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A088828 Nonsquare positive odd numbers.

%I A088828 #75 Jun 19 2024 10:46:53
%S A088828 3,5,7,11,13,15,17,19,21,23,27,29,31,33,35,37,39,41,43,45,47,51,53,55,
%T A088828 57,59,61,63,65,67,69,71,73,75,77,79,83,85,87,89,91,93,95,97,99,101,
%U A088828 103,105,107,109,111,113,115,117,119,123,125,127,129,131,133,135,137,139
%N A088828 Nonsquare positive odd numbers.
%C A088828 Odd numbers with even abundance: primes and some composites too.
%C A088828 Odd numbers with odd abundance are in A016754. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829.
%C A088828 Or, odd numbers without the squares. - _Gerald Hillier_, Apr 12 2009
%H A088828 Amiram Eldar, <a href="/A088828/b088828.txt">Table of n, a(n) for n = 1..10000</a>
%F A088828 a(n) = 2*n + s - ((s+1) mod 2) where s = round(sqrt(2*n-1)). - _Gerald Hillier_, Apr 15 2009
%F A088828 A005408 SETMINUS A016754. - _R. J. Mathar_, Jun 16 2018
%F A088828 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
%e A088828 n = p prime, abundance = 1 - p = even and negative;
%e A088828 n = 21, sigma = 1 + 3 + 7 + 21 = 32, abundance = 32 - 42 = -20.
%t A088828 Do[s=DivisorSigma[1, n]-2*n; If[ !OddQ[s]&&OddQ[n], Print[{n, s}]], {n, 1, 1000}]
%t A088828 Select[Range[1, 500, 2], EvenQ[DivisorSigma[1, #] - 2 #] &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)
%o A088828 (Magma) [ n: n in [1..140 by 2] | IsEven(SumOfDivisors(n)-2*n) ]; // _Klaus Brockhaus_, Apr 15 2009
%o A088828 (PARI) isok(n) = (n>0) && (n % 2) && ! issquare(n); \\ _Michel Marcus_, Aug 28 2013
%o A088828 (Python)
%o A088828 from itertools import count, islice
%o A088828 from sympy.ntheory.primetest import is_square
%o A088828 def A088828_gen(startvalue=1): # generator of terms >= startvalue
%o A088828     return filter(lambda n:not is_square(n),count(max(startvalue+(startvalue&1^1),1),2))
%o A088828 A088828_list = list(islice(A088828_gen(),30)) # _Chai Wah Wu_, Jul 06 2023
%o A088828 (Python)
%o A088828 from math import isqrt
%o A088828 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
%Y A088828 Cf. A005408, A014105, A016754, A088827, A088829, A157502.
%K A088828 nonn,easy
%O A088828 1,1
%A A088828 _Labos Elemer_, Oct 28 2003
%E A088828 Entry revised by _N. J. A. Sloane_, Jan 31 2014 at the suggestion of _Omar E. Pol_