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

1,2

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, ...

Links

Table of n, a(n) for n=1..35.

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.

Example

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.

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

Crossrefs

Cf. A005153.

Similar sequences: A306747, A306748, A306953.

Sequence in context: A052447 A226458 A364383 * A364707 A342152 A153537

Adjacent sequences: A330867 A330868 A330869 * A330871 A330872 A330873

Keyword

nonn

Author

Amiram Eldar, Apr 29 2020

Status

approved