A098719 -id:A098719 - cp's OEIS frontend

A052886 Expansion of e.g.f.: (1/2) - (1/2)(5 - 4exp(x))^(1/2).

0, 1, 3, 19, 207, 3211, 64383, 1581259, 45948927, 1541641771, 58645296063, 2494091717899, 117258952478847, 6038838138717931, 338082244882740543, 20443414320405268939, 1327850160592214291967, 92200405122521276427691, 6815359767190023358085823, 534337135055010788022858379

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

From the symmetry present in Vladeta Jovovic's Feb 02 2005 formula, it is easy to see that there are no positive even numbers in this sequence. Question: are there any multiples of 5 after the initial zero? Compare also to the comments in A366884. - Antti Karttunen, Jan 02 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 859.
Index entries for reversions of series

Cf. A000108, A054867, A277120, A293379, A366377, A366884.
Cf. A371316, A371317.
Cf. A371329, A376036, A376037.

spec := [S,{C=Set(Z,1 <= card),S=Prod(B,C),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/2-1/2*(5-4*E^x)^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
a[n_] := Sum[k! StirlingS2[n, k] CatalanNumber[k - 1], {k, 1, n}];
Array[a, 20, 0] (* Peter Luschny, Apr 30 2020 *)

N=66; x='x+O('x^N);
concat([0],Vec(serlaplace(serreverse(log(1+x-x^2)))))
\\ Joerg Arndt, May 25 2011

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = 1+ sum(k=1, n-1, binomial(n,k)*va[k]*va[n-k]);); concat(0, va);} \\ Michel Marcus, Apr 30 2020

A000108(n) = binomial(2*n, n)/(n+1);
A052886(n) = sum(k=1,n,k!*stirling(n,k,2)*A000108(k-1)); \\ Antti Karttunen, Jan 02 2024

E.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).
a(n) = 1 + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - Vladeta Jovovic, Feb 02 2005
a(n) = Sum_{k=1..n} k!*Stirling2(n,k)*C(k-1), where C(k) = Catalan numbers (A000108). - Vladimir Kruchinin, Sep 15 2010
a(n) ~ sqrt(5/2)/2 * n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
Equals the logarithmic derivative of A293379. - Paul D. Hanna, Oct 22 2017
O.g.f.: Sum_{k>=1} C(k-1)*Product_{r=1..k} r*x/(1-r*x), where C = A000108. - Petros Hadjicostas, Jun 12 2020
a(n) = A366377(A000040(n)) = A366884(A098719(n)). - Antti Karttunen, Jan 02 2024
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (exp(x) - 1) / (1 - A(x)).
E.g.f.: Series_Reversion( log(1 + x * (1 - x)) ). (End)

New name using e.g.f. by Vaclav Kotesovec, Sep 30 2013

A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 4, 1, 4, 3, 1, 3, 3, 1, 1, 5, 1, 4, 4, 1, 5, 4, 1, 4, 5, 2, 1, 6, 1, 5, 6, 1, 6, 5, 1, 5, 7, 3, 1, 7, 1, 6, 8, 1, 5, 8, 4, 1, 7, 6, 1, 4, 6, 4, 1, 1, 6, 9, 1, 6, 9, 4, 1, 8, 1, 7, 10, 1, 6, 11, 6, 1, 8, 7, 1, 5, 9, 7, 2, 1, 7, 12, 1, 7, 11, 5, 1, 9, 1, 8, 12, 1, 7, 14

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the A025487(n)th row (see below) also gives the number of divisors of the k-th power of A025487(n).
Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.
T(n,0)=1.

			Rows begin:
  1;
  1,1;
  1,2;
  1,2,1;
  1,3;
  1,3,2;
  1,4;
  1,4,3;...
36's 9 divisors include 1 divisor with 0 distinct prime factors (1); 4 with 1 (2, 3, 4 and 9); and 4 with 2 (6, 12, 18 and 36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 4, 4). These are the positive coefficients of the polynomial equation 1 + 4k + 4k^2 = (1 + 2k)(1 + 2k), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of distinct prime factors of n, see A001221.
Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146289. Also cf. A146291, A146292.

If A025487(n)'s canonical factorization into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + ek).

A306802 Position of highly composite numbers in the sequence of products of primorials.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 12, 13, 17, 20, 24, 27, 34, 36, 43, 47, 55, 67, 77, 84, 95, 102, 107, 112, 129, 133, 138, 154, 166, 183, 198, 211, 220, 245, 252, 261, 264, 294, 314, 348, 369, 390, 406, 446, 457, 476, 500, 533, 555, 582, 634, 652, 676, 726, 756, 822

Offset: 1

Michael De Vlieger, Mar 12 2019

nonn

Indices of A002182 in A025487. All terms in A002182 are products of terms in A002110; A025487 lists products of terms in A002110.

The first 28 terms of this sequence and those of A293635 are identical since the smallest 28 terms of A002182 and A004394 are the same.

