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 A330870 Practical numbers with a record gap to the next practical number. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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: A231495 A052447 A226458 * A342152 A153537 A153565 Adjacent sequences:  A330867 A330868 A330869 * A330871 A330872 A330873 KEYWORD nonn AUTHOR Amiram Eldar, Apr 29 2020 STATUS approved

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Last modified December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)