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A330870
Practical numbers with a record gap to the next practical number.
4
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 29 2020
STATUS
approved