A113824 -id:A113824 - cp's OEIS frontend

A151891 a(n) = Pi(A113824(n+1)).

Original entry on oeis.org

2, 4, 9, 36, 6561, 252252704150178, 1650016588712720468

Offset: 1

Artur Jasinski, Apr 04 2008

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824.

a(6)-a(7) using Kim Walisch's primecount, from Amiram Eldar, Mar 13 2020

A113878 a(1)=0; a(n+1) is the least number > a(n) such that Sum_{k=1..n+1} 2^a(k) is not composite.

Original entry on oeis.org

0, 1, 2, 4, 7, 16, 53, 66, 207, 1752, 5041, 6310

Offset: 1

Artur Jasinski, Jan 27 2006

Base-2 logarithms of A073924.

a(13) > 50000. - Don Reble

Cf. A073924, A080355, A080567, A113824.

a[1] = 0; a[n_] := a[n] = Block[{k = a[n - 1] + 1, s = Plus @@ (2^Array[a, n - 1])}, While[ !PrimeQ[s + 2^k], k++ ]; k]; Array[a, 12] (* Robert G. Wilson v *)

from sympy import isprime
def afind(limit):
    print("0, 1", end=", ")
    s, pow2 = 2**0 + 2**1, 2**2
    for m in range(2, limit+1):
        if isprime(s+pow2): print(m, end=", "); s += pow2
        pow2 *= 2
afind(2000) # Michael S. Branicky, Jul 11 2021

Edited by Don Reble, Feb 17 2006

A113914 (1,2,3) Jasinski-like positive power sequence.

Original entry on oeis.org

1, 5, 13, 29, 61, 131, 271, 569, 1381, 2789, 5581, 11171, 22369, 44741, 89491, 185543, 373273, 766229, 1532701, 3065411, 6130849, 12261701, 24700549, 49401101, 98802211, 202387391, 409557751, 819116231, 1638232471, 3276464969

Offset: 1

Jonathan Vos Post, Jan 29 2006

In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. The first differences of such sequences are powers of d, with no closed-form known upper bound.

			a(1) = 1 by definition.
a(2) = 2*1 + 3^1 = 5.
a(3) = 2*5 + 3^1 = 13.
a(4) = 2*13 + 3^1 = 29.
a(5) = 2*29 + 3^1 = 61.
a(6) = 2*61 + 3^2 = 271.
a(7) = 2*271 + 3^2 = 569.
a(32) = 2*6553461379 + 3^49 = 239299329230630636512841. Here 49 is a record value for the exponent.

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824.

a(1) = 1, a(n+1) = the least prime p such that p = 2*a(n) + 3^k for integer k>0.

A113879 a(n) = 1+A113878(n).

Original entry on oeis.org

1, 2, 3, 5, 8, 17, 54, 67, 208, 1753, 5042, 6311

Offset: 1

Artur Jasinski, Jan 27 2006

nonn

Cf. A113824.

Sum_{n>=1} 1/(2^a(n)) = 1/2+1/(2^2)+1/(2^3)+1/(2^5)+1/(2^8)+... = 0.910163879394531305517927948...

Simplified definition - R. J. Mathar, Oct 01 2009

A113927 a(1)=1, and recursively a(n+1) is the smallest prime p of the form p = 2*a(n) + 5^k for some k>0.

Original entry on oeis.org

1, 7, 19, 43, 211, 547, 4219, 8443, 17011, 34147, 71419, 142963, 1220989051, 3662681227, 19080811690579, 38161623381163, 76324467465451, 152648936884027, 305299094471179, 4656613483675581520483

Offset: 1

Jonathan Vos Post, Jan 30 2006

Note that last digits cycle 7, 9, 3, 1; 7, 9, 3, 1. Note that the exponent k of 5^k is always odd. This follows from taking this sequence mod 6.
Since the first prime value a(2) = 7 == 1 mod 6, all values a(n) thereafter are primes of the form 6*d+1. Hence a(n+1) = [2*(6*d+1) + 5^2] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; a(n+1) = [2*(6*d+1) + 5^4] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; and so for all even exponents.
In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. A113914 is the (1,2,3) Jasinski-like positive power sequence, and this here the (1,2,5) Jasinski-like power sequence.

			a(1) = 1 by definition.
a(2) = 2*1 + 5^1 = 7.
a(3) = 2*7 + 5^1 = 19.
a(4) = 2*19 + 5^1 = 43.
a(5) = 2*43 + 5^3 = 211.
a(6) = 2*211 + 5^3 = 547.
a(7) = 2*547 + 5^5 = 4219.
a(13) = 2*142963 + 5^13 = 1220989051.
a(20) = 2*305299094471179 + 5^31 = 4656613483675581520483, where 31 is a record exponent.
a(22) = 2*9313226967351163119091 + 5^45 = 28421709449030461369547296941307 and 45 is the new record exponent.

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824, A113914.

cp's OEIS Frontend

A151891 a(n) = Pi(A113824(n+1)).

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A113878 a(1)=0; a(n+1) is the least number > a(n) such that Sum_{k=1..n+1} 2^a(k) is not composite.

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A113914 (1,2,3) Jasinski-like positive power sequence.

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A113879 a(n) = 1+A113878(n).

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A113927 a(1)=1, and recursively a(n+1) is the smallest prime p of the form p = 2*a(n) + 5^k for some k>0.

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