cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A331412 Unitary abundant numbers k such that k + 1 is also unitary abundant.

Original entry on oeis.org

8857357509, 10783550414, 15197873690, 23620285689, 25537083494, 34736070369, 60326914934, 64139567205, 73969772954, 75776483145, 77509981185, 83968675790, 93092467754, 100012014465, 112236593469, 113606741534, 116519300534, 118905484334, 132584489114, 134889106065
Offset: 1

Views

Author

Amiram Eldar and Giovanni Resta, Jan 18 2020

Keywords

Comments

Apparently most of the terms are squarefree. Up to 10^13 there are 1150 terms, for only 17 terms k either k or k + 1 is nonsquarefree, and there are no terms k such that both k and k + 1 are nonsquarefree. The first nonsquarefree term is a(32) = 285491549265.

Examples

			8857357509 is a term since usigma(8857357509) = 17766604800 > 2 * 8857357509, and usigma(8857357510) = 17851083264 > 2 * 8857357510, where usigma is the sum of unitary divisors function (A034448).

Links

Crossrefs

Cf. A034448, A034683, A292704.
Analogous sequences: A096399 (regular abundant), A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary).

A327942 Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).

Original entry on oeis.org

165375, 893024, 1047375, 1576575, 2282175, 2304224, 2858624, 3614624, 4068224, 4096575, 4597424, 4975424, 6591375, 7574175, 8555624, 9511424, 10446975, 10749375, 10872224, 11477024, 12535424, 13773375, 13946624, 14277375, 15926624, 16041375, 16505775, 16769024
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2019

Keywords

Examples

			165375 is in the sequence since both 165375 and 165376 are nonunitary abundant: nusigma(165375) = 179280 > 165375, and nusigma(165376) = 183600 > 165376 (nusigma is the sum of nonunitary divisors, A048146).

Links

Crossrefs

Cf. A048146, A064597, A094889, A307823.
Cf. A096399, A292704, A318167, A327635.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); nuabQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (1 + Power @@@ FactorInteger[n]) > n; s = {}; q1 = False; Do[q2 = nuabQ[n]; If[q1 && q2, AppendTo[s, n - 1]]; q1 = q2, {n, 2, 10^7}]; s

A333951 Numbers k such that both k and k+1 are recursive abundant numbers (A333928).

Original entry on oeis.org

56924, 82004, 84524, 109395, 158235, 241604, 261260, 266475, 285075, 361844, 442035, 445004, 469755, 611324, 666315, 694484, 712844, 922635, 968715, 971684, 1102724, 1172115, 1190475, 1199835, 1239524, 1304324, 1338435, 1430715, 1442924, 1486275, 1523115, 1550835
Offset: 1

Views

Author

Amiram Eldar, Apr 11 2020

Keywords

Examples

			56924 is a term since A333926(56924) = 120960 > 2 * 56924, and A333926(56925) = 116064 > 2 * 56925.

Links

Crossrefs

Subsequence of A333928.
Cf. A333926, A333950.
Analogous sequences: A096399, A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary), A331412 (unitary).

Programs

  • Mathematica
    recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); recAbQ[n_] := recDivSum[n] > 2*n; Select[Range[2*10^5], recAbQ[#] && recAbQ[# + 1] &]

A357608 Numbers k such that k and k+1 are both in A357605.

Original entry on oeis.org

76544, 104895, 126224, 165375, 170624, 174824, 201824, 245024, 257984, 271215, 273104, 316575, 338624, 387855, 447615, 469664, 477224, 540224, 618975, 633555, 641024, 659295, 705375, 752895, 770175, 842624, 843975, 862784, 870975, 893024, 913275, 957824, 1047375
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

Numbers k such that A162296(k) > 2*k and A162296(k+1) > 2*(k+1).

Examples

			76544 is a term since 76544 and 76545 are both in A357605: A162296(76544) = 170688 > 2*76544 and A162296(76545) = 157248 > 2*76545.

Links

Crossrefs

Cf. A162296.
Subsequence of A013929, A096399 and A357605.
Similar sequences: A096399, A283418, A318167, A327635, A327942, A331412, A333951.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 10^6], q[#] && q[#+1] &]

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

Views

Author

Amiram Eldar, Dec 23 2020

Keywords

Examples

			2282175 is a term since 2282175 and 2282176 are both coreful abundant numbers.

Links

Crossrefs

Subsequence of A308053.
Similar sequences: A096399, A283418, A318167, A327635, A327942, A330872, A331412, A333951.

Programs

  • Mathematica
    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

A348276 Numbers k such that k and k+1 are both noninfinitary abundant numbers (A348274).

