cp's OEIS Frontend

A374781 Numbers whose divisors have a record value of the mean abundancy index.

%I A374781 #6 Jul 19 2024 14:29:09
%S A374781 1,2,4,6,8,12,24,48,72,96,120,144,240,288,360,480,576,720,1440,2160,
%T A374781 2880,4320,5760,8640,12960,15120,17280,20160,25920,30240,40320,51840,
%U A374781 60480,90720,120960,181440,241920,302400,362880,483840,604800,725760,907200,1209600
%N A374781 Numbers whose divisors have a record value of the mean abundancy index.
%C A374781 Positions of records in {f(k) | k = 1, 2, ...}, where f(k) = (Sum_{d|k} sigma(d)/d) / tau(k) = A374777(k)/A374778(k), i.e., numbers k such that f(k) > f(m) for all m < k.
%C A374781 Similarly to the sequence of abundancy indices {sigma(k)/k} = {A017665(k)/A017666(k)}, the record values occur at indices that are least numbers of each prime signature (A025487). For the abundancy indices the records occur at the superabundant numbers (A004394).
%C A374781 The least number k such that A374777(k)/A374778(k) > m for m = 2, 3, ..., is 144, 10886400, 532197314288640000, 2205754988720678629683817883074560000000, ... .
%H A374781 Amiram Eldar, <a href="/A374781/b374781.txt">Table of n, a(n) for n = 1..778</a>
%e A374781 The values of A374777(k)/A374778(k) for k = 1..6 are {1, 5/4, 7/6, 17/12, 11/10, 35/24} = {1, 1.25, 1.166..., 1.416..., 1.1, 1.458...}. The records values, {1, 1.25, 1.416..., 1.458...} occur at k = 1, 2, 4, and 6, the first 4 terms of this sequence.
%t A374781 f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{v = {}, smax = 0, s1}, Do[s1 = s[k]; If[s1 > smax, AppendTo[v, k]; smax = s1], {k, 1, kmax}]; v]; seq[10^5]
%o A374781 (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2));}
%o A374781 lista(kmax) = {my(smax = 0, s1); for(k = 1, kmax, s1 = s(k); if(s1 > smax, print1(k, ", "); smax = s1));}
%Y A374781 Subsequence of A025487.
%Y A374781 Cf. A004394, A017665, A017666, A374777, A374778.
%K A374781 nonn
%O A374781 1,2
%A A374781 _Amiram Eldar_, Jul 19 2024