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 1-9 of 9 results.

A066874 Number of partitions of n into unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 8, 2, 2, 2, 11, 2, 12, 2, 14, 14, 2, 2, 17, 2, 17, 18, 20, 2, 20, 2, 23, 2, 22, 2, 742, 2, 2, 26, 29, 26, 27, 2, 32, 30, 29, 2, 1654, 2, 32, 32, 38, 2, 36, 2, 41, 38, 37, 2, 44, 38, 38, 42, 47, 2, 3004, 2, 50, 42, 2, 44, 5257, 2, 47, 50, 5066, 2, 47, 2, 59, 54, 52, 50
Offset: 1

Views

Author

Naohiro Nomoto, Jan 26 2002

Keywords

Examples

			a(12) = 12 because the unitary divisors of 12 are 1, 3, 4 and 12; and the partitions are 12, 4+4+4, 4+4+3+1, 4+4+(4x1), 4+3+3+1+1, 4+3+(5x1), 4+(8x1), 3+3+3+3, 3+3+3+1+1+1, 3+3+(6x1), 3+(9x1) and 12x1.

Links

Crossrefs

Cf. A018818, A034444, A077610, A285614, A286852.

Programs

  • PARI
    unitary_divisors(n) = select(d -> (1==gcd(d,n/d)), divisors(n));
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A066874(n) = partitions_into(n,vecsort(unitary_divisors(n), , 4)); \\ Antti Karttunen, Aug 06 2018
  • PARI
    See Corneth link. \\ David A. Corneth, Aug 12 2018

Extensions

More terms from David Wasserman, Nov 21 2002

A292983 Bi-unitary highly abundant numbers: numbers n such that bsigma(n) > bsigma(m) for all m < n, where bsigma is the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 21, 22, 24, 30, 40, 42, 48, 54, 66, 72, 78, 88, 96, 120, 160, 168, 210, 216, 240, 264, 312, 330, 360, 378, 384, 408, 456, 480, 600, 648, 672, 840, 1056, 1080, 1320, 1512, 1560, 1680, 1848, 1920, 2040, 2184, 2280, 2640
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2017

Keywords

Comments

Analogous to highly abundant numbers (A002093) with bi-unitary sigma (A188999) instead of sigma (A000203).

Links

Crossrefs

Cf. A002093, A188999, A285614.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; bmax = 0; Do[b = bsigma[n]; If[b > bmax, AppendTo[a, n]; bmax = b], {n, 3000}]; a (* after Michael De Vlieger at A188999 *)

A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 21, 22, 24, 30, 40, 42, 54, 66, 72, 78, 88, 96, 102, 114, 120, 168, 210, 216, 264, 312, 330, 360, 378, 384, 408, 456, 480, 510, 546, 552, 600, 672, 690, 696, 744, 840, 1080, 1320, 1512, 1560, 1848, 1920, 2040, 2184, 2280, 2688
Offset: 1

Views

Author

Amiram Eldar, Sep 20 2019

Keywords

Comments

The infinitary version of A002093.

Examples

			The first 10 values of isigma(k) for k = 1 to 10 are: 1, 3, 4, 5, 6, 12, 8, 15, 10, 18. Record values are reached for all these values of k except for 7 and 9, therefore the sequence begins with 1, 2, 3, 4, 5, 6, 8, 10, ...

Links

Crossrefs

Cf. A002093, A037992, A049417, A285614, A292983.

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); seq = {};sm = 0; Do[s = isigma[n]; If[s > sm, sm = s; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A328134 Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 12, 16, 18, 20, 28, 36, 52, 60, 68, 72, 84, 92, 100, 124, 132, 140, 144, 180, 244, 252, 300, 324, 360, 396, 468, 588, 612, 684, 828, 900, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988
Offset: 1

Views

Author

Amiram Eldar, Oct 04 2019

Keywords

Comments

The exponential version of A002093.

Examples

			The first 10 values of esigma(k) for k = 1 to 10 are 1, 2, 3, 6, 5, 6, 7, 10, 12, 10. The record values are reached for 1, 2, 3, 4, 7, 8, 9.

Links

Crossrefs

Cf. A002093, A051377, A285614 (unitary), A292983 (bi-unitary), A327634 (infinitary).

