A106701 -id:A106701 - cp's OEIS frontend

A112416 Next-to-most-significant binary digit of the n-th prime.

0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Leroy Quet, Dec 09 2005

The length of the run of zeros pi(2^n+2^(n-1))-pi(2^n) (A095765): 1, 1, 1, 3, 4, 6, 12, 22, 38, 70, 130, 237, 441, ... and the length of the run of ones pi(2^n-1)-pi(2^n-2^(n-2)-1) (A095766): 1, 1, 1, 2, 3, 7, 11, 21, 37, 67, 125, 227, 431, ..., . - Robert G. Wilson v

			The 9th prime is 23 (in decimal), which is 10111 in binary. So a(9) = 0, the next-to-most significant binary digit of 23.

Cf. A004676, A106701.

f[n_] := IntegerDigits[Prime@n, 2][[2]]; Array[f, 105] (* Robert G. Wilson v *)

a(n) = floor((p(n) - 2^m)/2^(m-1)), where p(n) is the n-th prime and m = floor(log(p(n))/log(2)).

More terms from Robert G. Wilson v, Jan 24 2006

A113858 Difference between A095765 and A095766.

Original entry on oeis.org

0, 0, 0, 1, 1, -1, 1, 1, 1, 3, 5, 10, 10, 38, 48, 85, 157, 280, 477, 900, 1540, 2894, 5464, 9762, 18132, 33450, 62364, 116657, 217500, 407235, 763246, 1432291, 2698847, 5087838, 9620802, 18210979, 34487702, 65417618, 124353014, 236558508

Offset: 1

Robert G. Wilson v, Jan 24 2006

A095765: Number of primes whose binary expansion begins '10' less A095766: Number of primes whose binary expansion begins '11'.
The difference between successive runs of A112416.
A112416: a(n) = next-to-most-significant binary digit of n-th prime.

Cf. A106701, A112416, A095765-A095766.

f[n_] := PrimePi[2^n + 2^(n - 1)] - PrimePi[2^n] - PrimePi[2^(n + 1)] + PrimePi[2^n + 2^(n - 1) - 1]; Array[f, 40]

cp's OEIS Frontend

A112416 Next-to-most-significant binary digit of the n-th prime.

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