A024412 -id:A024412 - cp's OEIS frontend

A024413 Positions of odd numbers in A024412.

1, 2, 4, 6, 7, 8, 10, 12, 13, 14, 15, 19, 20, 22, 25, 26, 29, 30, 31, 35, 36, 38, 41, 43, 45, 47, 48, 51, 52, 56, 57, 61, 62, 64, 66, 68, 69, 72, 73, 76, 77, 78, 79, 82, 83, 85, 87, 88, 91, 92, 93, 97, 98, 101, 102, 103, 106, 107, 108, 111, 112, 115, 116, 119, 120, 124, 125, 128, 129, 132, 133, 137

Offset: 1

Clark Kimberling

nonn

a(23) onward corrected by Sean A. Irvine, Jul 08 2019

A024414 Positions of even numbers in A024412.

Original entry on oeis.org

0, 3, 5, 9, 11, 16, 17, 18, 21, 23, 24, 27, 28, 32, 33, 34, 37, 39, 40, 42, 44, 46, 49, 50, 53, 54, 55, 58, 59, 60, 63, 65, 67, 70, 71, 74, 75, 80, 81, 84, 86, 89, 90, 94, 95, 96, 99, 100, 104, 105, 109, 110, 113, 114, 117, 118, 121, 122, 123, 126, 127, 130, 131, 134, 135, 136, 138, 140

Offset: 1

Clark Kimberling

nonn

s2 = Union[Flatten[Table[Table[StirlingS2[n, k], {k, 1, 200}], {n, 1, 200}]]]; Select[Range[200], EvenQ[s2[[#]]] &] - 1 (* Vaclav Kotesovec, Jul 09 2019 *)

Missing a(1)=0 inserted and later terms corrected by Sean A. Irvine, Jul 08 2019

A024415 Position of max{s(n,k): k=1,2,...,n} in A024412, n >= 1, where s(n,k) = Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 37, 77, 155, 342, 775, 1770, 4486

Offset: 1

Clark Kimberling

Cf. A024412.

a(8)-a(9) corrected and a(10)-a(13) from Sean A. Irvine, Jul 09 2019

A024416 a(n) = position of next-to-largest s(n,k), for k=1,2,...,n, in A024412, n >= 3, where s(n,k) = Stirling numbers of the second kind.

Original entry on oeis.org

1, 3, 6, 15, 35, 63, 147, 308, 668, 1735, 4034

Offset: 1

Clark Kimberling

Cf. A008277, A024412.

a(6)-a(7) corrected and a(8)-a(11) from Sean A. Irvine, Jul 09 2019

A374796 Positive integers that can be written as a Stirling number of the second kind in more than one way.

Original entry on oeis.org

1, 15, 4095, 66066

Offset: 1

Pontus von Brömssen, Jul 20 2024

k*(k+1)/2 is a term if k is a term of A215797 and k != 0,2.

a(5) > 10^8 (if it exists).

			With S2(n,k) = A008277(n,k):
      1 = S2( 1, 1) = S2(  2,  1) (for example);
     15 = S2( 5, 2) = S2(  6,  5);
   4095 = S2(13, 2) = S2( 91, 90);
  66066 = S2(14,11) = S2(364,363).

Cf. A008277, A024412, A215797.

A193246 Ordered unsigned Stirling numbers |S1(n,k)|.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 11, 15, 21, 24, 28, 35, 36, 45, 50, 55, 66, 78, 85, 91, 105, 120, 136, 153, 171, 175, 190, 210, 225, 231, 253, 274, 276, 300, 322, 325, 351, 378, 406, 435, 465, 496, 528, 546, 561, 595, 630, 666, 703, 720, 735, 741, 780, 820, 861, 870, 903

Offset: 1

Vaclav Kotesovec, Jan 01 2013

nonn

List all unsigned Stirling numbers of the first kind, sort and remove duplicates.

The only numbers below 10^8 that appear in more than one place in the table A130534 are 1, 6, and 120. - Pontus von Brömssen, Jul 20 2024

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

Cf. A193245, A024412, A130534.

s1 = Union[Flatten[Table[Table[Abs[StirlingS1[n, k]], {k, 1, 100}], {n, 1, 100}]]]; Table[s1[[j]], {j, 1, 100}]

A375999 Narayana numbers (A001263), sorted, duplicates removed.

Original entry on oeis.org

1, 3, 6, 10, 15, 20, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 175, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

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

Cf. A001263, A006987, A024412, A376000, A376001.

from bisect import insort
from itertools import islice
def A375999_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        yield N0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0: break
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if n == 2*k-1:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A375999_list(nmax):
    return list(islice(A375999_generator(),nmax))

A193245 Ordered Stirling numbers S1(n,k) >= 0.

Original entry on oeis.org

0, 1, 2, 11, 24, 35, 85, 175, 274, 322, 546, 720, 870, 1320, 1624, 1925, 2717, 3731, 5005, 6580, 6769, 8500, 10812, 13068, 13566, 16815, 20615, 22449, 25025, 30107, 35926, 40320, 42550, 50050, 58500, 63273, 67977, 78561, 90335, 103385, 117800, 118124, 133672

Offset: 1

Vaclav Kotesovec, Jan 01 2013

nonn

List all Stirling numbers of the first kind >=0, sort and remove duplicates.

Cf. A193246, A024412.

s1 = Select[Union[Flatten[Table[Table[StirlingS1[n,k], {k,1,100}], {n,1,100}]]], #>=0&]; Table[s1[[j]], {j, 1, 100}]

A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

Original entry on oeis.org

2, 4, 20, 884, 528844, 3460086044, 340672148731996, 477782556719729075524, 11694209380474301218263758996, 4967476846044415922850025924897606724, 43298471669920632729336800855543564573041217668, 7790810575556906457316064931238939360882160372451591124244

Offset: 1

Gus Wiseman, May 07 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
                              2: {1}
                              4: {1,1}
                             20: {1,1,3}
                            884: {1,1,6,7}
                         528844: {1,1,10,15,25}
                     3460086044: {1,1,15,31,65,90}
                340672148731996: {1,1,21,63,140,301,350}
          477782556719729075524: {1,1,28,127,266,966,1050,1701}
  11694209380474301218263758996: {1,1,36,255,462,2646,3025,6951,7770}

Index entries for sequences related to Heinz numbers

Cf. A000040, A001222, A002870, A008277, A024412, A056239, A112798, A215366, A325500, A325502.

Times@@@Table[Prime[StirlingS2[n,k]],{n,1,10},{k,1,n}]

a(n) = Product_{i = 1..n} prime(A008277(n,i)).
A061395(a(n)) = A002870(n).
A056239(a(n)) = A000110(n).

cp's OEIS Frontend

A024413 Positions of odd numbers in A024412.

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A024414 Positions of even numbers in A024412.

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A024415 Position of max{s(n,k): k=1,2,...,n} in A024412, n >= 1, where s(n,k) = Stirling numbers of the second kind.

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A024416 a(n) = position of next-to-largest s(n,k), for k=1,2,...,n, in A024412, n >= 3, where s(n,k) = Stirling numbers of the second kind.

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A374796 Positive integers that can be written as a Stirling number of the second kind in more than one way.

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A193246 Ordered unsigned Stirling numbers |S1(n,k)|.

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Mathematica

A375999 Narayana numbers (A001263), sorted, duplicates removed.

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A193245 Ordered Stirling numbers S1(n,k) >= 0.

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A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

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