A053850 Odd numbers divisible by a square > 1.

9, 25, 27, 45, 49, 63, 75, 81, 99, 117, 121, 125, 135, 147, 153, 169, 171, 175, 189, 207, 225, 243, 245, 261, 275, 279, 289, 297, 315, 325, 333, 343, 351, 361, 363, 369, 375, 387, 405, 423, 425, 441, 459, 475, 477, 495, 507, 513, 525, 529, 531, 539, 549, 567

Offset

1,1

Comments

Odd n such that sum(k=1,n-1,floor(k^3/n)) is different from (1/4)*(n-2)*(n^2-1) (equality holds for n prime as well as for "1" union "A024556" ). - Benoit Cloitre, Dec 08 2002

Odd nonsquarefree numbers, odd terms of A013929. - Zak Seidov, Aug 16 2006

The asymptotic density of this sequence is 1/2 - 4/Pi^2 = 0.094715... - Amiram Eldar, Nov 21 2020

Odd composites x with at least one prime factor with exponent >= 2, which means Omega(x) > omega(x). All such x admit factors y with 1 <= y <= floor(x/4) for which x*y is a square number. Among them, those with omega(x) >= 2 also admit factors y with 2 <= y <= floor(x/4) for which x*y is an oblong number. - Charles Kusniec, Nov 17 2025

Links

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

Eric Weisstein's World of Mathematics, Prime Sums.

Formula

Intersection of A071904 and (A360769 disjoint union A244623). - Charles Kusniec, Nov 17 2025

Mathematica

Select[Range[1, 500, 2], !SquareFreeQ[#] &] (* Amiram Eldar, Nov 21 2020 *)

Prog

(PARI) lista(nn) = {forstep(n=1, nn, 2, if (! issquarefree(n), print1(n, ", "))); } \\ Michel Marcus, Jun 06 2014
(Python)
from math import isqrt
from sympy import mobius
def A053850(n):
    def f(x): return n+x+sum(mobius(k)*(x//k**2+1>>1) for k in range(3, isqrt(x)+1, 2))
    return bisection(f, n, n) # Chai Wah Wu, Dec 01 2025

Crossrefs

Cf. A078837, A013929, A024556.

Equals A071904 \ A024556.

Odd terms of A013929.

Cf. A360769, A244623.

Sequence in context: A377702 A155109 A268576 * A225498 A020210 A275196

Adjacent sequences: A053847 A053848 A053849 * A053851 A053852 A053853

Keyword

easy,nonn

Author

Enoch Haga, Mar 28 2000

Status

approved