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

24, 40, 54, 56, 88, 104, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 459, 472, 488, 513, 536, 568, 584, 621, 632, 664, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029, 1048, 1096, 1107, 1112

Offset: 1

Author

Alford Arnold, Nov 04 2001

Keywords

• easy
• nonn

Comments

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.

Or, the numbers n such that 20=number of perfect partitions of n. - Juri-Stepan Gerasimov, Sep 26 2009

Examples

			a(4)= 56 since 56 = 2*2*2*7.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index to sequences related to prime signature

Crossrefs

Cf. A000007, A000040, A001248, A002033, A006881, A030078, A054753, A064839, A089233.

Programs

Mathematica

Select[ Range[1500], Sort[ Transpose[ FactorInteger[ # ]] [[2]]] == {1, 3} & ]
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 *)

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

PARI

is(n)=my(f=factor(n)[,2]); f==[3,1]~||f==[1,3]~ \\ Charles R Greathouse IV, Oct 15 2015

Python

from sympy import primepi, primerange, integer_nthroot
def A065036(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

A002033(a(n)) = 20. - Juri-Stepan Gerasimov, Sep 26 2009

A089233(a(n)) = 3. - Reinhard Zumkeller, Sep 04 2013

A000005(a(n)) = 8. - Altug Alkan, Nov 11 2015