A362054 - OEIS

A362054

Primitive unitary abundant numbers k (A302573) whose unitary abundancy index usigma(k)/k has a record low value.

70, 1092, 1428, 1596, 4030, 5830, 50388, 133042, 216300, 269990, 437745, 442365, 4199030, 22982388, 124540390, 361745930, 507298090, 541900788, 624032630, 1113445430, 3002432810, 6771402960, 13455037365, 17242767300, 60428265370

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OFFSET

1,1

COMMENTS

The unitary abundancy index of an integer k is usigma(k)/k, where usigma is the sum-of-unitary-divisors function (A034448).

Terms k of A302573 such that usigma(k)/k < usigma(m)/m for all smaller terms m < k of A302573.

LINKS

Table of n, a(n) for n=1..25.

EXAMPLE

The unitary abundancy indices of the first terms are 72/35 > 80/39 > 240/119 > 800/399 > 4032/2015 > 5832/2915 > ... > 2.

MATHEMATICA

f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);

(* Returns the unitary abundancy index of n if n is primitive unitary abundant, and 0 otherwise: *)

uabIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; uabIndex[1] = 0;

seq[kmax_] := Module[{s = {}, uab, uabm = 3}, Do[If[0 < (uab = uabIndex[k]) < uabm, uabm = uab; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]

PROG

(PARI) uabindex(n) = {my(f = factor(n), r); r = prod(i = 1, #f~, 1 + 1/f[i, 1]^f[i, 2]); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, f[i, 1]^f[i, 2]/(f[i, 1]^f[i, 2] + 1))) * r < 2, r, 0); } \\ Returns the unitary abundancy index of n if n is primitive unitary abundant, and 0 otherwise.

lista(kmax) = {my(uab, uabm = 3); for(k = 1, kmax, uab = uabindex(k); if(uab > 0 && uab < uabm, uabm = uab; print1(k, ", "))); }

CROSSREFS

The unitary version of A362053.

Cf. A034448, A302573.

Sequence in context: A125114 A281649 A189413 * A229735 A254472 A361610

Adjacent sequences: A362051 A362052 A362053 * A362055 A362056 A362057

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Apr 06 2023

STATUS

approved