cp's OEIS Frontend

A028983 Numbers whose sum of divisors is even.

%I A028983 #74 Aug 23 2024 02:10:16
%S A028983 3,5,6,7,10,11,12,13,14,15,17,19,20,21,22,23,24,26,27,28,29,30,31,33,
%T A028983 34,35,37,38,39,40,41,42,43,44,45,46,47,48,51,52,53,54,55,56,57,58,59,
%U A028983 60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,82
%N A028983 Numbers whose sum of divisors is even.
%C A028983 The even terms of this sequence are the even terms appearing in A178910. [Edited by _M. F. Hasler_, Oct 02 2014]
%C A028983 A071324(a(n)) is even. - _Reinhard Zumkeller_, Jul 03 2008
%C A028983 Sigma(a(n)) = A000203(a(n)) = A152678(n). - _Jaroslav Krizek_, Oct 06 2009
%C A028983 A083207 is a subsequence. - _Reinhard Zumkeller_, Jul 19 2010
%C A028983 Numbers k such that the number of odd divisors of k (A001227) is even. - _Omar E. Pol_, Apr 04 2016
%C A028983 Numbers k such that the sum of odd divisors of k (A000593) is even. - _Omar E. Pol_, Jul 05 2016
%C A028983 Numbers with a squarefree part greater than 2. - _Peter Munn_, Apr 26 2020
%C A028983 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
%C A028983 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
%C A028983 Numbers whose number of middle divisors is not odd (cf. A067742). - _Omar E. Pol_, Aug 02 2022
%H A028983 T. D. Noe, <a href="/A028983/b028983.txt">Table of n, a(n) for n = 1..1000</a>
%F A028983 a(n) ~ n. - _Charles R Greathouse IV_, Jan 11 2013
%F A028983 a(n) = n + (1 + sqrt(2)/2)*sqrt(n) + O(1). - _Charles R Greathouse IV_, Sep 01 2015
%F A028983 A007913(a(n)) > 2. - _Peter Munn_, May 05 2020
%t A028983 Select[Range[82],EvenQ[DivisorSigma[1,#]]&] (* _Jayanta Basu_, Jun 05 2013 *)
%o A028983 (PARI) is(n)=!issquare(n)&&!issquare(n/2) \\ _Charles R Greathouse IV_, Jan 11 2013
%o A028983 (Python)
%o A028983 from math import isqrt
%o A028983 def A028983(n):
%o A028983     def f(x): return n-1+isqrt(x)+isqrt(x>>1)
%o A028983     kmin, kmax = 1,2
%o A028983     while f(kmax) >= kmax:
%o A028983         kmax <<= 1
%o A028983     while True:
%o A028983         kmid = kmax+kmin>>1
%o A028983         if f(kmid) < kmid:
%o A028983             kmax = kmid
%o A028983         else:
%o A028983             kmin = kmid
%o A028983         if kmax-kmin <= 1:
%o A028983             break
%o A028983     return kmax # _Chai Wah Wu_, Aug 22 2024
%Y A028983 The complement is A028982 = A000290 U A001105.
%Y A028983 Cf. A000203, A000593, A001227, A007913, A178910, A152678, A067742.
%Y A028983 Subsequences: A083207, A091067, A145204\{0}, A225838, A225858.
%Y A028983 Cf. A334748 (a permutation).
%Y A028983 Related to A008586 via A225546.
%Y A028983 Ranks the partitions counted by A347448, complement A119620.
%Y A028983 Cf. A030059, A335433, A335448, A339890, A344607, A347438, A347443, A347445, A347446, A347452, A347453, A347465.
%K A028983 nonn,easy
%O A028983 1,1
%A A028983 _Patrick De Geest_