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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A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 8, 6, 15, 13, 19, 25, 33, 36, 54, 58, 80, 96, 122, 141, 188, 217, 274, 326, 408, 474, 600, 695, 859, 1012, 1233, 1440, 1763, 2050, 2475, 2899, 3476, 4045, 4850, 5630, 6695, 7797, 9216, 10689, 12628, 14611, 17162, 19875, 23253
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2025

Keywords

Examples

			The a(3) = 1 through a(10) = 8 partitions:
  (21)  (31)  (41)  (51)    (61)   (71)    (81)      (91)
                    (321)   (421)  (431)   (531)     (541)
                    (2211)         (521)   (621)     (631)
                                   (3311)  (32211)   (721)
                                           (222111)  (4321)
                                                     (4411)
                                                     (33211)
                                                     (42211)

Crossrefs

First differences of A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A360014.
Equal case of A381544 (ranks A381439).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A051904, A091605, A116861, A237984, A239455, A362608, A363724, A381079, A381437, A381438.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,1]==Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A384084 Numbers whose prime signatures are self-conjugate.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153
Offset: 1

Views

Author

Hal M. Switkay, May 18 2025

Keywords

Comments

The implied partition corresponding to k is the partition of bigomega(k) (A001222) formed by the prime exponents. For example, bigomega(18) = 3, which is partitioned as 2 + 1, because 18 = (3^2)(2^1), and 2 + 1 is a self-conjugate partition of 3. In contrast, while bigomega(42) = 3, 3 is partitioned as 1 + 1 + 1, because 42 = (2^1)(3^1)(7^1), and 1 + 1 + 1 is not a self-conjugate partition of 3.
This sequence is very similar to, but ultimately different from, A212166. The first difference is a(342) = 1083, whereas A212166(342) = 1080.
This sequence is a subsequence of A212166.
It includes 1 (empty partition) and all primes (A000040: partition 1), as well as numbers of the form (p^2)q, where p and q are distinct primes (A054753: partition 2 + 1).
k is a term in this sequence if and only if A046523(k) is a term in A181825.

Examples

			120 is a term; its prime factorization (2^3)(3^2)(5^1) is self-conjugate.
24 is not a term; its prime factorization (2^3)(3^1) is not self-conjugate.

Links

Crossrefs

Cf. A001222, A046523, A054753, A181825, A212166.

Programs

  • Mathematica
    selfQ[p_] := ResourceFunction["ConjugatePartition"][p] == p; q[n_] := selfQ[Sort[FactorInteger[n][[;;, 2]], Greater]]; Select[Range[200], q] (* Amiram Eldar, May 26 2025 *)

A385216 Greatest Heinz number of a sparse submultiset of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 4, 13, 14, 5, 16, 17, 9, 19, 20, 21, 22, 23, 8, 25, 26, 27, 28, 29, 10, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 9, 46, 47, 16, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 33, 67, 68, 69
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2025

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 12 are {1,1,2}, with sparse submultisets {{},{1},{2},{1,1}}, with Heinz numbers {1,2,3,4}, so a(12) = 4.
The prime indices of 36 are {1,1,2,2}, with sparse submultisets {{},{1},{2},{1,1},{2,2}}, with Heinz numbers {1,2,3,4,9}, so a(36) = 9.
The prime indices of 462 are {1,2,4,5}, with sparse submultisets {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with Heinz numbers {1,2,3,7,11,14,21,22,33}, so a(462) = 33.

Crossrefs

Sparse submultisets are counted by A166469, maximal A385215.
The union is A319630 (Heinz numbers of sparse multisets), complement A104210.
For binary instead of prime indices we have A374356, see A245564, A384883.
A000005 counts divisors (or submultisets of prime indices).
A001222 counts prime factors, distinct A001221.
A051903 gives greatest prime exponent, least A051904, counted by A091602.
A055396 gives least prime index, greatest A061395, counted by A008284.
A056239 adds up prime indices, row sums of A112798.
A212166 ranks partitions with max multiplicity = length, counted by A239964.
A381542 ranks partitions with max part = max multiplicity, counted by A240312.
Cf. A000720, A005117, A122111, A130091, A268193, A316476.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@Select[Divisors[n],FreeQ[Differences[prix[#]],1]&],{n,100}]

Formula

a(n) = n iff n belongs to A319630.

A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 41, 56, 72, 94, 124, 158, 205, 262, 331, 419, 531, 663, 829, 1033, 1281, 1581, 1954, 2393, 2936, 3584, 4366, 5300, 6433, 7764, 9374, 11277, 13548, 16225, 19425, 23166, 27623, 32842, 39004, 46212, 54719, 64610, 76251
Offset: 0

Views

Author

Gus Wiseman, Apr 05 2025

Keywords

Examples

			The a(2) = 1 through a(9) = 12 partitions:
  (2)  (3)  (4)   (5)    (6)    (7)     (8)      (9)
            (22)  (32)   (33)   (43)    (44)     (54)
                  (221)  (42)   (52)    (53)     (63)
                         (222)  (322)   (62)     (72)
                                (331)   (332)    (333)
                                (2221)  (422)    (432)
                                        (2222)   (441)
                                        (3221)   (522)
                                        (22211)  (3222)
                                                 (3321)
                                                 (4221)
                                                 (22221)

Crossrefs

The complement (greater than or equal to) is A241131 except first, ranks A360015.
The opposite version (greater than) is A241131 shifted except first, ranks A360013.
These partitions have ranks A382856, complement A360015.
The weak version (less than or equal to) is A381544, ranks A381439.
For equality we have A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,1]
Previous Showing 31-34 of 34 results.

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