A095008 -id:A095008 - cp's OEIS frontend

A095357 Ratio A095107(n)/A095008(n) rounded down.

2, 6, 10, 18, 35, 49, 108, 181, 346, 651, 1236, 2348, 4240, 8454, 16537, 30963, 60986, 118814, 225337, 438305, 854049

Offset: 1

Antti Karttunen, Jun 12 2004

This is the average length of maximum Dyck path prefix (i.e. non-diving portion) found in the "Legendre-vectors" of all 4k+3 primes in range ]2^n,2^(n+1)]. See A095104-A095105.
The ratios before rounding are: 2, 6, 10, 18, 35.333333, 49.428571, 108.461538, 181.545455, 346.702703, 651.295775, 1236.34375, 2348.779221, 4240.445455, 8454.26518, 16537.703752, 30963.160864, 60986.990505, 118814.20247, 225337.874981, 438305.90522, 854049.74263.
Ratio (A095107(n)/A095008(n))/(A095110(n)/A000079(n-2)) starts as follows: 1, 1, 0.5, 0.75, 0.375, 0.4375, 0.40625, 0.34375, 0.289062, 0.277344, 0.25, 0.225586, 0.214844, 0.197021, 0.185425, 0.175293, 0.16391, 0.155701, 0.14756, 0.140224.

A095358 gives the same ratios rounded to nearest integer. A095361 gives similar ratios computed for all 4k+3 integers.

a(n) = floor(A095107(n)/A095008(n)).

A095358 Ratio A095107(n)/A095008(n) rounded to nearest integer.

Original entry on oeis.org

2, 6, 10, 18, 35, 49, 108, 182, 347, 651, 1236, 2349, 4240, 8454, 16538, 30963, 60987, 118814, 225338, 438306, 854050

Offset: 1

Antti Karttunen, Jun 12 2004

Cf. A095357, where the same ratios are given round down.

a(n) = round(A095107(n)/A095008(n)).

A036378 Number of primes p between powers of 2, 2^n < p <= 2^(n+1).

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 13, 23, 43, 75, 137, 255, 464, 872, 1612, 3030, 5709, 10749, 20390, 38635, 73586, 140336, 268216, 513708, 985818, 1894120, 3645744, 7027290, 13561907, 26207278, 50697537, 98182656, 190335585, 369323305, 717267168, 1394192236, 2712103833

Offset: 0

Labos Elemer

nonn

Number of primes whose binary order (A029837) is n+1, i.e., those with ceiling(log_2(p)) = n+1. [corrected by Jon E. Schoenfield, May 13 2018]
First differences of A007053. This sequence illustrates how far the Bertrand postulate is oversatisfied.
Scaled for Ramanujan primes as in A190501, A190502.
This sequence appears complete such that any nonnegative number can be written as a sum of distinct terms of this sequence. The sequence has been checked for completeness up to the gap between 2^46 and 2^47. Assuming that after 2^46 the formula x/log(x) is a good approximation to primepi(x), it can be proved that 2*a(n) > a(n+1) for all n >= 46, which is a sufficient condition for completeness. [Frank M Jackson, Feb 02 2012]

			The 7 primes for which A029837(p)=6 are 37, 41, 43, 47, 53, 59, 61.

Ray Chandler, Table of n, a(n) for n = 0..91 (using data from A007053; n = 0..74 by T. D. Noe, n = 75..85 by Gord Palameta, n = 86..89 by David Baugh)
Paul D. Beale, A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, arXiv:1411.2484 [physics.comp-ph], 2014-2015.
Paul D. Beale and Jetanat Datephanyawat, Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers, arXiv:1811.11629 [cs.CR], 2018.
Seung-Hoon Lee, Mario Gerla, Hugo Krawczyk, Kang-Won Lee, and Elizabeth A. Quaglia, Performance Evaluation of Secure Network Coding using Homomorphic Signature, 2011 International Symposium on Networking Coding.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A000720, A190501, A190502, A190568, A007053.

