A372841 4-full numbers that are not prime powers.

1296, 2592, 3888, 5184, 7776, 10000, 10368, 11664, 15552, 20000, 20736, 23328, 31104, 34992, 38416, 40000, 41472, 46656, 50000, 50625, 62208, 69984, 76832, 80000, 82944, 93312, 100000, 104976, 124416, 139968, 151875, 153664, 160000, 165888, 186624, 194481, 200000

Offset

1,1

Comments

Numbers k such that rad(k)^4 | k and omega(k) > 1. In other words, numbers with at least 2 distinct prime factors whose prime power factors have exponents that exceed 3.

Proper subset of the following sequences: A001694, A036966, A036967, A126706, A286708, A372695.

Smallest term k with omega(k) = m is k = A002110(m)^4.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Intersection of A036967 and A024619.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) - Sum_{p prime} 1/(p^3*(p-1)) - 1 = 0.0026996042121456100761... . - Amiram Eldar, May 17 2024

Example

Table of smallest 12 terms:
   n      a(n)
  -----------------------
   1     1296 = 2^4 * 3^4
   2     2592 = 2^5 * 3^4
   3     3888 = 2^4 * 3^5
   4     5184 = 2^6 * 3^4
   5     7776 = 2^5 * 3^5
   6    10000 = 2^4 * 5^4
   7    10368 = 2^7 * 3^4
   8    11664 = 2^4 * 3^6
   9    15552 = 2^6 * 3^5
  10    20000 = 2^5 * 5^4
  11    20736 = 2^8 * 3^4
  12    23328 = 2^5 * 3^6

Mathematica

With[{nn = 200000}, Rest@ Select[Union@ Flatten@ Table[a^7 * b^6 * c^5 * d^4, {d, Surd[nn, 4]}, {c, Surd[nn/(d^4), 5]}, {b, Surd[nn/(c^5 * d^4), 6]}, {a, Surd[nn/(b^6 * c^5 * d^4), 7]}], Not@*PrimePowerQ]]

Prog

(Python)
from math import gcd
from sympy import primepi, integer_nthroot, factorint
def A372841(n):
    def f(x):
        c = n+x+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(4, x.bit_length()))
        for u in range(1, integer_nthroot(x, 7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1, integer_nthroot(a:=x//u**7, 6)[0]+1):
                    if gcd(w, u)==1 and all(d<=1 for d in factorint(w).values()):
                        for y in range(1, integer_nthroot(z:=a//w**6, 5)[0]+1):
                            if gcd(w, y)==1 and gcd(u, y)==1 and all(d<=1 for d in factorint(y).values()):
                                c -= integer_nthroot(z//y**5, 4)[0]
        return c
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f, n, n) # Chai Wah Wu, Sep 10 2024

Crossrefs

Cf. A001694, A007947, A024619, A036966, A036967, A126706, A286708, A372695.

Sequence in context: A186485 A281399 A043372 * A217908 A223508 A250810

Adjacent sequences: A372838 A372839 A372840 * A372842 A372843 A372844

Keyword

nonn

Author

Michael De Vlieger, May 14 2024

Status

approved