A334883 Primitive practical numbers (A267124) with a record gap to the next primitive practical number.

1, 2, 6, 42, 104, 140, 1036, 1590, 2730, 7900, 10374, 19180, 22660, 23180, 26418, 105868, 114960, 139060, 295780, 403524, 482250, 1294144, 2468944, 4799058, 5379282, 19035500, 20233936, 21803860, 112406992, 789190976, 3520928922

Offset: 1

Author

Amiram Eldar, May 14 2020

Keywords

• nonn
• more

Comments

The record gap values are 1, 4, 14, 24, 36, 64, 74, 82, 84, 104, 106, 112, 120, 132, 154, 188, 204, 224, 236, 246, 258, 308, 326, 360, 418, 440, 452, 508, 674, 804, 846, ...

Examples

			The first 8 primitive practical numbers are 1, 2, 6, 20, 28, 30, 42 and 66. The differences between these terms are 1, 4, 14, 8, 2, 12 and 24. The record gaps are 1, 4, 14 and 24, which occur after the terms 1, 2, 6 and 42.
		

Crossrefs

Cf. A267124, A330870, A334882.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), ?(# > 1 &)]) == {}; pracTestQ[fct, k_] := Module[{f = fct}, f[[k, 2]] -= 1; pracQ[f]]; primPracQ[n_] := Module[{fct = FactorInteger[n]}, pracQ[fct] && AllTrue[Range@Length[fct], fct[[#, 2]] == 1 || ! pracTestQ[fct, #] &]]; seq = {1}; m = 2; dm = 1; Do[If[primPracQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 4, 10^5, 2}]; seq