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; em = 0; Do[e = esigma[n]; If[e > em, em = e; AppendTo[s, n]], {n, 1, 3000}]; 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

A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

Original entry on oeis.org

1, 6, 10, 12, 18, 24, 30, 42, 48, 54, 60, 78, 84, 90, 96, 120, 168, 192, 210, 240, 270, 312, 330, 360, 384, 420, 480, 630, 672, 840, 960, 1056, 1080, 1248, 1320, 1440, 1560, 1680, 1890, 1920, 2280, 2310, 2400, 2520, 2640, 2688, 3000, 3120, 3240, 3360, 4200, 4320
Offset: 1

Views

Author

Amiram Eldar, Oct 26 2021

Keywords

Comments

The corresponding record values are 1, 6, 8, 10, 15, 30, 42, 54, 58, 60, 78, ... (see the link for more values).

Examples

			The first 6 values of nesigma(k), for k = 1 to 6 are 1, 1, 1, 1, 1 and 6. The record values, 1 and 6, occur at 1 and 6, the first 2 terms of this sequence.

Links

Crossrefs

Cf. A160135, A348604.
The nonexponential version of A002093.
Similar sequences: A285614, A292983, A327634, A328134, A329883, A348272.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; sm = -1; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A349112 Powerful highly abundant numbers: numbers m such that psigma(m) > psigma(k) for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 72, 108, 128, 144, 200, 216, 256, 288, 392, 400, 432, 576, 648, 800, 864, 1152, 1296, 1728, 1944, 2304, 2592, 3456, 3888, 5184, 6912, 7776, 10000, 10368, 11664, 13824, 15552, 20000, 20736, 23328, 27000, 27648, 31104, 34992, 40000, 41472
Offset: 1

Views

Author

Amiram Eldar, Nov 08 2021

Keywords

Comments

The corresponding record values are 1, 5, 13, 29, 37, 61, 125, 130, 185, 253, ...

Examples

			The first 8 terms of A183097 are 1, 1, 1, 5, 1, 1, 1 and 13. The record values, 1, 5 and 13, occur at 1, 4 and 8, the first 3 terms of this sequence.

Links

Crossrefs

A349111 is a subsequence.
Cf. A005934, A183097.
Similar sequences: A285614, A292983, A327634, A328134, A329883, A348272.

Programs

  • Mathematica
    f[p_,e_] := (p^(e+1)-1)/(p-1) - p; s[1] = 1; s[n_] := Times @@ f @@@FactorInteger[n]; seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A290490 Numbers k such that (sum of proper unitary divisors of k) > (sum of proper unitary divisors of m) for all m < k.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 102, 114, 138, 150, 174, 186, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6510, 7410, 7590, 7770
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 03 2017

Keywords

Comments

Numbers k such that A034460(k) > A034460(m) for all m < k.

Links

Crossrefs

Cf. A002182, A002093, A034090, A034448, A034460, A063919, A280013, A281782, A285614.

Programs

  • Mathematica
    mx = -1; t = {}; Do[u = DivisorSum[n, # &, GCD[#, n/#] == 1 &] - n; If[u > mx, mx = u; AppendTo[t, n]], {n, 8000}]; t
  • PARI
    sumud(n) = sumdiv(n, d, if (gcd(d, n/d)==1, d)) - n;
lista(nn) = {lasts = -1; for (n=1, nn, if ((news = sumud(n)) > lasts, print1(n, ", "); lasts = news););} \\ Michel Marcus, Aug 04 2017

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 12, 16, 24, 32, 36, 48, 56, 64, 72, 96, 108, 128, 144, 192, 200, 216, 288, 360, 400, 432, 504, 576, 648, 720, 792, 800, 864, 1008, 1080, 1152, 1296, 1440, 1512, 1584, 1728, 1800, 1944, 2016, 2160, 2304, 2592, 2880, 3024, 3240, 3456, 3600
Offset: 1

Views

Author

Amiram Eldar, Dec 28 2020

Keywords

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
Analogous to highly abundant numbers (A002093) with the sum of the coreful divisors function (A057723) instead of the sum of divisors function (A000203).

Examples

			The first 10 values of A057723(n) for n=1..10 are: 1, 2, 3, 6, 5, 6, 7, 14, 12, 10. The record values, 1, 2, 3, 6, 7 and 14 occur at 1, 2, 3, 4, 7 and 8, the first 6 terms of this sequence.

Links

Crossrefs

Cf. A007947, A057723, A283052, A308053.
Similar sequences: A002093, A285614, A292983, A327634, A328134, A329883.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 3600}]; seq
Showing 1-9 of 9 results.

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