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A117221 Number of partitions of 3-smooth numbers.

%I A117221 #19 Sep 16 2024 15:55:41
%S A117221 1,2,3,5,11,22,30,77,231,385,1575,3010,8349,17977,147273,386155,
%T A117221 1741630,5392783,18004327,118114304,483502844,4351078600,22540654445,
%U A117221 129913904637,1987276856363,15285151248481,133978259344888,365749566870782,3925922161489422,49005643635237875
%N A117221 Number of partitions of 3-smooth numbers.
%H A117221 Andrew Howroyd, <a href="/A117221/b117221.txt">Table of n, a(n) for n = 1..100</a>
%F A117221 a(n) = A000041(A003586(n)).
%t A117221 PartitionsP /@ Select[Range[1000], AllTrue[FactorInteger[#][[All, 1]], # <= 3 &] &] (* _Jean-François Alcover_, Oct 13 2021 *)
%o A117221 (PARI) \\ here is(n) is test for A003586 inclusion.
%o A117221 is(n)={forprime(p=2, 3, n/=p^valuation(n,p)); n==1}
%o A117221 seq(n)={my(A=List(), i=0); while(#A<n, i++; if(is(i), listput(A, numbpart(i)))); Vec(A)} \\ _Andrew Howroyd_, Jan 02 2020
%o A117221 (Python)
%o A117221 from sympy import integer_log, npartitions
%o A117221 def A117221(n):
%o A117221     def bisection(f,kmin=0,kmax=1):
%o A117221         while f(kmax) > kmax: kmax <<= 1
%o A117221         while kmax-kmin > 1:
%o A117221             kmid = kmax+kmin>>1
%o A117221             if f(kmid) <= kmid:
%o A117221                 kmax = kmid
%o A117221             else:
%o A117221                 kmin = kmid
%o A117221         return kmax
%o A117221     def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
%o A117221     return npartitions(bisection(f,n,n)) # _Chai Wah Wu_, Sep 16 2024
%Y A117221 Cf. A000041, A003586, A090184, A117220, A117222.
%K A117221 nonn
%O A117221 1,2
%A A117221 _Reinhard Zumkeller_, Mar 04 2006
%E A117221 More terms from _Michel Marcus_, Apr 20 2019