A348742 - OEIS

%I #28 Nov 29 2024 08:11:25

%S 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089,1225,

%T 1369,1521,1681,1849,2025,2205,2209,2401,2601,2809,3025,3249,3481,

%U 3721,3969,4225,4489,4761,5041,5329,5625,5929,6241,6561,6889,7225,7569,7921,8281,8649,9025,9409,9801,10201,10609,11025,11449,11881,12321

%N Odd numbers k for which A161942(k) >= k, where A161942 is the odd part of sigma.

%C All odd squares (A016754) are present, but not all terms are squares. A348743 gives the nonsquare terms.

%C Odd terms of A336702 form a subsequence. Also all odd terms of A005820 would be present here, as well as any hypothetical quasi-perfect numbers (see comments and references in A332223, A336700), both in A016754. - _Antti Karttunen_, Nov 28 2024

%H <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%p q:= n-> (t-> is(t/2^padic[ordp](t,2)>=n))(numtheory[sigma](n)):

%p select(q, [2*i-1$i=1..10000])[];  # _Alois P. Heinz_, Nov 28 2024

%t odd[n_] := n/2^IntegerExponent[n, 2]; Select[Range[1, 10^4, 2], odd[DivisorSigma[1, #]] >= # &] (* _Amiram Eldar_, Nov 02 2021, edited (because of the changed definition) by _Antti Karttunen_, Nov 28 2024 *)

%o (PARI)

%o A000265(n) = (n >> valuation(n, 2));

%o isA348742(n) = ((n%2)&&A000265(sigma(n))>=n); \\ revised by _Antti Karttunen_, Nov 28 2024

%Y Union of A016754 and A348743.

%Y Cf. A161942, A162284 (subsequence), A336702, A348741 (complement among the odd numbers).

%Y Cf. also A005820, A332223, A336700, A348739.

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 02 2021

%E a(1) = 1 inserted as the initial term, because of the changed definition (from > to >=) - _Antti Karttunen_, Nov 28 2024