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

1,1

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Index to sequences related to prime signature

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A375076 A297401 A065127 * A329880 A303359 A340747

Adjacent sequences: A065033 A065034 A065035 * A065037 A065038 A065039

Keyword

easy,nonn,changed

Author

Alford Arnold, Nov 04 2001

Status

approved