A334883
Primitive practical numbers (A267124) with a record gap to the next primitive practical number.
Original entry on oeis.org
1, 2, 6, 42, 104, 140, 1036, 1590, 2730, 7900, 10374, 19180, 22660, 23180, 26418, 105868, 114960, 139060, 295780, 403524, 482250, 1294144, 2468944, 4799058, 5379282, 19035500, 20233936, 21803860, 112406992, 789190976, 3520928922
Offset: 1
The first 8 primitive practical numbers are 1, 2, 6, 20, 28, 30, 42 and 66. The differences between these terms are 1, 4, 14, 8, 2, 12 and 24. The record gaps are 1, 4, 14 and 24, which occur after the terms 1, 2, 6 and 42.
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f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), ?(# > 1 &)]) == {}; pracTestQ[fct, k_] := Module[{f = fct}, f[[k, 2]] -= 1; pracQ[f]]; primPracQ[n_] := Module[{fct = FactorInteger[n]}, pracQ[fct] && AllTrue[Range@Length[fct], fct[[#, 2]] == 1 || ! pracTestQ[fct, #] &]]; seq = {1}; m = 2; dm = 1; Do[If[primPracQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 4, 10^5, 2}]; seq
A364706
a(n) is the least number k such that the k-th difference between consecutive practical numbers, A179651(k), equals 2*n, or -1 if no such k exists.
Original entry on oeis.org
2, 5, 16, 33, 85, 46, 331, 188, 171, 300, 1986, 962, 3321, 968, 2316, 6514, 9974, 3219, 12162, 3831, 4588, 20585, 30099, 22005, 30465, 33485, 28874, 35901, 136396, 48483, 120127, 34145, 140589, 233364, 126080, 185421, 607164, 279989, 359002, 327768, 609867, 354143
Offset: 1
a(1) = 2 since A179651(2) = 2 = 2*1.
a(2) = 5 since A179651(5) = 4 = 2*2.
a(3) = 16 since A179651(16) = 6 = 2*3.
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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[len_, nmax_] := Module[{s = Table[0, {len}], n = 2, prev = 2, k = 2, c = 0, i}, While[c < len && n <= nmax, n+=2; If[pracQ[n], i = (n - prev)/2; If[i <= len && s[[i]] == 0, c++; s[[i]] = k]; prev= n; k++]]; s]; seq[20, 10^6]
A364707
a(n) is the least practical number A005153(k) such that A005153(k+1) - A005153(k) = 2*n, or -1 if no such number exists.
Original entry on oeis.org
2, 8, 42, 112, 368, 180, 1806, 936, 840, 1600, 14168, 6216, 25120, 6272, 16770, 52668, 83720, 24240, 103840, 29440, 35910, 184140, 278334, 197912, 282150, 313040, 266112, 337840, 1438722, 468540, 1254016, 319808, 1486584, 2566432, 1321376, 2003688, 7163646, 3121328
Offset: 1
a(1) = 2 since A005153(2) = 2 and A005153(3) = 4 = 2 + 2*1.
a(2) = 8 since A005153(5) = 8 and A005153(6) = 12 = 8 + 2*2.
a(3) = 42 since A005153(16) = 42 and A005153(17) = 48 = 42 + 2*3.
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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[len_, nmax_] := Module[{s = Table[0, {len}], n = 2, prev = 2, c = 0, i}, While[c < len && n <= nmax, n+=2; If[pracQ[n], i = (n - prev)/2; If[i <= len && s[[i]] == 0, c++; s[[i]] = prev]; prev= n]]; s]; seq[20, 10^6]
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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