A330870
Practical numbers with a record gap to the next practical number.
Original entry on oeis.org
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
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.
- 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.
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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
A334419
Primitive abundant numbers (A071395) with a record gap to the next primitive abundant number.
Original entry on oeis.org
20, 104, 945, 2210, 2584, 8415, 10184, 12104, 15368, 86272, 133484, 135470, 140668, 643336, 700256, 1149952, 2410816, 2434888, 5924032, 6100605, 7623872, 8531144, 8760424, 9405045, 10471755, 14803216, 16283085, 21506432, 26919250, 34441946, 35622016, 36064964
Offset: 1
The first 5 terms of A071395 are 20, 70, 88, 104 and 272. The differences between these terms are 50, 18, 16, and 168. The record gaps are 50 and 168, which occur after the terms 20 and 104.
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primAbQ[n_] := DivisorSigma[1, n] > 2 n && AllTrue[Most @ Rest @ Divisors[n], DivisorSigma[1, #] < 2*# &]; seq = {}; m = 20; dm = 0; Do[If[primAbQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 21, 10^6}]; seq
A334418
Primitive abundant numbers (A091191) with a record gap to the next primitive abundant number.
Original entry on oeis.org
12, 20, 30, 42, 114, 138, 678, 1758, 8296, 10052, 12966, 13076, 14862, 19635, 38950, 50802, 77118, 94108, 218334, 439134, 478194, 746202, 1128174, 2028198, 6934398, 7750146, 8330924, 10030804, 33467106, 36205482, 60716562, 65183838, 69334698, 81757564, 84010614
Offset: 1
The first 6 terms of A091191 are 12, 18, 20, 30, 42 and 56. The differences between these terms are 6, 2, 10, 12 and 14. The record gaps are 6, 10, 12 and 14, which occur after the terms 12, 20, 30 and 42.
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primAbQ[n_] := DivisorSigma[1, n] > 2 n && AllTrue[Most @ Rest @ Divisors[n], DivisorSigma[1, #] <= 2*# &]; seq = {}; m = 12; dm = 0; Do[If[primAbQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 13, 10^6}]; seq
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