A334418 -id:A334418 - cp's OEIS frontend

A334419 Primitive abundant numbers (A071395) with a record gap to the next primitive abundant number.

20, 104, 945, 2210, 2584, 8415, 10184, 12104, 15368, 86272, 133484, 135470, 140668, 643336, 700256, 1149952, 2410816, 2434888, 5924032, 6100605, 7623872, 8531144, 8760424, 9405045, 10471755, 14803216, 16283085, 21506432, 26919250, 34441946, 35622016, 36064964

Offset: 1

Amiram Eldar, Apr 29 2020

nonn

The record gap values are 50, 168, 239, 260, 406, 510, ... (see the link for more values).

			The first 5 terms of A071395 are 20, 70, 88, 104 and 272. The differences between these terms are 50, 18, 16, and 168. The record gaps are 50 and 168, which occur after the terms 20 and 104.

Amiram Eldar, Table of n, a(n) for n = 1..59
Amiram Eldar, Table of n, a(n), gap(n) for n = 1..59

Cf. A071395, A330872, A334418.

Similar sequences: A306747, A306748, A306953.

primAbQ[n_] := DivisorSigma[1, n] > 2 n && AllTrue[Most @ Rest @ Divisors[n], DivisorSigma[1, #] < 2*# &]; seq = {}; m = 20; dm = 0; Do[If[primAbQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 21, 10^6}]; seq

A364975 Admirable numbers (A111592) with a record gap to the next admirable number.

Original entry on oeis.org

12, 30, 42, 88, 120, 140, 186, 534, 678, 6774, 7962, 77118, 94108, 152826, 478194, 662154, 935564, 1128174, 2028198, 6934398, 7750146, 8330924, 9984738, 10030804, 22956114, 62062566, 151040622, 284791602, 732988732, 804394974, 1151476732, 9040886574, 31302713634

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

The corresponding record gaps are 8, 10, 12, 14, 18, 34, 36, 48, 84, 132, 204, 216, 254, 312, 348, 360, 392, 468, 516, 528, 552, 598, 624, 638, 828, 852, 936, 1056, 1082, 1128, 1454, 1692, 1752, ... .

			The first 5 admirable numbers are 12, 20, 24, 30 and 40. The differences between these terms are 8, 4, 6 and 10. The record gaps, 8 and 10, occur after the terms 12 and 30, which are the first two terms of this sequence.

Cf. A111592, A364727.

Similar sequences: A306953, A330870, A334418, A334419, A334883, A363296.

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
seq[kmax_] := Module[{s = {}, m = 12, dm = 0}, Do[If[admQ[k], d = k - m; If[d > dm, dm = d; AppendTo[s, m]]; m = k], {k, m + 1, kmax}]; s]; seq[10^6]

isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0; }
lista(kmax) = {my(m = 12, dm = 0); for(k = m+1, kmax, if(isadm(k), d = k - m; if(d > dm, dm = d; print1(m, ", ")); m = k));}

cp's OEIS Frontend

A334419 Primitive abundant numbers (A071395) with a record gap to the next primitive abundant number.

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