A083873 -id:A083873 - cp's OEIS frontend

A362053 Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.

20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752, 1848964, 8353792, 8378368, 8382464, 35021696, 45335936, 120888092, 134193152, 775397948, 1845991216, 2146926592, 2146992128, 3381872252

Offset: 1

Amiram Eldar, Apr 06 2023

nonn

The abundancy index of an integer k is sigma(k)/k, where sigma is the sum-of-divisors function (A000203).

Terms k of A071395 such that sigma(k)/k < sigma(m)/m for all smaller terms m < k of A071395.

			The abundancy indices of the first terms are 21/10 > 72/35 > 45/22 > 105/52 > 465/232 > 651/325 > 945/472 > ... > 2.

Cf. A000203, A071395, A071927.

Other sequences related to records in A071395: A083873, A334419.

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
(* Returns the abundancy index of n if n is primitive abundant, and 0 otherwise: *)
abIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; abIndex[1] = 0;
seq[kmax_] := Module[{s = {}, ab, abm = 3}, Do[If[0 < (ab = abIndex[k]) < abm, abm = ab; AppendTo[s, k]], {k, 1,  kmax}]; s]; seq[10^6]

abindex(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2, r, 0);} \\ Returns the abundancy index of n if n is primitive abundant, and 0 otherwise.
lista(kmax) = {my(ab, abm = 3); for(k = 1, kmax, ab = abindex(k); if(ab > 0 && ab < abm, abm = ab; print1(k, ", "))); }

A357921 Primitive abundant numbers for which there is no smaller primitive abundant number having the same ordered prime signature.

Original entry on oeis.org

20, 70, 88, 272, 550, 572, 945, 1184, 1430, 1575, 2205, 3465, 4288, 5775, 7425, 8085, 12705, 15015, 16768, 24272, 28215, 47025, 49875, 65792, 69825, 78975, 81081, 103455, 131625, 152224, 153153, 182325, 189189, 266752, 297297, 342225, 351351, 363375, 387345, 392445, 474045

Offset: 1

David A. Corneth, Oct 20 2022

nonn

			550 is in the sequence as it is primitive abundant and it is the smallest primitive abundant number having its prime signature (in this case (1,2,1)). No positive integer < 550 is a primitive abundant and has prime signature (1,2,1).

Supersequence of A083873.

Cf. A025487, A071395.

upto(n) = { m = Map(); res = List(); forfactored(i = 1, n, if(!mapisdefined(m, i[2][,2]), if(isA071395(i[1]), mapput(m, i[2][,2], i[1]); listput(res, i[1]) ) ) ); res }
isA071395(v) = {if (sigma(v) <= 2*v, return (0)); fordiv (v, d, if ((d != v) && (sigma(d) >= 2*d), return (0)); ); return (1); } \\ isA071395 from Michel Marcus at A071395

cp's OEIS Frontend

A362053 Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.

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