A330870 Practical numbers with a record gap to the next practical number.

1, 2, 8, 42, 112, 180, 840, 1600, 6216, 6272, 16770, 24240, 29440, 35910, 184140, 197912, 266112, 319808, 1321376, 2003688, 3121328, 3696480, 4017216, 4543672, 5300910, 5791302, 11582680, 12142088, 27631300, 31187592, 31243040, 64181800, 106366560, 307350504, 1255812780

Offset: 1

Author

Amiram Eldar, Apr 29 2020

Keywords

• nonn

Comments

The record gap values are 1, 2, 4, 6, 8, 12, 18, 20, 24, 28, 30, 36, 40, 42, 44, 48, 54, 64, 70, 72, 76, 80, 84, 88, 90, 98, 100, 112, 122, 124, 128, 136, 160, 180, 192, ...

Examples

			The first 6 practical numbers are 1, 2, 4, 6, 8 and 12. The differences between these terms are 1, 2, 2, 2 and 4. The record gaps are 1, 2 and 4, which occur after the terms 1, 2 and 8.
		

Links

• Miriam Hausman and Harold N. Shapiro, On practical numbers, Communications on Pure and Applied Mathematics, Vol. 37, No. 5 (1984), pp. 705-713, section 4.
• Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Politec. Torino, Vol. 53, No. 4 (1995), pp. 347-359, section 5.

Crossrefs

Cf. A005153.

Similar sequences: A306747, A306748, A306953.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most @ fct]), _?(# > 1 &)]) == {}; seq = {}; m = 1; dm = 0; Do[If[pracQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 2, 10^6}]; seq