A085724 Numbers k such that 2^k - 1 is a semiprime (A001358).

4, 9, 11, 23, 37, 41, 49, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997, 1061, 1063

Offset: 1

Author

Jason Earls, Jul 20 2003

Keywords

• nonn
• more

Comments

Subsequence of A000430. Apart from 4, 9, and 49 composites in this sequence are greater than 1.9e7. - Charles R Greathouse IV, Jun 05 2013

1427 and 1487 are also terms. 1277 is the only remaining unknown below them. - Charles R Greathouse IV, Jun 05 2013

Among the known terms only 11, 23, 83 and 131 are in A002515, that is, they are the only known values for n such that (2^n - 1)/(2*n + 1) is prime. - Jianing Song, Jan 22 2019

Either a(n) is a prime, or the square of a Mersenne prime exponent. - M. F. Hasler, Jun 23 2025

Examples

			11 is a member because 2^11 - 1 = 23*89.
		

References

• J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 56-60. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]
• J. Earls, "Cole Semiprimes," Mathematical Bliss, Pleroma Publications, 2009, pages 56-60. ASIN: B002ACVZ6O [From Jason Earls, Nov 25 2009]

Links

• S. S. Wagstaff, Jr., The Cunningham Project
• Eric Weisstein's World of Mathematics, Mersenne Number
• Eric Weisstein's World of Mathematics, Semiprime

Crossrefs

Cf. A092558, A092559, A092561, A092562.

Programs

Mathematica

SemiPrimeQ[n_]:=(n>1) && (2==Plus@@(Transpose[FactorInteger[n]][[2]])); Select[Range[100],SemiPrimeQ[2^#-1]&] (Noe)
Select[Range[1100],PrimeOmega[2^#-1]==2&] (* Harvey P. Dale, Feb 18 2018 *)
Select[Range[250], Total[Last /@ FactorInteger[2^# - 1, 3]] == 2 &] (* Eric W. Weisstein, Jul 28 2022 *)

PARI

issemi(n)=bigomega(n)==2
is(n)=if(isprime(n), issemi(2^n-1), my(q); isprimepower(n,&q)==2 && ispseudoprime(2^q-1) && ispseudoprime((2^n-1)/(2^q-1))) \\ Charles R Greathouse IV, Jun 05 2013

Extensions

More terms from Zak Seidov, Feb 27 2004

More terms from Cunningham project, Mar 23 2004

More terms from the Cunningham project sent by Robert G. Wilson v and T. D. Noe, Feb 22 2006

a(41)-a(42) from Charles R Greathouse IV, Jun 05 2013