Original entry on oeis.org

64198575, 84909824, 86424975, 110238975, 113223824, 191206575, 211266224, 224722575, 231058575, 231800624, 240069375, 240584175, 245383424, 262648575, 262911824, 279597824, 293893424, 297774224, 333773055, 338676975, 340250624, 340829775, 347244975, 372683024
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Examples

			64198575 is a term since A348271(64198575) = 69470136 > 64198575 and A348271(64198576) = 65363424 > 64198576.

Crossrefs

Cf. A348271.
Subsequence of A096399 and A348274.
Similar sequences: A318167, A327635, A327942, A331412.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := DivisorSigma[1,n] - isigma[n] > n; seq = {}; q1 = q[1]; Do[q2 = q[n]; If[q1 && q2, AppendTo[seq, n-1]]; q1=q2 ,{n,2,10^8}]; seq

A348606 Numbers k such that k and k+1 are both nonexponential abundant numbers.

Original entry on oeis.org

21735, 76544, 170624, 301664, 345344, 348704, 382304, 739935, 862784, 1218944, 1262624, 1272704, 1314495, 1370655, 1376864, 1539615, 1558304, 1707615, 1718144, 1761375, 1845375, 1890944, 1926015, 2100735, 2132864, 2223584, 2415104, 2463615, 2581215, 2675295, 2747744
Offset: 1

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Examples

			21735 is a term since A160135(21735) = 21930 > 21735 and A160135(21736) = 23230 > 21736.

Links

Crossrefs

Cf. A160135.
Subsequence of A096399 and A348604.
Similar sequences: A318167, A327635, A327942, A331412, A348276.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 3*10^6], q[#] && q[#+1] &]

A358483 Numbers k such that k, k+1 and k+2 are all infinitary abundant numbers (A129656).

Original entry on oeis.org

2666847104, 2695309694, 8207456894, 8967010688, 12147283070, 12491149670, 13911605630, 14126720894, 17238119624, 17238704768, 18420223094, 20922243110, 21786026624, 25118874494, 26079705728, 26979164288, 27257009624, 30000503168, 30478990904, 30832299134, 32892108248
Offset: 1

Views

Author

Amiram Eldar, Nov 18 2022

Keywords

Examples

			2666847104 is in the sequence since 2666847104, 2666847105 and 2666847106 are all infinitary abundant numbers (A129656): isigma(2666847104) = 5401952640 > 2 * 2666847104, isigma(2666847105) = 5374656000 > 2 * 2666847105, and isigma(2666847106) = 5419376640 > 2 * 2666847106 (isigma is the sum of infinitary divisors, A049417).

Links

Crossrefs

Subsequence of A096536, A129656 and A327635.
Cf. A049417.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); abQ[n_] := isigma[n] > 2*n; v = Cases[Import["https://oeis.org/A096536/b096536.txt", "Table"], {, }][[;; , 2]]; Select[v, And @@ abQ /@ (# + {0, 1, 2}) &]

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

Views

Author

Amiram Eldar, Apr 25 2024

Keywords

Comments

The corresponding sequence with definition 2 (A372299) coincides with this sequence for the first 24 terms.

Crossrefs

Subsequence of A129656, A327635 and A372298.
Cf. A372299.
Similar sequences: A283418, A330872, A361935.

Programs

  • PARI
    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);}

A380933 Numbers k such that k and k+1 are both in A380929.

Original entry on oeis.org

121643775, 157390064, 161019495, 275734304, 584899875, 1493214975, 1614323655, 2043708975, 3081783375, 3118599224, 3426851295, 3902652495, 3947893424, 5849043375, 11731509855, 12138531615, 13008843224, 14598032624, 17588484584, 19782621495, 20191564575, 20759209064
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Numbers k such that A380845(k) > 2*k and A380845(k+1) > 2*(k+1).

Examples

			121643775 is a term since A380845(121643775) = 244722015 > 2 * 121643775 = 243287550, and A380845(121643776) = 256456081 > 2 * 121643776 = 243287552.

Links

Crossrefs

Subsequence of A096399 and A380929.
Cf. A380845, A380932.
Similar sequences: A283418, A318167, A327635, A327942, A331412, A333951, A357608, A364727, A364861.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k];
seq[lim_] := Module[{s = {}}, Do[If[q[k], If[q[k-1], AppendTo[s, k-1]]; If[q[k+1], AppendTo[s, k]]], {k, 3, lim, 2}]; s];
seq[3*10^8]
  • PARI
    isab(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}
list(lim) = forstep(k = 3, lim, 2, if(isab(k), if(isab(k-1), print1(k-1, ", ")); if(isab(k+1), print1(k, ", "))));
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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