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 1-6 of 6 results.

A130839 Sequence related to factorizations and prime signatures: a(1) = 1; for n>1,a(n) = A108464(n) - 2*A050322(n).

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 2, 7, 5, 5, 8, 14, 14, 8, 20, 26, 32, 15, 40, 40, 45, 22, 47, 65, 23, 77, 94, 75, 63, 98, 122, 37, 135, 196, 120, 148, 188, 117, 216, 55, 226, 231, 187, 206, 377, 187, 310, 338, 286, 366, 83, 483, 375, 99, 454, 405, 683, 284, 598
Offset: 1

Views

Author

Alford Arnold, Jul 19 2007

Keywords

Crossrefs

Cf. A131802 A050322.

A057567 Number of partitions of n where the product of parts divides n.

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 2, 7, 4, 5, 2, 11, 2, 5, 5, 12, 2, 11, 2, 11, 5, 5, 2, 21, 4, 5, 7, 11, 2, 15, 2, 19, 5, 5, 5, 26, 2, 5, 5, 21, 2, 15, 2, 11, 11, 5, 2, 38, 4, 11, 5, 11, 2, 21, 5, 21, 5, 5, 2, 36, 2, 5, 11, 30, 5, 15, 2, 11, 5, 15, 2, 52, 2, 5, 11, 11, 5, 15, 2, 38, 12, 5, 2, 36, 5, 5, 5, 21
Offset: 1

Views

Author

Leroy Quet, Oct 04 2000

Keywords

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 03 2005

Examples

			From _Gus Wiseman_, Jul 04 2019: (Start)
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326155.
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (321)     (1111111)  (4211)
                    (211)            (3111)               (22211)
                    (1111)           (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)
(End)

Links

Crossrefs

Cf. A000070, A000110, A001055 (Mobius transform), A002110, A025487, A057568, A108464, A113309, A131802.
Any prime numbered column of array A108461.
Cf. A028422, A096276, A114324, A318950, A319000, A319005, A326152, A326155.

Programs

  • Mathematica
    Table[Function[m, Count[Map[Times @@ # &, IntegerPartitions[m]], P_ /; Divisible[m, P]] - Boole[n == 1]]@ Apply[Times, #] &@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, 88}] (* Michael De Vlieger, Aug 16 2017 *)
  • PARI
    fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s}
A001055(n) = fcnt(n, n) \\ This function from Michael B. Porter, Oct 29 2009
A057567(n) = sumdiv(n, d, A001055(d)); \\ After Jovovic's formula. Antti Karttunen, May 25 2017
  • Python
    from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n <= m else 0
    A = (d for d in divisors(n) if 1 < d < n and d <= m)
    s = sum(T(n // d, d) for d in A)
    return s + 1 if n <= m else s
def a001055(n): return T(n, n)
def a(n): return sum(a001055(d) for d in divisors(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 19 2017

Formula

a(n) = Sum_{d|n} A001055(d). - Vladeta Jovovic, Nov 19 2000
a(A025487(n)) = A108464(n).
a(p^k) = A000070(k).
a(A002110(n)) = A000110(n+1).
Dirichlet g.f.: zeta(s) * Product_{k>=2} 1/(1 - 1/k^s). - Ilya Gutkovskiy, Nov 03 2020

Extensions

More terms from James Sellers, Oct 09 2000
More terms from Vladeta Jovovic, Nov 19 2000

A131802 Sequence related to factorizations and prime signatures: a(1) = 1; for n>1, a(n) = A057567(n) - 2*A001055(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 7, 0, 1, 1, 3, 0, 5, 0, 5, 1, 1, 1, 8, 0, 1, 1, 7, 0, 5, 0, 3, 3, 1, 0, 14, 0, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 14, 0, 1, 3, 8, 1, 5, 0, 3, 1, 5, 0, 20, 0, 1, 3, 3, 1, 5, 0, 14, 2, 1, 0, 14, 1, 1, 1, 7
Offset: 1

Views

Author

Alford Arnold, Jul 18 2007

Keywords

Examples

			A001055(12) = 4 and A057567(12) = 11 so a(12) = 11 - 2*4 = 3

Links

Crossrefs

Cf. A001055, A025487, A057567, A077569, A108464, A122819.

