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

1, 2, 4, 6, 8, 12, 24, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1440, 2160, 2880, 4320, 5760, 8640, 12960, 15120, 17280, 20160, 25920, 30240, 40320, 51840, 60480, 90720, 120960, 181440, 241920, 302400, 362880, 483840, 604800, 725760, 907200, 1209600

Offset: 1

Author

Amiram Eldar, Jul 19 2024

Keywords

• nonn

Comments

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.

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).

The least number k such that A374777(k)/A374778(k) > m for m = 2, 3, ..., is 144, 10886400, 532197314288640000, 2205754988720678629683817883074560000000, ... .

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..778

Crossrefs

Subsequence of A025487.

Cf. A004394, A017665, A017666, A374777, A374778.

Programs

Mathematica

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]

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));}
lista(kmax) = {my(smax = 0, s1); for(k = 1, kmax, s1 = s(k); if(s1 > smax, print1(k, ", "); smax = s1));}