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.

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A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 4, 7, 5, 7, 9, 12, 7, 11, 11, 16, 11, 19, 16, 21, 15, 29, 11, 12, 26, 30, 15, 31, 38, 22, 21, 47, 26, 29, 52, 45, 36, 57, 26, 64, 19, 30, 52, 77, 52, 36, 57, 98, 21, 67, 38, 74, 97, 66, 105, 47, 42, 36, 109, 118, 98, 92, 109, 52, 171, 30
Offset: 1

Views

Author

Gus Wiseman, Jan 10 2020

Keywords

Comments

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. Factorizations are counted by A001055.
The unsorted prime signature of A055932(n) is given by row n of A124829.

Examples

			The a(1) = 1 through a(11) = 7 factorizations:
  {}  2  4    6    8      12     16       18     24       30     32
         2*2  2*3  2*4    2*6    2*8      2*9    3*8      5*6    4*8
                   2*2*2  3*4    4*4      3*6    4*6      2*15   2*16
                          2*2*3  2*2*4    2*3*3  2*12     3*10   2*2*8
                                 2*2*2*2         2*2*6    2*3*5  2*4*4
                                                 2*3*4           2*2*2*4
                                                 2*2*2*3         2*2*2*2*2

Crossrefs

The sorted-signature version is A050322.
This sequence has range A045782.
Factorizations are A001055.
Cf. A025487, A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length@*facs/@First/@GatherBy[Range[1500],If[#==1,{},Last/@FactorInteger[#]]&]

Formula

a(n) = A001055(A055932(n)).

A382325 Numbers with a record ratio of proper factorizations to nontrivial divisors.

Original entry on oeis.org

4, 16, 32, 64, 128, 192, 256, 384, 512, 576, 768, 864, 1024, 1152, 1536, 1728, 2304, 3456, 4608, 5184, 5760, 6912, 8640, 9216, 10368, 11520, 13824, 17280, 20736, 23040, 25920, 27648, 34560, 41472, 51840, 62208, 69120, 82944, 103680, 138240, 165888, 172800
Offset: 1

Views

Author

Charles L. Hohn, Mar 21 2025

Keywords

Comments

Numbers k that give a record value for A028422(k)/A070824(k).
a(n) = 0 (mod 4), and with prime factors of terms clustering around the smallest primes, it is observed that as n increases, the gcd of a(n)..a(oo) remains among the largest divisors of a(n).

Examples

			a(1)=4: |{{2, 2}}| / |{2}| = 1/1.
a(2)=16: |{{2, 2, 2, 2}, {2, 2, 4}, {2, 8}, {4, 4}}| / |{2, 4, 8}| = 4/3.
a(3)=32: |{{2, 2, 2, 2, 2}, {2, 2, 2, 4}, {2, 2, 8}, {2, 4, 4}, {2, 16}, {4, 8}}| / |{2, 4, 8, 16}| = 6/4.

Crossrefs

Cf. A028422, A070824, A033833, A002182, A382326, A382327.
Subsequence of A025487.

Programs

  • PARI
    f_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dmx, mx=m; print1(x, ", ")))

A382327 Numbers with a record ratio of proper factorizations to prime factors (counted with multiplicity).

Original entry on oeis.org

4, 8, 12, 24, 36, 48, 60, 72, 120, 144, 180, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 25200, 30240, 40320, 50400, 60480, 80640, 90720, 100800, 120960, 151200, 181440, 201600
Offset: 1

Views

Author

Charles L. Hohn, Mar 21 2025

Keywords

Comments

Numbers k that give a record value for A028422(k)/A001222(k).
a(n) = 0 (mod 4), and with prime factors of terms clustering around the smallest primes, it is observed that as n increases, the gcd of a(n)..a(oo) remains among the largest divisors of a(n).

Examples

			a(1) = 4: |{{2, 2}}| / |{2, 2}| = 1/2.
a(2) = 8: |{{2, 2, 2}, {2, 4}}| / |{2, 2, 2}| = 2/3.
a(3) = 12: |{{2, 2, 3}, {2, 6}, {3, 4}}| / |{2, 2, 3}| = 3/3.

