A073924 -id:A073924 - cp's OEIS frontend

A113824 a(1)=1; a(n+1) = the least prime greater than 2*a(n) which is a(n) plus a power of two.

1, 3, 7, 23, 151, 65687, 9007199254806679, 73795983494093013143, 205688069665150755269371147819668813122842057000180977011589271

Offset: 1

Artur Jasinski, Jan 23 2006

nonn

Next term 205688069665150755269371147819668813122842057000180977011589271 + 2^1752 is too large to include here.

Those powers of two are A073924.

			151 is there because 23 + 2^7 = 151 is prime.

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

Edited by Don Reble, Jan 25 2006

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

A073925 Powers of 2 rearranged such that every partial sum (n>1) is composite.

Original entry on oeis.org

1, 8, 16, 2, 64, 4, 256, 1024, 32, 128, 512, 2048, 8192, 4096, 16384, 32768, 131072, 65536, 2097152, 262144, 524288, 1048576, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 2147483648, 1073741824

Offset: 1

Amarnath Murthy, Aug 19 2002

nonn

			a(2)=8 because 1+2=3, 1+4=5 are primes and 1+8=9 is composite.

Robert Israel, Table of n, a(n) for n = 1..3319

Cf. A073923, A073924.

N:= 100: # for terms before the first > 2^N
Cands:= [seq(2^i,i=0..N)]:
Res:= NULL: s:= 0: nC:= N+1:
found:= true;
while found do
  found:= false;
  for i from 1 to nC do
    if not isprime(s+Cands[i]) then
      Res:= Res, Cands[i];
      s:= s + Cands[i];
      Cands:= subsop(i=NULL, Cands);
      nC:= nC-1;
      found:= true;
      break
    fi
  od
od:
Res; # Robert Israel, Apr 05 2020

More terms from Sascha Kurz, Feb 01 2003

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.

A073926 Powers of 2 rearranged so that every partial sum after 1 is a proper prime power.

Original entry on oeis.org

1, 8, 16, 2

Offset: 1

Amarnath Murthy, Aug 19 2002

If a(5) exists -- that is, if there is some k for which 2^k + 27 is a proper prime power -- then k > 100000. Or can it be proved that 2^k + 27 is either prime or has more than one distinct prime factor for all k? - Ryan Propper, Jun 17 2006

Cf. A073923, A073924, A073925.

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.

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

cp's OEIS Frontend

A113824 a(1)=1; a(n+1) = the least prime greater than 2*a(n) which is a(n) plus a power of two.

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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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A073925 Powers of 2 rearranged such that every partial sum (n>1) is composite.

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

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A073926 Powers of 2 rearranged so that every partial sum after 1 is a proper prime power.

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

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