A088828 Nonsquare positive odd numbers.

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Author

Labos Elemer, Oct 28 2003

Keywords

• nonn
• easy

Comments

Odd numbers with even abundance: primes and some composites too.

Odd numbers with odd abundance are in A016754. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829.

Or, odd numbers without the squares. - Gerald Hillier, Apr 12 2009

Examples

			n = p prime, abundance = 1 - p = even and negative;
n = 21, sigma = 1 + 3 + 7 + 21 = 32, abundance = 32 - 42 = -20.
		

Links

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

Crossrefs

Cf. A005408, A014105, A016754, A088827, A088829, A157502.

Programs

Magma

[ n: n in [1..140 by 2] | IsEven(SumOfDivisors(n)-2*n) ]; // Klaus Brockhaus, Apr 15 2009

Mathematica

Do[s=DivisorSigma[1, n]-2*n; If[ !OddQ[s]&&OddQ[n], Print[{n, s}]], {n, 1, 1000}]
Select[Range[1, 500, 2], EvenQ[DivisorSigma[1, #] - 2 #] &] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

PARI

isok(n) = (n>0) && (n % 2) && ! issquare(n); \\ Michel Marcus, Aug 28 2013

Python

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A088828_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not is_square(n),count(max(startvalue+(startvalue&1^1),1),2))
A088828_list = list(islice(A088828_gen(),30)) # Chai Wah Wu, Jul 06 2023

Python

from math import isqrt
def A088828(n): return (s:=(m:=isqrt(k:=(n<<1)-1))+(k-m*(m+1)>=1))+k+(s&1) # Chai Wah Wu, Jun 19 2024

Formula

a(n) = 2*n + s - ((s+1) mod 2) where s = round(sqrt(2*n-1)). - Gerald Hillier, Apr 15 2009

A005408 SETMINUS A016754. - R. J. Mathar, Jun 16 2018

a(n) = 2*(n+h) + 1 where h = floor((1/4)*(sqrt(8*n) - 1)) is the largest value such that A014105(h) < n. - John Tyler Rascoe, Jul 05 2022

Extensions

Entry revised by N. J. A. Sloane, Jan 31 2014 at the suggestion of Omar E. Pol