[1,1] cat [#PrimesInInterval(2^n, 2^(n+1)): n in [2..29]]; // Vincenzo Librandi, Nov 18 2014

t = Table[PrimePi[2^n], {n, 0, 20}]; Rest@t - Most@t (* Robert G. Wilson v, Mar 20 2006 *)

a(n) = primepi(1<<(n+1))-primepi(1<

a(n) = primepi(2^(n+1)) - primepi(2^n).

a(n) = A095005(n)+A095006(n) = A095007(n) + A095008(n) = A095013(n) + A095014(n) = A095015(n) + A095016(n) (for n > 1) = A095021(n) + A095022(n) + A095023(n) + A095024(n) = A095019(n) + A095054(n) = A095020(n) + A095055(n) = A095060(n) + A095061(n) = A095063(n) + A095064(n) = A095094(n) + A095095(n).

More terms from Labos Elemer, May 13 2004

Entries checked by Robert G. Wilson v, Mar 20 2006

A095093 Number of 4k+3 primes whose Legendre-vector is not Dyck-path (A095103) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 12, 21, 41, 77, 143, 287, 530, 1010, 1967, 3711, 7125, 13806, 26525, 51126

Offset: 1

Antti Karttunen, Jun 01 2004

A. 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. A095103.

is(m) = {my(s=0); if(isprime(m), for(i=1, m-1, if((s+=kronecker(i, m))<0, return(1)))); 0; }
a(n) = {my(c=0); forstep(m=2^n+3, 2^(n+1), 4, c+=is(m)); c; } \\ Jinyuan Wang, Jul 20 2020

a(n) = A095008(n) - A095092(n).

A095092 Number of 4k+3 primes whose Legendre-vector is a Dyck-path (A095102) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 7, 10, 16, 30, 51, 88, 153, 277, 509, 905, 1660, 3079, 5535, 10234, 19053

Offset: 1

Antti Karttunen, Jun 01 2004

A. 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. A095090, A095102.

is(m) = {if(!isprime(m), return(0)); my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; }
a(n) = {my(c=0); forstep(m=2^n+3*(n>1), 2^(n+1), 4, c+=is(m)); c; } \\ Jinyuan Wang, Jul 20 2020

a(n) = A095008(n) - A095093(n).

A095007 Number of 4k+1 primes (A002144) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 10, 21, 38, 66, 127, 233, 432, 805, 1511, 2837, 5378, 10186, 19294, 36827, 70157, 133975, 256852, 492882, 946848, 1823129, 3513599, 6780412, 13103462, 25348870, 49090415, 95167380, 184662052, 358630671, 697097364, 1356051342, 2639870329, 5142823877

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John 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. A002144, A036378, A095008, A095009, A095011.

a(n) = A036378(n) - A095008(n) = A095009(n) + A095011(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A095010 Number of 8k+3 primes (A007520) in range [2^n,2^(n+1)].

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 10, 20, 35, 66, 113, 218, 412, 746, 1460, 2672, 5104, 9651, 18375, 35105, 67165, 128410, 246453, 473535, 911489, 1756670, 3390856, 6552449, 12673142, 24546849, 47583904, 92330578, 179317889, 348548185, 678029708, 1319939685, 2571409639

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John 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. A007520, A091127, A095008, A095011, A095012, A095014.

a(n) = A095014(n) - A095011(n) = A095008(n) - A095012(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

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

A095357 Ratio A095107(n)/A095008(n) rounded down.

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A095358 Ratio A095107(n)/A095008(n) rounded to nearest integer.

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A036378 Number of primes p between powers of 2, 2^n < p <= 2^(n+1).

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A095093 Number of 4k+3 primes whose Legendre-vector is not Dyck-path (A095103) in range ]2^n,2^(n+1)].

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A095092 Number of 4k+3 primes whose Legendre-vector is a Dyck-path (A095102) in range ]2^n,2^(n+1)].

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A095007 Number of 4k+1 primes (A002144) in range ]2^n,2^(n+1)].

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A095010 Number of 8k+3 primes (A007520) in range [2^n,2^(n+1)].

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Extensions

A096369 Triangle read by rows, 0<=k

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