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.
-
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
A362053
Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.
Original entry on oeis.org
20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752, 1848964, 8353792, 8378368, 8382464, 35021696, 45335936, 120888092, 134193152, 775397948, 1845991216, 2146926592, 2146992128, 3381872252
Offset: 1
The abundancy indices of the first terms are 21/10 > 72/35 > 45/22 > 105/52 > 465/232 > 651/325 > 945/472 > ... > 2.
-
f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
(* Returns the abundancy index of n if n is primitive abundant, and 0 otherwise: *)
abIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; abIndex[1] = 0;
seq[kmax_] := Module[{s = {}, ab, abm = 3}, Do[If[0 < (ab = abIndex[k]) < abm, abm = ab; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]
-
abindex(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2, r, 0);} \\ Returns the abundancy index of n if n is primitive abundant, and 0 otherwise.
lista(kmax) = {my(ab, abm = 3); for(k = 1, kmax, ab = abindex(k); if(ab > 0 && ab < abm, abm = ab; print1(k, ", "))); }
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.
-
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]
-
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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