A123359 -id:A123359 - cp's OEIS frontend

A159266 Least positive integer m such that 2^n+3^m is prime.

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 4, 1, 3, 3, 1, 1, 8, 1, 4, 8, 8, 6, 2, 7, 27, 6, 13, 1, 10, 1, 6, 4, 8, 18, 9, 19, 2, 15, 29, 3, 3, 17, 10, 3, 11, 6, 2, 5, 20, 34, 4, 12, 10, 26, 1, 4, 2, 9, 29, 29, 10, 34, 13, 4, 8, 2, 1, 8, 10, 26, 50, 19, 12, 10, 8, 13, 27, 17, 9, 33, 4, 2, 17, 1, 7, 3, 5, 61, 26

Offset: 1

M. F. Hasler, Apr 07 2009

nonn

In contrast to A123340 which allows m=0, a(0) does not exist for this sequence.

			a(1)=1 is the least m>0 such that 2^1+3^m (=5) is prime.
a(2)=1 is the least m>0 such that 2^2+3^m (=7) is prime.
a(5)=2 is the least m>0 such that 2^5+3^m (=41) is prime.

Robert Israel, Table of n, a(n) for n = 1..2000
M. F. Hasler, Primes of the form (x+1)^p-x^p, Apr 7, 2009.
Maximilian Hasler, Mike Oakes, Mark Underwood, David Broadhurst and others, Primes of the form (x+1)^p-x^p, digest of 22 messages in primenumbers Yahoo group, Apr 5 - May 7, 2009. [Cached copy]

Cf. A123340 (allows for m=0), A123359 (roles of 2 and 3 exchanged).

f:= proc(n) local t,m;
  t:= 2^n;
  for m from 1 do if isprime(t+3^m) then return m fi od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 18 2018

a[n_] := Module[{m, t = 2^n}, For[m = 1, True, m++, If[PrimeQ[t + 3^m], Return[m]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 01 2023 *)

A159266(n,m=0)=until( is/*pseudo*/prime(2^n+3^m++),);m /* 2nd optional arg allows us to resume search after a given m and thus (when set to previous result) the list of all m yielding primes */

a(n) = min { m>0 | 2^n+3^m is prime } = A123340(n) whenever the latter is > 1.

A159625 Numbers n such that 2^x + 3^y is never prime when max(x,y) = n.

Original entry on oeis.org

1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 6578, 7251, 7406, 7642, 8218, 8331, 9475, 9578, 9749, 10735

Offset: 1

David Broadhurst, Apr 17 2009

Mark Underwood found that for each nonnegative integer n < 1421 there is at least one prime of the form 2^m + 3^n or 2^n + 3^m with m not exceeding n.
This sequence consists of numbers for which there is no such prime.
David Broadhurst estimated that a fraction in excess of 1/800 of the natural numbers belongs to this sequence and found 17 instances with n < 10^4.
For each of the remaining 9983 nonnegative integers n < 10^4, a prime or probable prime of the form 2^x + 3^y was found with max(x,y) = n.
Each probable prime was subjected to a combination of strong Fermat and strong Lucas tests.

			a(3) = 4980, since there is no prime of the form 2^m + 3^4980 or 2^4980 + 3^m with m < 4981 and 4980 is the third number n such that 2^x + 3^y is never prime when max(x,y) = n.

Broadhurst's heuristic in the PrimeNumbers list. [Broken link]
Maximilian Hasler, Mike Oakes, Mark Underwood, David Broadhurst and others, Primes of the form (x+1)^p-x^p, digest of 22 messages in primenumbers Yahoo group, Apr 5 - May 7, 2009. [Cached copy]
Underwood's posting in the PrimeNumbers list
A list of 9983 primes or probable primes for the excluded cases with n < 10^4

Cf. A159270, A159266, A123359.

a(18) from Giovanni Resta, Apr 09 2014

A159270 Number of positive integers m<=n such that 2^m+3^n or 2^n+3^m is prime.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 5, 3, 5, 3, 6, 6, 4, 4, 7, 6, 8, 4, 4, 4, 4, 6, 7, 2, 4, 5, 6, 6, 8, 6, 10, 3, 3, 5, 6, 4, 9, 6, 9, 7, 5, 5, 9, 6, 8, 7, 7, 10, 4, 5, 8, 9, 1, 8, 6, 6, 7, 7, 10, 5, 5, 4, 10, 8, 7, 8, 8, 2, 3, 8, 8, 8, 5, 6, 7, 5, 10, 6, 7, 7, 8, 10, 10, 9, 10, 5, 7, 5, 5, 6, 9, 6, 5, 5, 12, 3, 7, 6, 8, 9

Offset: 0

M. F. Hasler, Apr 09 2009

nonn

Zeros occur at n = 0 and for n in A159625.

For the values in A159625 both A159266(x) and A123359(x) are greater than x. - Michel Marcus, Apr 07 2014

Michel Marcus, Table of n, a(n) for n = 0..2000
David Broadhurst, 2^m+3^n and 2^n+3^m, Prime numbers and primality testing Group, Apr 11 2009. [Broken link]
Maximilian Hasler, Mike Oakes, Mark Underwood, David Broadhurst and others, Primes of the form (x+1)^p-x^p, digest of 22 messages in primenumbers Yahoo group, Apr 5 - May 7, 2009. [Cached copy]
Mark Underwood, Re: primes of the form (x+1)^p-x^p, Prime numbers and primality testing Group, Apr 07 2009. [Broken link]

Cf. A159266, A123359, A159625.

A159270(n)=sum(m=1,n, ispseudoprime(2^n+3^m) || ispseudoprime(3^n+2^m))

A159269 Least positive integer such that 2^m+3^n or 2^n+3^m is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 3, 1, 1, 1, 6, 1, 4, 3, 8, 2, 2, 1, 7, 1, 4, 1, 10, 1, 4, 4, 8, 15, 6, 1, 2, 3, 8, 3, 3, 2, 6, 3, 11, 6, 2, 5, 4, 18, 4, 12, 6, 26, 1, 4, 2, 9, 6, 4, 10, 18, 1, 4, 6, 2, 1, 8, 10, 26, 12, 17, 12, 10, 4, 13, 3, 7, 9, 11, 4, 2, 17, 1, 7, 3, 2, 3, 26, 22, 6, 12, 8, 9

Offset: 0

M. F. Hasler, Apr 08 2009

nonn

It seems that a(n)<=n for all n>0.

			a(0)=1 since 2^1+3^0=3 is prime.
a(1)=1 since 2^1+3^1=5 is prime.
a(2)=1 since 2^2+3^1=7, or 2^1+3^2=11, is prime. (Only one prime is required).
a(3)=1 since 2^3+3^1=11 and also 2^1+3^3=29, are prime.
a(4)=1 since 2^4+3^1=19 (and also 2^1+3^4=83) are prime.
a(5)=2 is the least integer m such that 2^5+3^m (=41) is prime and 2^m+3^5 is not prime until A159267(5)=4.

A159269(n,m=0)=until( is/*pseudo*/prime(2^n+3^m++) || is/*pseudo*/prime(3^n+2^m),);m

a(n) = min { A123359(n), A159266(n) }

cp's OEIS Frontend

A159266 Least positive integer m such that 2^n+3^m is prime.

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A159625 Numbers n such that 2^x + 3^y is never prime when max(x,y) = n.

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A159270 Number of positive integers m<=n such that 2^m+3^n or 2^n+3^m is prime.

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PARI

A159269 Least positive integer such that 2^m+3^n or 2^n+3^m is prime.

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