A095060 -id:A095060 - cp's OEIS frontend

A095062 Number of fib00 primes (A095082) in range [2^n,2^(n+1)].

1, 1, 2, 1, 2, 7, 12, 14, 27, 50, 91, 178, 335, 611, 1156, 2147, 4042, 7831, 14724, 28227, 53736, 102482, 196303, 376121, 723408, 1393572, 2683465, 5180304, 10009707, 19366479, 37509260, 72706948, 141074303, 273975483, 532538340

Offset: 1

Antti Karttunen, Jun 01 2004

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. A095060, A095065, A095067, A095068, A095082.

a(n) = A095060(n) - A095067(n) = A095065(n) + A095068(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

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

A095061 Number of fibodd primes (A095081) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 2, 4, 4, 7, 18, 25, 54, 105, 178, 332, 637, 1165, 2194, 4161, 7770, 14800, 28100, 53525, 102394, 195938, 377301, 723938, 1391620, 2684760, 5178439, 10010119, 19362205, 37501838, 72702221, 141062816, 273985225, 532514962

Offset: 1

Antti Karttunen, Jun 01 2004

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. A036378, A095060, A095066, A095069, A095081.

a(n) = A036378(n) - A095060(n) = A095066(n) + A095069(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095067 Number of fib010 primes (A095087) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 4, 11, 23, 33, 59, 108, 205, 364, 709, 1368, 2546, 4789, 9111, 17259, 33075, 63340, 121467, 232396, 446774, 860552, 1659065, 3203164, 6187452, 11968853, 23171558, 44926416, 87186186, 169306460, 329138934

Offset: 1

Antti Karttunen, Jun 01 2004

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. A095060, A095062, A095087.

a(n) = A095060(n) - A095062(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095080 Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 47, 71, 73, 79, 83, 89, 97, 107, 109, 113, 131, 139, 149, 151, 157, 167, 173, 181, 191, 193, 199, 223, 227, 233, 241, 251, 257, 269, 277, 283, 293, 311, 317, 337, 353, 359, 367, 379, 397, 401, 409, 419, 421

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A022342. Union of A095082 and A095087. Cf. A095060, A095081.

F:= combinat[fibonacci]:
b:= proc(n) option remember; local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if b(p)::even then break fi
      od; p
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 27 2016

F = Fibonacci;
b[n_] := b[n] = Module[{j},
     If[n == 0, 0, For[j = 2, F[j + 1] <= n, j++];
     b[n - F[j]] + 2^(j - 2)]];
a[n_] := a[n] = Module[{p},
     p = If[n == 1, 1, a[n - 1]]; While[True,
     p = NextPrime[p]; If[ EvenQ[b[p]], Break[]]]; p];
Array[a, 100] (* Jean-François Alcover, Jul 01 2021, after Alois P. Heinz *)

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n):
    return str(a(n))[-1]=="0"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 07 2017

A095290 Number of lower Wythoff primes (A095280) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 0, 1, 3, 5, 8, 14, 30, 40, 86, 162, 289, 541, 1017, 1881, 3527, 6652, 12641, 23855, 45455, 86753, 165844, 317363, 609942, 1171377, 2253588, 4343268, 8381084, 16198859, 31329311, 60683252, 117637523, 228259189

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

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)]

a(n) = A036378(n)-A095291(n). Cf. A095060, A095291.

cp's OEIS Frontend

A095062 Number of fib00 primes (A095082) in range [2^n,2^(n+1)].

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

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A095061 Number of fibodd primes (A095081) in range [2^n,2^(n+1)].

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A095067 Number of fib010 primes (A095087) in range [2^n,2^(n+1)].

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A095080 Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.

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Maple

Mathematica

Python

A095290 Number of lower Wythoff primes (A095280) in range ]2^n,2^(n+1)].

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