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 14 results. Next

A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

Original entry on oeis.org

36, 48, 80, 144, 180, 240, 252, 288, 300, 324, 336, 396, 400, 432, 468, 528, 560, 576, 588, 612, 624, 684, 720, 768, 784, 816, 828, 880, 900, 912, 960, 1008, 1040, 1044, 1104, 1116, 1200, 1232, 1260, 1280, 1296, 1332, 1360, 1392, 1440, 1456, 1476, 1488, 1520, 1548
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Comments

The first odd term is a(3577) = 99225.
The number of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 3, 31, 360, 3605, 36160, 360840, 3618980, 36144059, ... Apparently this sequence has an asymptotic density 0.0361...

Examples

			36 is a term since A348271(36) = 41 > 36.

Links

Crossrefs

Cf. A348271, A327633.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1500], s[#] > # &]

A348272 Noninfinitary highly abundant numbers: numbers m such that nisigma(m) > nisigma(k) for all k < m, where nisigma(k) is the sum of noninfinitary divisors of n (A348271).

Original entry on oeis.org

1, 4, 9, 12, 16, 28, 36, 48, 80, 100, 112, 144, 180, 240, 300, 324, 336, 396, 400, 432, 468, 528, 576, 684, 720, 900, 1008, 1200, 1296, 1584, 1872, 2160, 2268, 2304, 2448, 2736, 2880, 3312, 3600, 5040, 6300, 6480, 7056, 7920, 9072, 9360, 10800, 11088, 11520, 12240
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Comments

The corresponding record values are 0, 2, 3, 8, 14, 16, 41, 56, 84, 87, 112, ...

Examples

			The first 9 values of A348271(k) for k = 1 to 9 are: 0, 0, 0, 2, 0, 0, 0, 0 and 3. The record values, 0, 2 and 3, occur at 1, 4 and 9, the first 3 terms of this sequence.

Links

Crossrefs

Cf. A348271.
The noninfinitary version of A002093.
Similar sequences: A285614, A292983, A327634, A328134, A329883.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq={}; sm = -1; Do[s1 = s[n];If[s1 > sm, sm= s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A348343 Smaller member of a noninfinitary amicable pair: numbers (k, m) such that nisigma(k) = m and nisigma(m) = k, where nisigma(k) is the sum of the noninfinitary divisors of k (A348271).

Original entry on oeis.org

336, 1792, 5376, 6096, 21504, 32004, 97536, 34062336, 64512000, 118008576, 30064771072
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

The larger counterparts are in A348344.

Examples

			336 is a term since A348271(336) = 448 and A348271(448) = 336.

Crossrefs

Cf. A327633, A348271, A348344.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, n]], {n,1,10^4}]; seq

A348344 Larger member of a noninfinitary amicable pair: numbers (k, m) such that nisigma(k) = m and nisigma(m) = k, where nisigma(k) is the sum of the noninfinitary divisors of k (A348271).

Original entry on oeis.org

448, 2032, 8128, 7168, 24384, 41984, 130048, 41940480, 102222432, 221316608, 34359738352
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

The terms are ordered according to their smaller counterparts (A348343).

Examples

			448 is a term since A348271(448) = 336 and A348271(336) = 448.

Crossrefs

Cf. A327633, A348271, A348343.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, m]], {n,1,10^4}]; seq

A348346 Numbers k such that k and k+1 have the same positive sum of noninfinitary divisors (A348271).

Original entry on oeis.org

20150, 52767, 99296, 835515, 1241504, 2199392, 6294015, 11158496, 12770450, 17016416, 19127907, 20128544, 23686748, 24790688, 26580554, 33366015, 34385247, 39687651, 42106976, 44157087, 45466676, 59825349, 60832449, 73780244, 75268775, 81654650, 84696849, 111457213
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

Numbers k such that A348271(k) = A348271(k+1) > 0.
The terms are restricted to have a positive sum of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

Examples

			20150 is a term since A348271(20150) = A348271(20151) = 6720.

Links

Crossrefs

Cf. A036537, A348271, A348345.
Subsequence of A162643.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[10^5], (s1 = s[#]) > 0 && s1 == s[# + 1] &]

A348273 Noninfinitary superabundant numbers: numbers m such that nisigma(m)/m > nisigma(k)/k for all k < m, where nisigma(m) is the sum of noninfinitary divisors of m (A348271).

