A318167 -id:A318167 - cp's OEIS frontend

A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

21735, 21944, 43064, 58695, 188055, 262184, 414855, 520695, 567944, 611415, 687015, 764504, 792855, 809864, 812889, 833624, 874664, 911624, 945944, 976184, 991304, 1019655, 1026375, 1065015, 1073709, 1157624, 1201095, 1218944, 1248344, 1254015, 1272375, 1272704

Offset: 1

Amiram Eldar, Sep 20 2019

nonn

The least k such that k, k+1 and k+2 are all infinitary abundant numbers is a(75976) = 2666847104.

			21735 is in the sequence since both 21735 and 21736 are infinitary abundant: isigma(21735) = 46080 > 2 * 21735, and isigma(21736) = 50400 > 2 * 21736 (isigma is the sum of infinitary divisors, A049417).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049417, A127666, A129656.

Cf. A096399, A292704, A318167.

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] > 2n; s={}; ab1 = 0; Do[ab2 = abQ[n]; If[ab1 && ab2, AppendTo[s, n-1]]; ab1 = ab2, {n, 2, 10^5}]; s

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

Amiram Eldar and Giovanni Resta, Jan 18 2020

nonn

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.

			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).

Giovanni Resta, Table of n, a(n) for n = 1..1150 (terms below 10^13)

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

Amiram Eldar, Sep 30 2019

nonn

			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).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048146, A064597, A094889, A307823.

Cf. A096399, A292704, A318167, A327635.

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

Amiram Eldar, Apr 11 2020

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Amiram Eldar, Oct 06 2022

nonn

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Amiram Eldar, Dec 23 2020

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A308053.

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

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

Amiram Eldar, Oct 09 2021

nonn

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

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

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

Amiram Eldar, Oct 25 2021

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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] &]

A358484 Numbers k such that k, k+1 and k+2 are all bi-unitary abundant numbers (A292982).

Original entry on oeis.org

268005374, 600350750, 2666847104, 2683146464, 2695309694, 2849458688, 3904592768, 4112553248, 5368737374, 6554410784, 6955574624, 8207456894, 8967010688, 9220179968, 9868366430, 10529171288, 12147283070, 12411630944, 12491149670, 13911605630, 14126720894, 14396391008

Offset: 1

Amiram Eldar, Nov 18 2022

nonn

			268005374 is in the sequence since 268005374, 268005375 and 268005376 are all bi-unitary abundant numbers (A292982): bsigma(268005374) = 568995840 > 2 * 268005374, bsigma(268005375) = 540633600 > 2 * 268005375, and bsigma(268005376) = 541258200 > 2 * 268005376 (bsigma is the sum of bi-unitary divisors, A188999).

Amiram Eldar, Table of n, a(n) for n = 1..1546 (terms below 10^12)

Subsequence of A096536, A292982 and A318167.

Cf. A188999.

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

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

Amiram Eldar, Feb 08 2025

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

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

Amiram Eldar, Table of n, a(n) for n = 1..32

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

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]

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, ", "))));

cp's OEIS Frontend

A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

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

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

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

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

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A339937 Numbers k such that k and k+1 are both coreful abundant numbers (A308053).

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

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

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A358484 Numbers k such that k, k+1 and k+2 are all bi-unitary abundant numbers (A292982).

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