Programs

A131419 Central diagonal and left half of table A054225 which counts the partitions of n objects of 2 colors.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 7, 12, 16, 11, 19, 29, 31, 15, 30, 47, 57, 22, 45, 77, 97, 109, 30, 67, 118, 162, 189, 42, 97, 181, 257, 323, 339, 56, 139, 267, 401, 522, 589, 77, 195, 392, 608, 831, 975, 1043, 101, 272, 560, 907, 1279, 1576, 1752, 135, 373
Offset: 0

Views

Author

Alford Arnold, Jul 11 2007

Keywords

Comments

Every diagonal appears in sequence A129306.

Examples

			Table A054225 begins
1
1...1
2...2...2
3...4...4...3
5...7...9...7...5
7..12..16..16..12..7
The diagonals in the new table begin
1..1..2..3..5...7...11...15...22...30..42 A000041
.....2..4..7..12...19...30...45...67..97 A000070
........9.16..29...47...77..118..181
..........31..57...97..162..257
.............109..189..323
..................339

Crossrefs

Cf. A108464, A129306.

Extensions

More terms from R. J. Mathar, Sep 02 2007

A131420 A tabular sequence of arrays counting ordered factorizations over least prime signatures. The unordered version is described by sequence A129306.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 26, 16, 48, 132, 308, 541, 76, 176, 32, 112, 368, 1076, 2612, 4683, 208, 604, 1460, 252, 818, 64, 256, 876, 3408, 10404, 25988, 47293, 544, 1888, 5740, 14300, 768, 2316, 3172, 7880, 128, 576, 2496, 10096, 36848, 116180
Offset: 1

Views

Author

Alford Arnold, Jul 10 2007

Keywords

Comments

The display has 1 2 3 5 7 11 15 ... terms per column. (cf. A000041)
The arrays begin
1.....2.....4......8......16.....32.....64......128
......3.....8.....20......48....112....256......576
...........13.....44.....132....368....976.....2496
..................75.....308...1076...3408....10096
.........................541...2612..10404....36848
...............................4683..25988...116180
.....................................47293...296564
.............................................545835
..................26......76....208....544
.........................176....604...1888
...............................1460...5740
.....................................14300
................................252....768
......................................2316
................................818...3172
......................................7880
with column sums
1....5....25....173....1297....12225....124997 => A035341
Column i corresponds to partitions of i. The rows correspond successively to the partitions {i}, {i-1,1},{i-2,1,1},{i-3,1,1,1}, ..., {i-7,1,1,1,1,1,1,1}, {i-2,2}, {i-3,2,1}, {i-4,2,1,1}, {i-5,2,1,1,1}, {i-3,3}, {i-3,3,1}, {i-4,2,2}, {i-5,2,2,1}. - Roger Lipsett, Feb 26 2016

Examples

			36 = 2*2*3*3 and is in A025487. There are 26 ways to factor 36 so a(11) = 26.

Crossrefs

Cf. A000041, A000670, A002033, A025487, A035098, A035341, A050324, A074206, A095705, A098348, A104725, A108464, A129306.

Programs

  • Mathematica
    gozinta counts ordered factorizations of an integer, and if lst is a partition we have
gozinta[1] = 1;
gozinta[n_] := gozinta[n] = 1 + Sum[gozinta[n/i], {i, Rest@Most@Divisors@n}]
a[lst_] := gozinta[Times @@ (Array[Prime, Length@lst]^lst)] (* Roger Lipsett, Feb 26 2016 *)

Extensions

Corrected entries in table in comments section - Roger Lipsett, Feb 26 2016

A131997 Array read by rows in which the n-th row contains odd numbers of all possible prime signatures with n divisors.

Original entry on oeis.org

1, 3, 9, 27, 15, 81, 243, 45, 729, 2187, 135, 105, 6561, 225, 19683, 405, 59049, 177147, 1215, 315, 675
Offset: 1

Views

Author

Alford Arnold, Aug 18 2007

Keywords

Examples

			Row 12 of A131886 is 2048 96 72 60 so here row 12 is 177147 1215 315 675.

Crossrefs

Cf. A001055, A025487, A050322, A057567, A077569, A108464, A122819, A130839, A131802, A131886.
Showing 1-6 of 6 results.

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