Original entry on oeis.org

1, 4, 12, 16, 36, 48, 144, 720, 3600, 25200, 176400, 226800, 1587600, 1940400, 2494800, 17463600, 32432400, 192099600, 227026800, 2497294800, 3632428800, 32464832400, 39956716800
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Comments

The least term k with A348271(k)/k > m for m = 1, 2, 3, .... is 36, 3600, 1587600, ...

Crossrefs

Cf. A348271.
Subsequence of A348272.
The noninfinitary version of A004394.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742 (exponential), A292984, A329882.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq = {}; rm = -1; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]],{n, 1, 10^6}]; seq

A348521 Numbers k such that A348271(k) > 2*k.

Original entry on oeis.org

3600, 5040, 6480, 7056, 7920, 9072, 9360, 11088, 11520, 12240, 13680, 14400, 16128, 16560, 18000, 20880, 22320, 25200, 32400, 35280, 39600, 44100, 45360, 46800, 55440, 56700, 57600, 58320, 58800, 61200, 63504, 65520, 68400, 69300, 71280, 75600, 77616, 79380, 80640
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Comments

Odd terms exist (e.g., 349476304574870948475). What is the smallest odd term?

Examples

			3600 is a term since the sum of the noninfinitary divisors of 3600 is A348271(3600) = 8073 > 2*3600 = 7200.

Links

Crossrefs

Cf. A348271, A348274.
Similar sequence: A063846.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[10^5], s[#] > 2*# &]

A348341 a(n) is the number of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Examples

			a(4) = 1 since 4 has one noninfinitary divisor, 2.

Links

Crossrefs

Cf. A000005, A036537, A037445, A077609, A162643, A327576, A348271.
Similar sequences: A034444, A048105, A286324, A049419.

Programs

  • Mathematica
    a[1] = 0; a[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Array[a, 100]
  • PARI
    A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2]))); \\ Antti Karttunen, Oct 13 2021

Formula

a(n) = A000005(n) - A037445(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).
Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (1 - 2 * A327576) = 0.266749... . - Amiram Eldar, Dec 09 2022

A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

Original entry on oeis.org

99225, 1091475, 1289925, 1334025, 1576575, 1686825, 1715175, 1863225, 1885275, 2027025, 2061675, 2282175, 2304225, 2395575, 2401245, 2436525, 2480625, 2650725, 2723175, 2789325, 2877525, 2962575, 3031875, 3075975, 3132675, 3185325, 3186225, 3296475, 3353805, 3501225
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Comments

The number of terms below 10^k, for k = 5, 6, ..., are 1, 113, 630, 7771, 73685, ... Apparently this sequence has an asymptotic density 0.000007...

Examples

			99225 is a term since A348271(99225) = 107207 > 99225.

Links

Crossrefs

Cf. A348271.
Subsequence of A005231 and A348274.
Similar sequences: A094889, A127666, A129485, A293186, A321147.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1, 2*10^6, 2], s[#] > # &]

A348525 Noninfinitary weird numbers: noninfinitary abundant numbers (A348274) that are not equal to the sum of any subset of their noninfinitary divisors.

Original entry on oeis.org

3344, 12636, 88900, 95900, 109900, 116900, 121100, 181424, 271472, 365552, 476272, 504016, 975568, 1016048, 1354288, 1375504, 1407824, 1552304, 1628528, 1641904, 1862608, 1882672, 1902736, 1909424, 1929488, 1962928, 1982992, 2003056, 2009744, 2029808, 2049872
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Examples

			3344 is a term since the sum of its noninfinitary divisors, {2, 4, 8, 22, 38, 44, 76, 88, 152, 418, 836, 1672}, is 3360 > 3344, and no subset of these divisors sums to 3344.

Links

Crossrefs

Cf. A348271, A348274, A348341.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948.

Programs

  • Mathematica
    q[n_] := !IntegerQ@ Log2@ DivisorSigma[0, n]; nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; s = {}; Do[If[! q[n], Continue[]]; d = nidiv[n]; If[Total[d] <= n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 0, AppendTo[s, n]], {n, 1, 13000}]; s

Extensions

More terms from Amiram Eldar, Mar 25 2023
Showing 1-10 of 14 results. Next

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