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.

Previous Showing 21-26 of 26 results.

A028378 Concatenate rows of triangle in A028364 (removing duplicates).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 14, 19, 23, 28, 42, 56, 66, 76, 90, 132, 174, 202, 227, 255, 297, 429, 561, 645, 715, 785, 869, 1001, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 6292, 7150, 7810, 8398, 8986, 9646, 10504, 11934, 16796, 21658
Offset: 0

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Author

Wouter Meeussen

Keywords

Comments

a(n+1) is the number of subpartitions of the smallest partition of n into distinct parts (minimizing the size of the largest part). The sequence of partitions is [], [1], [2], [2,1], [3,1], [3,2], [3,2,1], [4,2,1], .... - Franklin T. Adams-Watters, Apr 11 2006

Crossrefs

Cf. A028364, A028376.
Cf. A115728, A115729.

A116473 Number of partitions with exactly n subpartitions.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 2, 6, 5, 4, 4, 6, 5, 8, 6, 5, 6, 9, 5, 10, 8, 6, 6, 12, 6, 8, 10, 8, 12, 10, 4, 12, 14, 12, 10, 13, 4, 10, 12, 10, 12, 12, 8, 16, 15, 10, 8, 16, 13, 16, 14, 9, 14, 22, 10, 12, 12, 8, 12, 20, 16, 14, 16, 19, 16, 14, 8, 17, 23, 10, 12, 16, 10, 25, 18, 14, 16, 22, 16, 22
Offset: 1

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Author

Franklin T. Adams-Watters, Mar 17 2006

Keywords

Comments

Conjecture that lim_{n->infinity} a(n) = infinity. Definitely a(n) > 2 for n > 8.

Links

Crossrefs

Cf. A115728, A115729.

A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 7, 4, 6, 6, 9, 7, 7, 9, 11, 10, 8, 12, 9, 13, 5, 10, 13, 9, 15, 14, 15, 9, 14, 16, 10, 18, 19, 17, 13, 18, 10, 19, 11, 16, 21, 12, 15, 24, 19, 17, 22, 16, 22, 12, 23, 24, 6, 19, 20, 29, 21, 21, 26, 22, 25, 13, 30, 27, 11, 26, 25, 19, 34
Offset: 1

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Author

Matthew Vandermast, Apr 28 2014

Keywords

Comments

Also the number of members of A025487 that divide A025487(n).

Examples

			5 members of A025487 divide A025487(6) = 12 (namely, 1, 2, 4, 6 and 12); therefore, a(6) = 5.

Links

Crossrefs

Cf. A085082, A025487, A181822, A322584.
Rearrangement of A115728, A115729 and A238690.
A116473(n) is the number of times n appears in the sequence.

Programs

  • Mathematica
    lpsQ[n_] := n == 1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]); lps = Select[Range[6000], lpsQ]; c[n_] := Count[Divisors[n], ?(MemberQ[lps, #] &)]; c /@ lps  (* _Amiram Eldar, Jan 21 2024 *)

Formula

a(n) = A085082(A025487(n)) = A085082(A181822(n)).
a(n) = A322584(A025487(n)). - Amiram Eldar, Jan 21 2024

A116480 Maximum number of subpartitions for any partition of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 19, 26, 33, 42, 56, 75, 94, 118, 145, 181, 230, 286, 356, 428, 522, 633, 774, 915, 1125, 1341, 1621, 1935, 2351
Offset: 0

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Author

Franklin T. Adams-Watters, Mar 19 2006

Keywords

Comments

The sequence grows roughly as an exponential in the square root of n. a(n) <= 1 + Sum_{0<=kA000108) subpartitions; m ~ sqrt(2n) and the Catalan numbers grow exponentially. Through n=30, there is either a unique partition with the maximum number of subpartitions, or a unique pair of conjugate partitions, except for n=10, where there is a 3-way between [5,3,1^2] and its conjugate [4,2^2,1^2] and two self-conjugate partitions: [4,3,2,1] and [5,2,1^3]. It is not clear what the limiting shape of the maximum partition is. The minimum number of subpartitions is n+1, for the conjugate partitions [n] and [1^n].

Examples

			The 5 partitions of 4 are [4], [3,1], [2^2], [2,1^2], [1^4]; these have respectively 5,7,6,7 and 5 subpartitions, so a(4) = 7, the largest of these.

Crossrefs

Cf. A115728, A115729, A000041, A000108.

A115965 Number of planar subpartitions of size n pyramidal planar partition.

Original entry on oeis.org

1, 2, 9, 96, 2498, 161422, 26217833, 10794429504
Offset: 0

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Author

Franklin T. Adams-Watters, Mar 14 2006

Keywords

Comments

This is a 2-dimensional analog of the Catalan numbers C_n (A000108). The number of subpartitions of the triangular partition [n,n-1,...,1] is C_{n+1}. The planar partition having its subpartitions counted is:
n n-1 ... 2 1
n-1 n-2 ... 1
... ...
2 1
1

Examples

			The 9 planar subpartitions of [2,1|1] are [], [1], [2], [1,1], [1|1], [2,1], [2|1], [1,1|1] and [2,1|1] itself, so a(2)=9. (Here "," separates values on the same line and "|" separates lines.)

Links

Crossrefs

Cf. A115728, A115729, A000219, A000108, A008793.

A120759 Eigensequence for subpartitions of a partition.

Original entry on oeis.org

1, 2, 5, 24, 527, 271156, 73452582161, 5395271857717411958088, 29108958418479344853405820427519529324955406, 847331460208759521535495911124086692972161538057881358236684093384849875943910959287454
Offset: 0

Views

Author

Franklin T. Adams-Watters and Paul D. Hanna, Jul 03 2006

Keywords

Comments

Let this sequence, A, be a partition P=A, then the total number of subpartitions of the partition P is equal to A. See A115728 for the definition of subpartitions of a partition.

Examples

			At n=4, the recurrence gives:
a(4) = a(3)^2 + 1 - Sum_{k=0..2} (-1)^(4-k)*a(k)*C(a(k),4-k)
= a(3)^2 + 1 - [a(0)*C(a(0),4) - a(1)*C(a(1),3) + a(2)*C(a(2),2)]
= 24^2 + 1 - [1*0 - 2*0 + 5*C(5,2)] = 24^2 + 1 - 5*10 = 527.
The recurrence extracts a(n) from the g.f.:
1/(1-x) = 1*(1-x) + 2*x*(1-x)^2 + 5*x^2*(1-x)^5 + 24*x^3*(1-x)^24 +...
+ a(n)*x^n*(1-x)^a(n) +...
The number of digits of a(n) base 10 begins:
[1,1,1,2,3,6,11,22,44,87,174,348,696,1391,...]

Crossrefs

Cf. A115728.

Programs

  • PARI
    a(n)=if(n==0,1,a(n-1)^2+1-sum(k=0,n-2,(-1)^(n-k)*a(k)*binomial(a(k),n-k)))
  • PARI
    a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^a(k)), n)

Formula

a(n) = a(n-1)^2 + 1 - Sum_{k=0..n-2} (-1)^(n-k)*a(k)*C(a(k),n-k) for n>=1, with a(0)=1.
G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^a(n).
Previous Showing 21-26 of 26 results.

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