A325240 Numbers whose minimum prime exponent is 2.

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset

1,1

Comments

Or barely powerful numbers, a subset of powerful numbers A001694.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).

Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

Example

The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

Mathematica

Select[Range[1000], Min@@FactorInteger[#][[All, 2]]==2&]

Prog

(PARI) is(n)={my(e=factor(n)[, 2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023
(Python)
from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    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
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        for w in range(1, integer_nthroot(x, 5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
                    if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4, 3)[0]
        return c
    return bisection(f, n, n**2) # Chai Wah Wu, Oct 02 2024

Crossrefs

Positions of 2's in A051904.

Maximum instead of minimum gives A067259.

Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.

Cf. A065483, A082695.

Sequence in context: A367406 A030140 A374458 * A355058 A153158 A111245

Adjacent sequences: A325237 A325238 A325239 * A325241 A325242 A325243

Keyword

nonn

Author

Gus Wiseman, Apr 15 2019

Status

approved