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A374781
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Numbers whose divisors have a record value of the mean abundancy index.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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, ... .
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]
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PROG
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(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)); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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