A095766 -id:A095766 - cp's OEIS frontend

A113858 Difference between A095765 and A095766.

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]

A095765 Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).

Original entry on oeis.org

0, 1, 1, 3, 4, 6, 12, 22, 38, 70, 130, 237, 441, 825, 1539, 2897, 5453, 10335, 19556, 37243, 70938, 135555, 259586, 497790, 956126, 1839597, 3544827, 6839282, 13212389, 25552386, 49472951, 95883938, 186011076, 361177503, 701906519

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

I.e., number of primes p such that 2^n < p < (2^n + 2^(n-1)).
Ratio a(n)/A036378(n) converges as follows: 0, 0.5, 0.5, 0.6, 0.571429, 0.461538, 0.521739, 0.511628, 0.506667, 0.510949, 0.509804, 0.510776, 0.505734, 0.511787, 0.507921, 0.507444, 0.507303, 0.506866, 0.506173, 0.506115, 0.505487, 0.505395, 0.505318, 0.504951, 0.504786, 0.504588, 0.504437, 0.504301, 0.50415, 0.504016, 0.503887, 0.503763, 0.503654
Ratio a(n)/A095766(n) converges as follows: 0, 1, 1, 1.5, 1.333333, 0.857143, 1.090909, 1.047619, 1.027027, 1.044776, 1.04, 1.044053, 1.023202, 1.048285, 1.032193, 1.030228, 1.029645, 1.027847, 1.025001, 1.024764, 1.022191, 1.021815, 1.021501, 1.020003, 1.019331, 1.01852, 1.017908, 1.017353, 1.016737, 1.016195, 1.015669, 1.015164, 1.014723
I think this explains also the bias present in ratios shown at A095297, A095298, etc.

			Table showing the derivation of the initial terms:
  n   2^n+1  2^(n+1)  a(n)   primes starting '10' in binary
  1     3       4      0       -
  2     5       8      1       5 = 101_2
  3     9      16      1      11 = 1011_2
  4    17      32      3      17 = 10001_2, 19 = 10011_2, 23 = 10111_2

Antti Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A036378, A080165, A095297, A095298, A095766.

a[n_] := PrimePi[2^n + 2^(n - 1) - 1] - PrimePi[2^n];
Array[a, 35] (* Robert G. Wilson v, Jan 24 2006 *)

a(n) = A036378(n)-A095766(n).

a(34) and a(35) from Robert G. Wilson v, Jan 24 2006

Edited, restoring meaning of name, by Peter Munn, Jun 27 2023

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

Original entry on oeis.org

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

A096369 Triangle read by rows, 0<=k

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 2, 1, 1, 2, 5, 3, 3, 2, 5, 7, 3, 4, 5, 3, 7, 13, 7, 6, 6, 4, 7, 13, 23, 13, 12, 9, 10, 12, 11, 23, 43, 22, 23, 22, 23, 22, 21, 21, 43, 75, 37, 37, 36, 40, 39, 38, 38, 37, 75, 137, 71, 71, 73, 66, 56, 71, 70, 66, 67, 137, 255, 128, 125, 130, 127, 132, 128, 130, 129, 126, 125, 255

Offset: 1

Reinhard Zumkeller, Jul 19 2004

T(n,0) = A036378(n-1) for n>1; T(n,n-1) = T(n,0) for n>2;
T(n,1) = A095008(n-1) for n>2;
T(n,n-2) = A095766(n-1) for n>1;
conjecture: T(n,k) > 0 for n>1.

			prime(12) = 37 -> 1 0 0 1 0 1 ..... n = 6
prime(13) = 41 -> 1 0 1 0 0 1 ..... all primes p:
prime(14) = 43 -> 1 0 1 0 1 1 ..... 2^(6-1) <= p < 2^6
prime(15) = 47 -> 1 0 1 1 1 1
prime(16) = 53 -> 1 1 0 1 0 1
prime(17) = 59 -> 1 1 1 0 1 1
prime(18) = 61 -> 1 1 1 1 0 1
col-sums of bits: 7 3 5 4 3 7 : T(6,5)=7, T(6,4)=3, T(6,3)=5,
...

S[n_] := S[n] = IntegerDigits[Select[Range[2^(n-1), 2^n], PrimeQ], 2] // Transpose;
T[1, 1] = 0;
T[n_, k_] := S[n][[n-k+1]] // Total;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 14 2021 *)

cp's OEIS Frontend

A113858 Difference between A095765 and A095766.

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A095765 Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).

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

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A096369 Triangle read by rows, 0<=k

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