A002734 - OEIS

A002734

Remove squares!
(Formerly M0611 N0222)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Intersection of A000037 and A007913.

Sequence in context: A175081 A023834 A084735 * A367417 A339949 A160100

Adjacent sequences: A002731 A002732 A002733 * A002735 A002736 A002737

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved