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A065036 Product of the cube of a prime (A030078) and a different prime.

%I A065036 #42 Feb 21 2025 16:47:08
%S A065036 24,40,54,56,88,104,135,136,152,184,189,232,248,250,296,297,328,344,
%T A065036 351,375,376,424,459,472,488,513,536,568,584,621,632,664,686,712,776,
%U A065036 783,808,824,837,856,872,875,904,999,1016,1029,1048,1096,1107,1112
%N A065036 Product of the cube of a prime (A030078) and a different prime.
%C A065036 This sequence appears on row 8 of the list illustrated in A064839 and is similar to A054753 which appears on row 6. Previous rows are generated by A000007, A000040, A001248, A006881, A030078 respectively.
%C A065036 Or, the numbers n such that 20=number of perfect partitions of n. - _Juri-Stepan Gerasimov_, Sep 26 2009
%H A065036 T. D. Noe, <a href="/A065036/b065036.txt">Table of n, a(n) for n = 1..1000</a>
%H A065036 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A065036 A002033(a(n)) = 20. - _Juri-Stepan Gerasimov_, Sep 26 2009
%F A065036 A089233(a(n)) = 3. - _Reinhard Zumkeller_, Sep 04 2013
%F A065036 A000005(a(n)) = 8. - _Altug Alkan_, Nov 11 2015
%e A065036 a(4)= 56 since 56 = 2*2*2*7.
%t A065036 Select[ Range[1500], Sort[ Transpose[ FactorInteger[ # ]] [[2]]] == {1, 3} & ]
%t A065036 Module[{upto=1200},Select[(Union[Flatten[{#[[1]]^3 #[[2]],#[[1]]#[[2]]^3}&/@Subsets[Prime[Range[upto/8]],{2}]]]),#<=upto&]] (* _Harvey P. Dale_, May 23 2015 *)
%o A065036 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/3),t=p^3; forprime(q=2,lim\t,if(p==q,next);listput(v,t*q)));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A065036 (PARI) is(n)=my(f=factor(n)[,2]); f==[3,1]~||f==[1,3]~ \\ _Charles R Greathouse IV_, Oct 15 2015
%o A065036 (Python)
%o A065036 from sympy import primepi, primerange, integer_nthroot
%o A065036 def A065036(n):
%o A065036     def bisection(f,kmin=0,kmax=1):
%o A065036         while f(kmax) > kmax: kmax <<= 1
%o A065036         kmin = kmax >> 1
%o A065036         while kmax-kmin > 1:
%o A065036             kmid = kmax+kmin>>1
%o A065036             if f(kmid) <= kmid:
%o A065036                 kmax = kmid
%o A065036             else:
%o A065036                 kmin = kmid
%o A065036         return kmax
%o A065036     def f(x): return n+x-sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,4)[0])
%o A065036     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A065036 Cf. A000007, A000040, A001248, A002033, A006881, A030078, A054753, A064839, A089233.
%K A065036 easy,nonn
%O A065036 1,1
%A A065036 _Alford Arnold_, Nov 04 2001