Crossrefs

Cf. A028422, A001222, A033833, A382325, A382326.
Subsequence of A025487.

Programs

  • PARI
    f_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dmx, mx=m; print1(x, ", ")))

A291928 Positions of records in A218320.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3360, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 60480, 75600, 90720, 100800, 110880, 120960
Offset: 1

Views

Author

Michael De Vlieger, Sep 06 2017

Keywords

Comments

Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases.
A291834(13) = 192 is the smallest term not in a(n).
Subsequence of A025487.

Links

Crossrefs

Cf. A002182, A007416, A025487, A033833, A218320.

Programs

  • Mathematica
    f[n_, i_, t_] := f[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; With[{s = Array[f[#, #, 4] &, 10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Sep 06 2017, after Alois P. Heinz at A218320 *)

A291927 Records transform of A218320.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 15, 16, 20, 25, 27, 33, 36, 46, 50, 53, 77, 86, 118, 145, 158, 173, 174, 184, 224, 270, 282, 304, 330, 422, 522, 625, 656, 820, 881, 899, 1030, 1218, 1276, 1416, 1529, 1590, 1722, 2012, 2106, 2161, 2369, 2478, 2994, 3132, 3361, 3484
Offset: 1

Views

Author

Michael De Vlieger, Sep 06 2017

Keywords

Examples

			A218320(n) for 1 <= n <= 24 is {1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7}; the records are {1, 2, 3, 4, 5, 7}, thus these are the first 6 terms of this sequence.

Crossrefs

Cf. A033833, A218320.

Programs

  • Mathematica
    f[n_, i_, t_] := f[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Union@ FoldList[Max, Array[f[#, #, 4] &, 10^5]] (* Michael De Vlieger, Sep 06 2017, after Alois P. Heinz at A218320 *)

A379926 Numbers with a record number of proper factorizations for which the sum of the squares of the factors is a square.

Original entry on oeis.org

1, 12, 48, 108, 240, 864, 1152, 6912, 23040, 34560, 43200, 55296, 57600, 103680, 138240, 241920, 311040, 414720, 552960, 645120, 691200, 829440, 907200, 967680, 1209600, 1814400, 2177280, 2903040, 3628800, 4838400, 7257600, 8709120, 10886400, 14515200, 19353600
Offset: 1

Views

Author

Charles L. Hohn, Jan 06 2025

Keywords

Comments

Also, numbers with a record number of proper factorizations that form the base lengths of Pythagorean hyperrectangles.
Though total factorization counts can serve as a rough predictor of Pythagorean counts, this sequence has significant non-overlap with A033833 (record total proper factorizations).

Examples

			a(1) = 1, 0 examples.
a(2) = 12, 1 example: {3, 4} (3 * 4 = 12 and 3^2 + 4^2 = 5^2; {2, 6} is not counted as 2^2 + 6^2 = 40 is not a perfect square).
a(3) = 48, 2 examples: {2, 2, 2, 2, 3} (2 * 2 * 2 * 2 * 3 = 48 and 2^2 + 2^2 + 2^2 + 2^2 + 3^2 = 5^2), {6, 8}.
a(4) = 108, 3 examples: {3, 6, 6}, {9, 12}, {2, 6, 9}.
a(5) = 240, 4 examples: {2, 2, 2, 3, 10}, {2, 2, 6, 10}, {2, 4, 5, 6}, {10, 24}.
a(6) = 864, 7 examples: {3, 12, 24}, {3, 8, 36}, {2, 3, 6, 24}, {2, 12, 36}, {6, 12, 12}, {4, 12, 18}, {8, 9, 12}.

Crossrefs

Cf. A033833.

Programs

  • PARI
    a379926_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(#f && issquare(sum(i=1, #f, f[i]^2)), 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dcmax, cmax=c; print(x)))

Extensions

a(1) = 1 from David A. Corneth, Mar 12 2025
Deleted an incorrect assertion and a misleading comment. - N. J. A. Sloane, Mar 14 2025
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