A165153 -id:A165153 - cp's OEIS frontend

A165416 Irregular array read by rows: The n-th row contains those distinct positive integers that each, when written in binary, occurs as a substring in binary n.

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

Offset: 1

Leroy Quet, Sep 17 2009

This is sequence A119709 with the 0's removed.

The n-th row of this sequence contains A122953(n) terms.

			6 in binary is 110. The distinct positive integers that occur as substrings in n when they and n are written in binary are: 1 (1 in binary), 2 (10 in binary), 3 (11 in binary), and 6 (110 in binary). So row 6 is (1,2,3,6).

Reinhard Zumkeller, Rows n = 1..512 of table, flattened

Cf. A119709, A122953.
Cf. A030308.
Cf. A165153 (row products), A225243 (subsequence).

a165416 n k = a165416_tabf !! (n-1) !! (k-1)
a165416_row n = a165416_tabf !! (n-1)
a165416_tabf = map (dropWhile (== 0)) $ tail a119709_tabf
-- Reinhard Zumkeller, Aug 14 2013

Extended by Ray Chandler, Mar 13 2010

A332030 a(n) is the product of the distinct positive numbers whose binary digits appear in order, but not necessarily as consecutive digits, in the binary representation of n.

Original entry on oeis.org

1, 1, 2, 3, 8, 30, 36, 21, 64, 1080, 7200, 2310, 1728, 16380, 3528, 315, 1024, 146880, 9331200, 1580040, 13824000, 1362160800, 170755200, 796950, 331776, 176904000, 2861913600, 72972900, 4741632, 99754200, 1587600, 9765, 32768, 77552640, 86294937600

Offset: 0

Rémy Sigrist, Feb 05 2020

This sequence is a variant of A165153.

For n > 0, a(n) is the product of the terms of the n-th row of A301983.

			For n = 9:
- the binary representation of 9 is "1001",
- the following positive binary strings appear in it: "1", "10", "11", "100", "101" and "1001",
- they correspond to: 1, 2, 3, 4, 5 and 9,
- so a(9) = 1 * 2 * 3 * 4 * 5 * 9 = 1080.

Rémy Sigrist, Table of n, a(n) for n = 0..4096

Cf. A005329, A006125, A165153, A301983, A328379 (additive variant).

a(n) = my (b=binary(n), s=[0]); for (i=1, #b, s=setunion(s, apply(m -> 2*m+b[i], s))); vecprod(s[2..#s])

a(n) >= A165153(n).
a(2^k) = A006125(k+1) for any k >= 0.
a(2^k-1) = A005329(k) for any k >= 0.

cp's OEIS Frontend

A165416 Irregular array read by rows: The n-th row contains those distinct positive integers that each, when written in binary, occurs as a substring in binary n.

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