A077065 Semiprimes of form prime - 1.

4, 6, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578, 2818, 2878, 2902

Offset

1,1

Comments

There are 670 semiprimes of form prime-1 below 10^5.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

a(n) = A005385(n) - 1 = 2*A005384(n).

A010051(A006093(a(n))/2) = A064911(A006093(a(n))) = 1. - Reinhard Zumkeller, Nov 22 2013

a(n) = A077068(n) - A232342(n). - Reinhard Zumkeller, Dec 16 2013

a(n) = A000010(A194593(n+1)). - Torlach Rush, Aug 23 2018

A000010((a(n)*2)+2) = A023900((a(n)*2)+2). - Torlach Rush, Aug 23 2018

Example

A001358(16) = 46 = 2*23 is a term as 46 = A000040(15) - 1 = 47 - 1.

Maple

q:= n-> (n::even) and andmap(isprime, [n+1, n/2]):
select(q, [$1..5000])[];  # Alois P. Heinz, Jul 19 2023

Mathematica

Select[Range[6000], Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#+1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[Range[3000], PrimeOmega[#]==2&&PrimeQ[#+1]&] (* Harvey P. Dale, Oct 16 2012 *)
Select[ Prime@ Range@ 430 - 1, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

Prog

(Magma) IsSemiprime:=func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [2..500] | IsSemiprime(s) where s is NthPrime(n)-1]; // Vincenzo Librandi, Oct 17 2012
(Haskell)
a077065 n = a077065_list !! (n-1)
a077065_list = filter ((== 1) . a010051' . (`div` 2)) a006093_list
-- Reinhard Zumkeller, Nov 22 2013, Oct 27 2012
(PARI) [x-1|x<-primes(10^4), bigomega(x-1)==2] \\ Charles R Greathouse IV, Nov 22 2013

Crossrefs

Intersection of A006093 and A001358.

Intersection of A006093 and A100484.

Cf. A077068, A232342, A064911.

Cf. A005384, A005385.

Cf. A000010, A023900, A194593.

Sequence in context: A185913 A243119 A277343 * A131867 A252656 A322961

Adjacent sequences: A077062 A077063 A077064 * A077066 A077067 A077068

Keyword

nonn

Author

Reinhard Zumkeller, Oct 23 2002

Status

approved