmega[#]==PrimeNu[#]==2&&PrimeQ[Total[ FactorInteger[ #][[;; , 1]]]]&] (* Harvey P. Dale, Apr 10 2023 *)

Prog

(PARI) list(lim)=my(v=List(), p=2); forprime(q=3, lim\2+1, if(q-p==2, listput(v, 2*p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

Crossrefs

Cf. A001358 semiprimes, A001359 lesser of twin primes, A101605 3-almost primes, A108606 semiprimes with prime sum of digits, A108607 intersection of A108605 and A108606.

Sequence in context: A082917 A001172 A339437 * A216049 A063765 A085712

Adjacent sequences: A108602 A108603 A108604 * A108606 A108607 A108608

Keyword

easy,nonn

Author

Zak Seidov, Jun 12 2005

Extensions

Changed division by 2 to multiplication by 2 in formula related to A001359. - R. J. Mathar, Nov 28 2008

Status

approved