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
A353544
Numbers k such that k and k+1 are both in A353543.
Original entry on oeis.org
285, 43214, 190773, 2676321, 3027002, 3209073, 3894638, 5344118, 8963306, 15059985, 16558005, 18619634, 35731857, 36233846, 36413385, 37601342, 43559714, 52596434, 70700145, 75135962, 81136418, 83557617, 90577994, 91667666, 99846201, 111263074, 124896045, 128709801
Offset: 1
285 is a term since both 285 and 286 are in A353543.
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q[n_] := DivisorSigma[-1, n] > Pi^2/6 && AllTrue[Most @ Divisors[n], DivisorSigma[-1, #] < Pi^2/6 &]; Position[Partition[Array[q, 4*10^6], 2, 1], {True, True}] // Flatten
A339937
Numbers k such that k and k+1 are both coreful abundant numbers (A308053).
Original entry on oeis.org
2282175, 33350624, 46734975, 86424975, 87152624, 105674624, 126114975, 169707824, 179762624, 214491375, 221370975, 235857824, 266022224, 270586575, 278524575, 297774224, 360021375, 372683024, 380858624, 395715375, 425840624, 470624175, 489873824, 503963775
Offset: 1
2282175 is a term since 2282175 and 2282176 are both coreful abundant numbers.
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f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); abQ[n_] := s[n] > 2*n; q1 = False; seq = {}; Do[q2 = abQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 10^8}]; seq
A361934
Numbers k such that k and k+1 are both primitive Zumkeller numbers (A180332).
Original entry on oeis.org
82004, 84524, 158235, 516704, 2921535, 5801984, 10846016, 12374144, 12603824, 18738224, 24252074, 24887655, 25691984, 32409530, 33696975, 35356544, 36149295, 41078114, 42541190, 43485584
Offset: 1
82004 is a term since 82004 and 82005 are both primitive Zumkeller numbers.
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q[n_, d_, s1_, m1_] := Module[{s = s1, m = m1}, If[m == 0, False, While[d[[m]] > n, s -= d[[m]]; m--]; d[[m]] == n || If[s > n, q[n - d[[m]], d, s - d[[m]], m - 1] || q[n, d, s - d[[m]], m - 1], n == s]]];
(* after M. F. Hasler's pari code at A006037 *)
zumQ[n_] := Module[{d = Most[Divisors[n]], m, s}, m = Length[d]; s = Total[d]; If[OddQ[s + n], False, q[(s + n)/2, d, s, m]]];
primZumQ[n_] := zumQ[n] && AllTrue[Most[Divisors[n]], ! zumQ[#] &];
seq[kmax_] := Module[{s = {}, zq1 = False, zq2}, Do[zq2 = primZumQ[k]; If[zq1 && zq2, AppendTo[s, k - 1]]; zq1 = zq2, {k, 2, kmax}]; s]; seq[3*10^6]
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is1(n,d,s,m) = {m||return; while(d[m]>n, s-=d[m]; m--||return); d[m]==n || if(nM. F. Hasler at A006037
isZum(n) = {my(d = divisors(n)[^-1], s = vecsum(d), m = #d); if((s+n)%2, return(0), is1((s+n)/2, d, s, m)); }
isPrimZum(n) = {if(!isZum(n), return(0)); fordiv(n, d, if(d < n && isZum(d), return(0))); 1;}
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isPrimZum(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}
A361935
Numbers k such that k and k+1 are both primitive unitary abundant numbers (definition 2, A302574).
Original entry on oeis.org
2457405145194, 2601523139214, 3320774552094, 3490250769005, 3733421997305, 3934651766045, 3954730124345, 4514767592334, 4553585751714, 4563327473705, 5226433847634
Offset: 1
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f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);
puabQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
Select[Import["https://oeis.org/A331412/b331412.txt", "Table"][[;; , 2]], puabQ[#] && puabQ[# + 1] &]
A372300
Numbers k such that k and k+1 are both primitive infinitary abundant numbers (definition 1, A372298).
Original entry on oeis.org
812889, 3181815, 20787584, 181480695, 183872535, 307510664, 337206344, 350158808, 523403264, 744074624, 868421504, 1063361144, 1955365125, 2076191864, 2578966215, 3672231255, 4185590408, 5032685384, 7158001304, 8348108535, 10784978295, 16264812135, 20917209495, 24514454055
Offset: 1
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isidiv(d, f) = {my(bne,bde); if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
isab(n) = isigma(n) > 2*n;
isprim(n) = select(x -> x= 2*x, idivs(n)) == [];
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isab(k); if(is1 && is2, if(isprim(k-1) && isprim(k), print1(k-1, ", "))); is1 = is2);}
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