A028983 Numbers whose sum of divisors is even.

3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

The even terms of this sequence are the even terms appearing in A178910. [Edited by M. F. Hasler, Oct 02 2014]

A071324(a(n)) is even. - Reinhard Zumkeller, Jul 03 2008

Sigma(a(n)) = A000203(a(n)) = A152678(n). - Jaroslav Krizek, Oct 06 2009

A083207 is a subsequence. - Reinhard Zumkeller, Jul 19 2010

Numbers k such that the number of odd divisors of k (A001227) is even. - Omar E. Pol, Apr 04 2016

Numbers k such that the sum of odd divisors of k (A000593) is even. - Omar E. Pol, Jul 05 2016

Numbers with a squarefree part greater than 2. - Peter Munn, Apr 26 2020

Equivalently, numbers whose odd part is nonsquare. Compare with the numbers whose square part is even (i.e., nonodd): these are the positive multiples of 4, A008586\{0}, and A225546 provides a self-inverse bijection between the two sets. - Peter Munn, Jul 19 2020

Also numbers whose reversed prime indices have alternating product > 1, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). Also Heinz numbers of the partitions counted by A347448. - Gus Wiseman, Oct 29 2021

Numbers whose number of middle divisors is not odd (cf. A067742). - Omar E. Pol, Aug 02 2022

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

The complement is A028982 = A000290 U A001105.

Cf. A000203, A000593, A001227, A007913, A178910, A152678, A067742.

Subsequences: A083207, A091067, A145204\{0}, A225838, A225858.

Cf. A334748 (a permutation).

Related to A008586 via A225546.

Ranks the partitions counted by A347448, complement A119620.

Cf. A030059, A335433, A335448, A339890, A344607, A347438, A347443, A347445, A347446, A347452, A347453, A347465.

Programs

Mathematica

Select[Range[82],EvenQ[DivisorSigma[1,#]]&] (* Jayanta Basu, Jun 05 2013 *)

PARI

is(n)=!issquare(n)&&!issquare(n/2) \\ Charles R Greathouse IV, Jan 11 2013

Python

from math import isqrt
def A028983(n):
    def f(x): return n-1+isqrt(x)+isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

Formula

a(n) ~ n. - Charles R Greathouse IV, Jan 11 2013

a(n) = n + (1 + sqrt(2)/2)*sqrt(n) + O(1). - Charles R Greathouse IV, Sep 01 2015

A007913(a(n)) > 2. - Peter Munn, May 05 2020