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A101048

Number of partitions of n into semiprimes (a(0) = 1 by convention).

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 3, 1, 5, 3, 5, 4, 7, 4, 9, 7, 10, 8, 13, 10, 17, 13, 18, 17, 25, 21, 29, 25, 34, 34, 43, 37, 51, 49, 61, 59, 73, 69, 89, 87, 103, 103, 124, 122, 148, 149, 172, 176, 206, 208, 244, 248, 281, 293, 337, 344, 391, 405, 456, 479, 537, 553

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OFFSET

0,11

COMMENTS

Semiprime analog of A000607. a(n) <= A002095(n). - Jonathan Vos Post, Oct 01 2007

Das, Robles, Zaharescu, & Zeindler give an asymptotic formula, see Links. - Charles R Greathouse IV, Jan 20 2023

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

Madhuparna Das, Nicolas Robles, Alexandru Zaharescu, and Dirk Zeindler, Partitions into semiprimes, arXiv preprint (2022). arXiv:2212.12489 [math.NT]

FORMULA

G.f.: 1/product(product(1-x^(p(i)p(j)), i = 1..j),j = 1..infinity), p(k) is the k-th prime. - Emeric Deutsch, Apr 04 2006

EXAMPLE

a(12) = #{6 + 6, 4 + 4 + 4} = #{2 * (2*3), 3 * (2*2)} = 2.

MAPLE

g:=1/product(product(1-x^(ithprime(i)*ithprime(j)), i=1..j), j=1..30): gser:=series(g, x=0, 75): seq(coeff(gser, x, n), n=1..71); # Emeric Deutsch, Apr 04 2006

# Alternative:

h:= proc(n) option remember; `if`(n=0, 0,

`if`(numtheory[bigomega](n)=2, n, h(n-1)))

end:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

`if`(i>n, 0, b(n-i, h(min(n-i, i))))+b(n, h(i-1))))

end:

a:= n-> b(n, h(n)):

seq(a(n), n=0..100); # Alois P. Heinz, May 19 2021

MATHEMATICA

terms = 100; CoefficientList[1/Product[1 - x^(Prime[i] Prime[j]), {i, 1, PrimePi[Ceiling[terms/2]]}, {j, 1, i}] + O[x]^terms, x] (* Jean-François Alcover, Aug 01 2018 *)

PROG

(Haskell)

a101048 = p a001358_list where

p _ 0 = 1

p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Mar 21 2014

(PARI) issemi(n)=if(n<4, return(0)); forprime(p=2, 97, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2

allsemi(v)=for(i=1, #v, if(!issemi(v[i]), return(0))); 1

a(n)=my(s); if(n<4, return(n==0)); forpart(k=n, if(allsemi(k), s++), [4, n]); s \\ Charles R Greathouse IV, Jan 20 2023

CROSSREFS

Cf. A000041, A000607, A101049, A001358, A064911, A002095.

Cf. A112020, A112021.