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

Author

Zak Seidov, Jun 12 2005

Keywords

• easy
• nonn

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

Examples

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

Links

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

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.

Programs

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 *)

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

Formula

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

Extensions

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