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

1,1

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

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A352830 A320056 A175679 * A348741 A348748 A182318

Adjacent sequences: A088825 A088826 A088827 * A088829 A088830 A088831

Keyword

nonn,easy

Author

Labos Elemer, Oct 28 2003

Extensions

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

Status

approved