A355634 Irregular triangle T(n, k), n > 0, k = 1..A093640(n), read by rows; the n-th row contains in ascending order the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25

Offset

1,3

Links

Paolo Xausa, Table of n, a(n) for n = 1..10019 (rows 1..2000 of the triangle, flattened).

Index entries for sequences related to binary expansion of n

Index entries for sequences related to divisors

Formula

T(n, 1) = 1.

T(n, A093640(n)) = n.

Sum_{k = 1..A093640(n)} T(n, k) = A355633(n).

Example

Triangle T(n, k) begins:
     1: [1]
     2: [1, 2]
     3: [1, 3]
     4: [1, 2, 4]
     5: [1, 5]
     6: [1, 2, 3, 6]
     7: [1, 7]
     8: [1, 2, 4, 8]
     9: [1, 9]
    10: [1, 2, 5, 10]
    11: [1, 11]
    12: [1, 2, 3, 4, 6, 12]
    13: [1, 13]
    14: [1, 2, 7, 14]
    15: [1, 3, 15]
    16: [1, 2, 4, 8, 16]

Mathematica

Table[Select[Divisors[n], StringContainsQ[IntegerString[n, 2], IntegerString[#, 2]] &], {n, 50}] (* Paolo Xausa, Jul 23 2024 *)

Prog

(PARI) row(n, base=2) = { my (d=digits(n, base), s=setbinop((i, j) -> fromdigits(d[i..j], base), [1..#d]), v=0); select(v -> v && n%v==0, s) }
(Python)
from sympy import divisors
def row(n):
    s = bin(n)[2:]
    return sorted(d for d in divisors(n, generator=True) if bin(d)[2:] in s)
def table(r): return [i for n in range(1, r+1) for i in row(n)]
print(table(25)) # Michael S. Branicky, Jul 11 2022

Crossrefs

Cf. A027750, A093640 (row lengths), A355632 (decimal analog), A355633 (row sums).

Sequence in context: A275055 A254679 A343651 * A275280 A319845 A319847

Adjacent sequences: A355631 A355632 A355633 * A355635 A355636 A355637

Keyword

nonn,base,tabf

Author

Rémy Sigrist, Jul 11 2022

Status

approved