A375570 -id:A375570 - cp's OEIS frontend

A375571 a(n) is the unique integer k such that A008949(A375570(n),k) = n.

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 4, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 5, 6, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Cf. A008949, A357327, A375570.

A008949(A375570(n),a(n)) = n.
a(2^n) = n.
a(2^n-1) = n-1.

A375572 Numbers occurring at least twice in Bernoulli's triangle A008949.

Original entry on oeis.org

1, 4, 7, 8, 11, 15, 16, 22, 26, 29, 31, 32, 37, 42, 46, 56, 57, 63, 64, 67, 79, 92, 93, 99, 106, 120, 121, 127, 128, 130, 137, 154, 163, 172, 176, 191, 211, 219, 232, 247, 254, 255, 256, 277, 299, 301, 326, 352, 378, 379, 382, 386, 407, 436, 466, 470, 497, 502

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Equivalently, 1 together with numbers occurring in columns k >= 2 of Bernoulli's triangle.

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

Cf. A006987, A008949, A375570, A375573.

isok(k) = my(nb=0); for (i=0, k, nb += #select(x->(x==k), vector(i+1, j, sum(jj=0, j-1, binomial(i, jj))))); nb >= 2; \\ Michel Marcus, Aug 22 2024

lista(nn) = my(v = vector(nn)); for (n=1, nn, my(w=vector(n+1, j, sum(jj=0, j-1, binomial(n, jj)))); for (i=1, #w, if (w[i] <= nn, v[w[i]]++));); Vec(select(x->(x>=2), v, 1)); \\ Michel Marcus, Aug 23 2024

from math import comb
from bisect import insort
def A375572_list(nmax):
    a_list = [1]
    if nmax == 1: return a_list
    nkb_list = [(2,2,4)] # List of triples (n,k,A008949(n,k)), sorted by the last element.
    while 1:
        b0 = nkb_list[0][2]
        a_list.append(b0)
        if len(a_list) == nmax: return a_list
        while 1:
            n,k,b = nkb_list[0]
            if b > b0: break
            del nkb_list[0]
            insort(nkb_list,(n+1,k,2*b-comb(n,k)),key=lambda x:x[2])
            if n == k:
                insort(nkb_list,(n+1,k+1,2**(k+1)),key=lambda x:x[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.

cp's OEIS Frontend

A375571 a(n) is the unique integer k such that A008949(A375570(n),k) = n.

Views

Author

Keywords

Crossrefs

Formula

A375572 Numbers occurring at least twice in Bernoulli's triangle A008949.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Python

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

Views

Author

Keywords

Comments

Crossrefs