A102730 -id:A102730 - cp's OEIS frontend

A103671 Smallest m such that the binary representation of n! does not contain m!.

4, 5, 5, 5, 6, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 5, 6, 5, 6, 5, 6, 5, 5, 5, 6, 6, 6, 5, 7, 6, 6, 6, 6, 6, 7, 6, 6, 5, 6, 6, 6, 6, 6, 6, 5, 6, 7, 6, 6, 5, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 7, 6, 7, 6, 6

Offset: 6

Reinhard Zumkeller, Feb 12 2005

Reinhard Zumkeller conjectures (at A102730) that this sequence is bounded. I conjecture the contrary, that for every k there is n with a(n) > k. - Charles R Greathouse IV, Apr 07 2013

Cf. A102730, A103672, A036603, A007088, A000142.

q[n_, m_] := StringContainsQ[IntegerString[n!, 2], IntegerString[m!, 2]]; a[n_] := Module[{m = 2}, While[q[n, m], m++]; m]; Array[a, 105, 6] (* Amiram Eldar, Apr 03 2025 *)

A103672 Greatest m < n such that the binary representation of n! contains m!.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 7, 4, 5, 4, 5, 4, 4, 4, 15, 4, 4, 4, 6, 4, 5, 6, 4, 5, 4, 5, 4, 5, 6, 6, 31, 5, 5, 5, 4, 6, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 5, 5, 5, 63, 6, 5, 5, 5, 5, 5, 6, 5, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5, 5, 6, 6, 5, 5, 5, 6, 7, 5, 6, 5, 5

Offset: 1

Reinhard Zumkeller, Feb 12 2005

Cf. A000079, A000225, A102730, A103671, A036603, A007088, A000142.

q[n_, m_] := StringContainsQ[IntegerString[n!, 2], IntegerString[m!, 2]]; a[n_] := Module[{m = n-1}, While[!q[n, m], m--]; m]; Array[a, 104] (* Amiram Eldar, Apr 03 2025 *)

a(2^k) = 2^k - 1, a(A000079(k)) = A000225(k).

cp's OEIS Frontend

A103671 Smallest m such that the binary representation of n! does not contain m!.

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