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A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.

%I A053624 #80 Sep 20 2024 14:15:23
%S A053624 1,3,9,15,45,105,225,315,945,1575,2835,3465,10395,17325,31185,45045,
%T A053624 121275,135135,225225,405405,675675,1576575,2027025,2297295,3828825,
%U A053624 6891885,11486475,26801775,34459425,43648605,72747675,130945815
%N A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.
%C A053624 Also numbers k such that the number of partitions of k into consecutive integers is a record. For example, 45 = 22+23 = 14+15+16 = 7+8+9+10+11 = 5+6+7+8+9+10 = 1+2+3+4+5+6+7+8+9, six such partitions, but all smaller terms have fewer such partitions (15 has four). See A000005 comments and A038547 formula. - _Rick L. Shepherd_, Apr 20 2008
%C A053624 From _Hartmut F. W. Hoft_, Mar 29 2022: (Start)
%C A053624 Also the odd parts of the numbers in A340506, see also comments in A250071.
%C A053624 A140864 is a subsequence. (End)
%C A053624 Positions of records in A001227, i.e., integers whose number of odd divisors sets a new record. - _Bernard Schott_, Jul 18 2022
%C A053624 Conjecture: all terms after the first three terms are congruent to 5 mod 10. - _Harvey P. Dale_, Jul 05 2023
%C A053624 From _Keith F. Lynch_, Jan 12 2024: (Start)
%C A053624 Dale's conjecture is correct.  a(n) can't be even, since then a(n)/2 would be a smaller number with the same number of odd divisors.  The respective powers of the successive odd primes can't increase, since if they did, swapping them would give a smaller number with the same number of divisors, e.g., 3^2 * 5^4 has the same number of divisors as 3^4 * 5^2, and the latter is smaller.  As such, every a(n) must be an odd multiple of 5, hence congruent to 5 mod 10, unless it's simply a power of 3.  But multiplying a power of 3 by 3 gives just one more divisor while multiplying a power of 3 by 5 doubles the number of divisors, so after a(n) = 9 all a(n) must be congruent to 5 mod 10, i.e., have a rightmost decimal digit of 5.
%C A053624 This has three equivalent definitions:
%C A053624 * Odd numbers with more divisors than any smaller odd number.
%C A053624 * Numbers with more odd divisors than any smaller number, i.e., record high values of A001227.
%C A053624 * Numbers with a greater excess of odd divisors over even divisors than any smaller number, i.e., record high values of A048272. (End)
%H A053624 Amiram Eldar, <a href="/A053624/b053624.txt">Table of n, a(n) for n = 1..363</a> (terms 1..170 from Ray Chandler)
%H A053624 David Ryan, <a href="https://arxiv.org/abs/1603.08904">Mathematical Harmony Analysis</a>, arXiv preprint arXiv:1603.08904 [cs.SD], 2016.
%e A053624 9 is in the sequence because 9 has 3 divisors {1, 3, 9}, which is more than any previous odd number.
%t A053624 nn = 10^6; maxd = 0;
%t A053624 Reap[For[n = 1, n <= nn, n += 2, If[(nd = DivisorSigma[0, n]) > maxd, Print[n]; Sow[n]; maxd = nd]]][[2, 1]] (* _Jean-François Alcover_, Sep 20 2018, from PARI *)
%t A053624 next[n_] := Module[{k=n, r=DivisorSigma[0, n]}, While[DivisorSigma[0, k]<=r, k+=2]; k]
%t A053624 a053624[n_] := NestList[next, 1, n-1]/; n>=1 (* returns n numbers *)
%t A053624 a053624[31] (* _Hartmut F. W. Hoft_, Mar 29 2022 *)
%t A053624 DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,1,131*10^6,2}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* _Harvey P. Dale_, Jul 05 2023 *)
%o A053624 (PARI) lista(nn) = {maxd = 0; forstep (n=1, nn, 2, if ((nd = numdiv(n)) > maxd, print1(n, ", "); maxd = nd;););} \\ _Michel Marcus_, Apr 21 2014
%Y A053624 Cf. A002182, A053640, A000005. Subsequence of A147516.
%Y A053624 Cf. A140864, A237593, A250071, A279387, A340506.
%Y A053624 Cf. A005408, A001227, A038547.
%K A053624 nonn,nice
%O A053624 1,2
%A A053624 Stefano Lanfranco (lastefano(AT)yahoo.it), Mar 21 2000