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
A363296
Unitary weird numbers (A064114) with a record gap to the next unitary weird number.
Original entry on oeis.org
70, 5830, 2197790, 902388130, 2013240110
Offset: 1
70 is in the sequence since it is the first unitary weird number and the next unitary weird number after it is 4030 = 70 + 3960. The next gap which is larger than 3960 is 4600 and it occurs at 5830 which is followed by 10430 = 5830 + 4600.
A364975
Admirable numbers (A111592) with a record gap to the next admirable number.
Original entry on oeis.org
12, 30, 42, 88, 120, 140, 186, 534, 678, 6774, 7962, 77118, 94108, 152826, 478194, 662154, 935564, 1128174, 2028198, 6934398, 7750146, 8330924, 9984738, 10030804, 22956114, 62062566, 151040622, 284791602, 732988732, 804394974, 1151476732, 9040886574, 31302713634
Offset: 1
The first 5 admirable numbers are 12, 20, 24, 30 and 40. The differences between these terms are 8, 4, 6 and 10. The record gaps, 8 and 10, occur after the terms 12 and 30, which are the first two terms of this sequence.
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admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
seq[kmax_] := Module[{s = {}, m = 12, dm = 0}, Do[If[admQ[k], d = k - m; If[d > dm, dm = d; AppendTo[s, m]]; m = k], {k, m + 1, kmax}]; s]; seq[10^6]
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isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0; }
lista(kmax) = {my(m = 12, dm = 0); for(k = m+1, kmax, if(isadm(k), d = k - m; if(d > dm, dm = d; print1(m, ", ")); m = k));}
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