A176504 a(n) = m + k where prime(m)*prime(k) = semiprime(n).

2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 7, 9, 8, 10, 8, 9, 8, 10, 11, 12, 9, 11, 13, 9, 14, 10, 15, 12, 10, 13, 16, 11, 17, 14, 12, 18, 11, 10, 19, 15, 16, 12, 20, 17, 21, 11, 13, 22, 14, 23, 18, 13, 24, 12, 19, 25, 20, 15, 12, 26, 21, 27, 14, 16, 28, 13, 22, 29, 17, 15, 30, 23, 13, 31

Offset

1,1

Links

Table of n, a(n) for n=1..76.

Formula

a(n) = A056239(A001358(n)) = A338912(n) + A338913(n). - Gus Wiseman, Dec 04 2020

sqrt(n/(log n log log n)) << a(n) << n/log log n. - Charles R Greathouse IV, Apr 17 2024

Example

From Gus Wiseman, Dec 04 2020: (Start)
A semiprime (A001358) is a product of any two prime numbers. The sequence of all semiprimes together with their prime indices and weights begins:
   4: 1 + 1 = 2
   6: 1 + 2 = 3
   9: 2 + 2 = 4
  10: 1 + 3 = 4
  14: 1 + 4 = 5
  15: 2 + 3 = 5
  21: 2 + 4 = 6
  22: 1 + 5 = 6
  25: 3 + 3 = 6
  26: 1 + 6 = 7
(End)

Maple

isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176504 := proc(n) numtheory[pi](A084126(n)) + numtheory[pi](A084127(n)) ; end proc: seq(A176504(n), n=1..80) ; # R. J. Mathar, Apr 20 2010

Mathematica

Table[If[SquareFreeQ[n], Total[PrimePi/@First/@FactorInteger[n]], 2*PrimePi[Sqrt[n]]], {n, Select[Range[100], PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange, factorint
from oeis_sequences.OEISsequences import bisection
def A176504(n):
    def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    return sum(primepi(p)*e for p, e in factorint(bisection(f, n, n)).items()) # Chai Wah Wu, Jan 02 2026

Crossrefs

A056239 is the version for not just semiprimes.

A087794 gives the product of the same two indices.

A176506 gives the difference of the same two indices.

A338904 puts the n-th semiprime in row a(n).

A001358 lists semiprimes.

A006881 lists squarefree semiprimes.

A338898/A338912/A338913 give the prime indices of semiprimes.

A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.

Cf. A001222, A046315, A065516, A084126, A084127, A100484, A112798, A115392, A128301, A338906/A338907.

Sequence in context: A027434 A319434 A174697 * A196162 A071940 A085883

Adjacent sequences: A176501 A176502 A176503 * A176505 A176506 A176507

Keyword

nonn

Author

Juri-Stepan Gerasimov, Apr 19 2010

Extensions

Entries checked by R. J. Mathar, Apr 20 2010

Status

approved