			The number 120 is 10th in the sequence of highly composite numbers, since it sets a record for the divisor counting function. The index of this number in A025487 is 17.

Michael De Vlieger, Table of n, a(n) for n = 1..511

Cf. A002182, A025487, A098719, A293635.

Block[{P = Product[Prime@ i, {i, 8}], s, t, u}, s = Array[DivisorSigma[0, #] &, P]; t = Array[If[# == 1, {0}, Sort[FactorInteger[#][[All, -1]], Greater]] &, P]; u = Values[PositionIndex@ t][[All, 1]]; Map[FirstPosition[u, #][[1]] &, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] ]

A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 4, 1, 2, 2, 4, 2, 4, 4, 4, 4, 6, 8, 6, 8, 1, 2, 2, 6, 6, 6, 10, 2, 4, 4, 6, 8, 6, 10, 4, 8, 6, 8, 12, 6, 10, 6, 8, 12, 10, 8, 12, 12, 10, 16, 12, 20, 1, 2, 6, 8, 10, 6, 10, 8, 10, 16, 14, 20, 2, 4, 12, 10, 10, 14, 10, 16, 12, 20, 6, 6, 10, 8, 10, 12, 20, 4, 8, 14, 14, 20, 14, 10, 16, 14, 24, 6, 12, 12

Offset: 1

Antti Karttunen, Feb 27 2019

nonn

A098719 gives the positions of ones in this sequence. See also comments in A324383.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (based on the b-file of A025487 as computed by Will Nicholes and Franklin T. Adams-Watters)
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Cf. A002110, A025487, A098719 (positions of ones), A276150, A324342.

Cf. A324382 for a subsequence, and A324383, A324386 for permutations of this sequence.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324387(n) = A276150(A025487(n));

a(n) = A276150(A025487(n)).

A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 2, 2, 2, 2

Offset: 1

Matthew Vandermast, Nov 11 2008

All rows are palindromic. T(n, 0) = T(n, A036041(n)) = 1.

Every row that appears in A146291 appears exactly once in the table. Rows appear in order of first appearance in A146291.

			Rows begin:
  1;
  1,1;
  1,1,1;
  1,2,1;
  1,1,1,1;
  1,2,2,1;
  1,1,1,1,1;...
36's 9 divisors include 1 divisor with 0 total prime factors (1);, 2 with 1 (2 and 3); 3 with 2 (4, 6 and 9); 2 with 3 (12 and 18); and 1 with 4 (36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 2, 3, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 3k^2 + 2k^3 + (1)k^4 = (1 + k + k^2)(1 + k + k^2), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of prime factors of n counted with multiplicity, see A001222.
Row sums equal A146288(n). T(n, 1) = A061394(n) for n>1.
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146291. Also cf. A146289, A146290.

If A025487(n)'s canonical factorization into prime powers is the product of p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (sum_{i=0..e} k^i).

A366884 Number of branching factorizations of the least integer of each prime signature (A025487).

Original entry on oeis.org

0, 1, 2, 3, 5, 11, 15, 45, 19, 51, 62, 195, 113, 188, 345, 873, 645, 731, 1890, 911, 3989, 207, 2405, 3585, 2950, 10221, 6525, 18483, 1709, 15775, 19569, 12235, 54718, 43545, 86515, 12405, 99215, 9332, 105447, 51822, 55885, 290611, 17753, 120075, 277203, 408105, 83505, 605135, 80565, 562739, 223191, 432975, 1533670

Offset: 1

Antti Karttunen, Jan 02 2024

nonn

Sequence appears to be injective, but can it be proved? This would prove also the conjectures given in A277120 and A366377.

Of the first 21001 terms, there are 701 terms ending with digit "0", 614 with "1", 68 with "2", 570 with "3", 0 with "4", 17795 with "5", 0 with "6", 550 with "7", 67 with "8", and 636 with "9". Why such an overrepresentation (~ 85% of the total) of the terms of form 10k+5? Do any terms of the form 10k+4 or 10k+6 exist? See also the comments in A052886.

Antti Karttunen, Table of n, a(n) for n = 1..21001
Michael De Vlieger, Plot k = a(n) mod 10 at (x,y) = (n mod 144, 1 + floor(n/144)), n = 1..20736, showing 0 in black, 1 in red, 2 in orange, 3 in yellow, 4 = dark green, 5 = bright green, 6 = cyan, 7 = light blue, 8 = dark blue, and 9 in purple.

Cf. A007317, A025487, A025488, A052886, A098719, A181815, A277120.

Permutation of A366377.

a(n) = A277120(A025487(n)).
a(n) = A366377(A181815(n)).
For all n >= 1, a(A025488(n)) = A007317(n), a(A098719(n)) = A052886(n).

