A002734 Remove squares!

2, 3, 5, 6, 7, 2, 10, 11, 3, 13, 14, 15, 17, 2, 19, 5, 21, 22, 23, 6, 26, 3, 7, 29, 30, 31, 2, 33, 34, 35, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77, 78, 79

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is a riddle. (Explanation: This lists the squarefree part of the nonsquares. - Amiram Eldar, Sep 05 2020)

A007913 with the ones removed.

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• H. Brocard, Notes élémentaires sur le problème de Peel, Nouvelle Correspondance Mathématique, Vol. 4 (1878), pp. 161-169.

Crossrefs

Intersection of A000037 and A007913.

Programs

Mathematica

core[n_] := Times @@ (First[#]^Mod[Last[#], 2] & /@ FactorInteger[n]); Select[core /@ Range[100], # > 1 &] (* Amiram Eldar, Sep 05 2020 *)

Python

from math import isqrt, prod
from sympy import factorint
def A002734(n):
    k = isqrt(n)
    return prod(p for p, e in factorint(n+k+int(n>k*(k+1))).items() if e&1) # Chai Wah Wu, Jun 05 2025

Formula

a(n) = A007913(A000037(n)). - Amiram Eldar, Sep 05 2020

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Feb 02 2024