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 1-4 of 4 results.

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

Original entry on oeis.org

24, 30, 42, 48, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 150, 156, 160, 162, 168, 174, 180, 186, 192, 198, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 300, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390
Offset: 1

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Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

The smallest odd term is a(1357) = 8505.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 13, 148, 1595, 15688, 158068, 1578957, 15762209, 157745113, 1577808429, ... Apparently this sequence has an asymptotic density 0.157...

Examples

			24 is a term since A160135(24) = 30 > 24.

Links

Crossrefs

Cf. A160135, A348601.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[400], q]

A348602 Smaller member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

Original entry on oeis.org

198, 18180, 142310, 1077890, 1156870, 1511930, 1669910, 2236570, 2728726, 3776580, 4246130, 4532710, 5123090, 5385310, 6993610, 7288930, 8619765, 8754130, 8826070, 9478910, 10254970, 14426230, 17041010, 17257695, 21448630, 30724694, 34256222, 35361326, 37784810
Offset: 1

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

The larger counterparts are in A348603.

Examples

			198 is a term since A160135(198) = 204 and A160135(204) = 198.

Crossrefs

Cf. A160135, A348601, A348603.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 1.7*10^6}]; seq

A348603 Larger member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

Original entry on oeis.org

204, 19332, 168730, 1099390, 1292570, 1598470, 2062570, 2429030, 3077354, 3903012, 4488910, 6135962, 5504110, 5812130, 7158710, 8221598, 9627915, 10893230, 10043690, 11049730, 10273670, 18087818, 19150222, 17578785, 23030090, 32174506, 35997346, 40117714, 39944086
Offset: 1

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

The terms are ordered according to their smaller counterparts (A348602).

Examples

			204 is a term since A160135(204) = 198 and A160135(198) = 204.

Crossrefs

Cf. A160135, A348601, A348602.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 1, 1.7*10^6}]; seq

A353901 Numbers k such that A353900(k) is divisible by k.

Original entry on oeis.org

1, 6, 28, 56, 1104, 2208, 2178540, 4357080, 6499584, 12999168
Offset: 1

Views

Author

Amiram Eldar, May 10 2022

Keywords

Comments

a(11) > 8*10^10, if it exists.
The corresponding ratios A353900(k)/k are 1, 2, 2, 1, 2, 1, 4, 2, 2, 1, ...

Examples

			6 is a term since A353900(6) = 12 is divisible by 6.
56 is a term since A353900(56) = 56 is divisible by 56.

Crossrefs

Cf. A353900.
Similar sequences: A007691, A189000, A327158, A348601.

Programs

  • Mathematica
    f[p_, e_] := 1 + Sum[p^(2^k), {k, 0, Floor[Log2[e]]}]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[6.5*10^6], Divisible[s[#], #] &]
Showing 1-4 of 4 results.

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

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