A096903 Least integer of each ordered prime signature (A055932) arranged by prime signature (each row starting with least integer of each prime signature, A025487).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 16, 24, 54, 30, 32, 36, 48, 162, 60, 90, 150, 64, 72, 108, 96, 486, 120, 270, 750, 128, 144, 324, 180, 300, 450, 192, 1458, 210, 216, 240, 810, 3750, 256, 288, 972, 360, 540, 600, 1350, 1500, 2250, 384, 4374, 420, 630, 1050, 1470, 432, 648

Offset: 0

Ray Chandler, Aug 01 2004

There are several other sequences closely related to a(n). A066099 and A108244 both list the associated exponents, while A108730 provides an elegant mapping to binary representations. - Alford Arnold, Mar 05 2006

			Sequence begins
1,
2,
4,
6,
8,
12,18,
16,
24,54,
30,
32,
36,
48,162,
60,90,150

Michael De Vlieger, Table of n, a(n) for n = 0..14264 (rows 0 <= n <= 487 = A098719(9) - 1.)
OEIS Wiki, Orderings of ordered prime signatures

Cf. A025487, A055932, A066099, A108244, A108730.

SortBy[#, First] &@ Map[Union@ Map[Times @@ MapIndexed[Prime[First@ #2]^#1 &, #] &, Permutations[#]] &, Map[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #] &, Import["https://oeis.org/A025487/b025487.txt", "Data"][[1 ;; 30, -1]] ] ] // Flatten (* Michael De Vlieger, Feb 06 2020, using b-file from A025487 *)

Edited by Daniel Forgues, Jan 24 2011

A098718 Position of n! in A025487.

Original entry on oeis.org

1, 2, 4, 8, 17, 34, 67, 125, 224, 391, 666, 1108, 1797, 2887, 4552, 7088, 10875, 16495, 24756, 36766, 54084, 78858, 114018, 163558, 232965, 329478, 462996, 646551, 897699, 1239395, 1702142, 2325845, 3162865, 4281304, 5769761, 7742941, 10348857, 13778106, 18275141

Offset: 1

Jeff Burch, Sep 29 2004

nonn

			A025487(34) = 720 = 6! so a(6) = 34. - _David A. Corneth_, Sep 19 2019

Cf. A000142, A025487, A098719.

A025487(a(n)) = n!. - Amiram Eldar, Jun 20 2019

More terms from Amiram Eldar, Jun 20 2019
a(35) corrected by Amiram Eldar, Jul 26 2019
a(36)-a(39) from David A. Corneth, Sep 19 2019

A369137 Inverse permutation to A369136.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 11, 7, 8, 9, 51, 10, 28, 14, 12, 15, 1602, 13, 194, 16, 18, 60, 307, 17, 23, 35, 20, 21, 681, 19

Offset: 1

Pontus von Brömssen, Jan 14 2024

A284456(a(n)) is the smallest number whose prime tower factorization tree has Matula-Göbel number n.

After a(31), the sequence continues 25, 71, 1726, 31, 22, 1304, 221, 44, 24, ?, 26, ?, 79, 27, 343, ?, 29, 47, 33, 1867, 50, ?, 32, 98, 34, 250, 739, ?, 30, ?, ?, 37, 42, 66, 91, ?, 1935, 381, 41, ... .

Index entries for sequences related to Matula-Göbel numbers.

Cf. A007097, A014221, A025488, A098719, A284456, A369015, A369099, A369136, A369138.

A369015(A284456(a(n))) = A369136(a(n)) = n.
A369099(n) = A284456(a(n)).
A369138(a(A007097(n))) = A025488(A014221(n-1)).
A369138(a(2^n)) = A098719(n+1).

A369138 Index of A284456(n) in A025487.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 12, 13, 14, 19, 20, 21, 22, 24, 29, 33, 34, 38, 39, 43, 44, 47, 54, 57, 61, 63, 66, 67, 68, 72, 73, 74, 88, 93, 94, 101, 102, 107, 108, 114, 118, 121, 128, 129, 131, 138, 140, 142, 145, 147, 149, 161, 172, 185, 186, 187, 188, 192, 198

Offset: 1

Pontus von Brömssen, Jan 14 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A007097, A014221, A025487, A025488, A085089, A098719, A284456, A369137.

a(n) = A085089(A284456(n)).
A025487(a(n)) = A284456(n).
a(A369137(A007097(n))) = A025488(A014221(n-1)).
a(A369137(2^n)) = A098719(n+1).

cp's OEIS Frontend

A052886 Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).

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A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

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A306802 Position of highly composite numbers in the sequence of products of primorials.

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Mathematica

A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

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A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

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A366884 Number of branching factorizations of the least integer of each prime signature (A025487).

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A096903 Least integer of each ordered prime signature (A055932) arranged by prime signature (each row starting with least integer of each prime signature, A025487).

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Mathematica

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A098718 Position of n! in A025487.

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A052886 Expansion of e.g.f.: (1/2) - (1/2)(5 - 4exp(x))^(1/2).