A375572 -id:A375572 - cp's OEIS frontend

A375570 Smallest m such that A008949(m,k) = n for some k.

0, 1, 2, 2, 4, 5, 3, 3, 8, 9, 4, 11, 12, 13, 4, 4, 16, 17, 18, 19, 20, 6, 22, 23, 24, 5, 26, 27, 7, 29, 5, 5, 32, 33, 34, 35, 8, 37, 38, 39, 40, 6, 42, 43, 44, 9, 46, 47, 48, 49, 50, 51, 52, 53, 54, 10, 6, 57, 58, 59, 60, 61, 6, 6, 64, 65, 11, 67, 68, 69, 70

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Cf. A008949, A058084, A375571, A375572, A375573.

A008949(a(n),A375571(n)) = n.
a(n) <= n-1.
a(2^n) = n.
a(2^n-1) = n for n >= 2.

A375573 Numbers occurring at least three times in Bernoulli's triangle A008949.

Original entry on oeis.org

1, 16, 64, 232, 256, 466, 562, 1024, 1486, 2048, 4096, 15226, 16384, 44552, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Equivalently, 1 together with numbers occurring at least three times in columns k >= 1 of Bernoulli's triangle.
Equivalently, 1 together with numbers occurring at least twice in columns k >= 2 of Bernoulli's triangle.
4^j is a term if j >= 0 and j != 1, because 4^j = A008949(2*j,2*j) = A008949(2*j+1,j) = A008949(4^j-1,1) for j >= 2 and A008949(i,0) = 1 for all i. Are 232, 466, 562, 1486, 2048, 15226, 44552 the only terms not of this form? There are no more such terms below 2^70.
Are 1 and 4096 = A008949(12,12) = A008949(13,6) = A008949(90,2) = A008949(4095,1) the only numbers that occur at least 4 times? There are no more such numbers below 2^70.

Cf. A003015, A008949, A374796, A375570, A375572.

A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

Original entry on oeis.org

1, 6, 10, 15, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 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, 1176, 1210

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

All Narayana numbers A001263(n,k) with n != 2*k-1, are terms since A001263(n,k) = A001263(n,n+1-k). In particular, all positive triangular numbers except 3 are terms. Are there any other terms, i.e., is there a number A001263(2*k-1,k), k >= 2, that can be written as a Narayana number in another way? Any such number would also be a term of A376001.

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

Cf. A000217, A001263, A006987, A098564, A374796, A375572, A375999, A376001.

from bisect import insort
from itertools import islice
def A376000_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]
        c = 0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0:
                if c >= 2: yield N0
                break
            central = n==2*k-1
            c += 2-central
            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 central:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A376000_list(nmax):
    return list(islice(A376000_generator(),nmax))

cp's OEIS Frontend

A375570 Smallest m such that A008949(m,k) = n for some k.

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A375573 Numbers occurring at least three times in Bernoulli's triangle A008949.

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A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

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