A167504 -id:A167504 - cp's OEIS frontend

A167505 Number of primes of the form 2^(n-m) 3^m + 1, 0 <= m <= n.

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 2, 0, 5, 1, 1, 6, 6, 2, 5, 1, 1, 2, 5, 0, 3, 1, 2, 2, 4, 2, 5, 3, 0, 1, 6, 2, 8, 2, 2, 3, 9, 1, 7, 4, 4, 0, 6, 0, 3, 3, 2, 7, 8, 1, 4, 4, 1, 2, 6, 0, 5, 2, 4, 2, 2, 1, 11, 4, 3, 1, 3, 0, 6, 0, 2, 3, 4, 1, 6, 0, 4, 3, 8, 2, 2, 2, 2, 1, 3, 1, 2, 3, 3, 1, 3, 0, 5, 1, 2, 1, 7, 0, 7, 2, 2

Offset: 1

M. F. Hasler, Nov 19 2009

nonn

M. Underwood, 2^a*3^b one away from a prime. Post to primenumbers group, Nov. 19, 2009.
Mark Underwood, Jens Kruse Andersen, 2^a*3^b one away from a prime, digest of 3 messages in primenumbers Yahoo group, Nov 19, 2009.

Cf. A167504, A167506.

A167505(n)=sum(b=0,n,ispseudoprime(3^b<<(n-b)+1))

A167506 Number of m >= 0, m <= n such that 2^(n-m)3^m + 1 or 2^(n-m)3^m - 1 is prime.

Original entry on oeis.org

2, 2, 3, 4, 5, 2, 6, 7, 6, 3, 5, 1, 10, 1, 3, 8, 10, 2, 7, 4, 3, 2, 9, 1, 5, 1, 5, 5, 6, 2, 13, 6, 3, 1, 9, 5, 10, 2, 5, 7, 13, 1, 11, 6, 4, 0, 12, 1, 8, 3, 7, 9, 11, 1, 7, 7, 4, 2, 11, 1, 11, 2, 9, 6, 6, 1, 13, 8, 8, 1, 9, 2, 13, 0, 5, 4, 12, 1, 11, 2, 10, 3, 13, 2, 8, 2, 4, 6, 9, 1, 6, 7, 4, 1, 8, 1, 9, 1

Offset: 1

M. F. Hasler, Nov 19 2009

nonn

M. Underwood observed that for all primes p < 3187 we have a(p) > 1, and asks whether there is a prime such that a(p) = 0. (This is equivalent to A167504(p) = A167505(p) = 0.)

Robert Israel, Table of n, a(n) for n = 1..2000
Mark Underwood, 2^a*3^b one away from a prime. Post to primenumbers group, Nov. 19, 2009.
Mark Underwood and Jens Kruse Andersen, 2^a*3^b one away from a prime, digest of 3 messages in primenumbers Yahoo group, Nov 19, 2009.

Cf. A167504, A167505.

g:= proc(n,m) local t; t:= 2^(n-m)*3^m; isprime(t+1) or isprime(t-1) end proc:
f:= proc(n) nops(select(m -> g(n,m), [$0..n])) end proc:
map(f, [$1..100]); # Robert Israel, Mar 11 2025

A167505(n)=sum( b=0,n, ispseudoprime(3^b<<(n-b)-1) || ispseudoprime(3^b<<(n-b)+1))

max { A167504(n), A167505(n) } <= a(n) <= A167504(n)+A167505(n).

A381970 Numbers k such that there are no primes of the form 2^(k-m)3^m + 1 or 2^(k-m)3^m - 1 for 0 <= m <= k.

Original entry on oeis.org

46, 74, 102, 118, 130, 142, 162, 165, 166, 186, 200, 234, 242, 252, 258, 306, 318, 358, 370, 374, 414, 462, 478, 494, 506, 518, 522, 538, 540, 550, 578, 594, 618, 630, 654, 662, 666, 672, 690, 738, 750, 768, 778, 780, 790, 802, 810, 826, 834, 858, 886, 902, 912, 938, 942, 958, 982, 990, 1002

Offset: 1

Robert Israel, Mar 11 2025

nonn

Numbers k such that A167506(k) = 0.
Mark Underwood observed that there are no primes < 3187 in this sequence.
The first two odd terms are 165 and 3615.
One reason why there are so many more even terms than odd terms is that if k is even, one of 2^(k-m)*3^m + 1 and 2^(k-m)*3^m - 1 is always divisible by 5, while if k is odd, neither is divisible by 5.

Robert Israel, Table of n, a(n) for n = 1..200
Mark Underwood and Jens Kruse Andersen, 2^a*3^b one away from a prime, digest of 3 messages in primenumbers Yahoo group, Nov 19, 2009.

Cf. A167506.

g:= proc(n,m) local t; t:= 2^(n-m)*3^m; isprime(t+1) or isprime(t-1) end proc:
filter:= proc(n) not ormap(m -> g(n,m), [$0..n]) end proc:
select(filter, [$1..1000]);

cp's OEIS Frontend

A167505 Number of primes of the form 2^(n-m) 3^m + 1, 0 <= m <= n.

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A167506 Number of m >= 0, m <= n such that 2^(n-m)3^m + 1 or 2^(n-m)3^m - 1 is prime.

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A381970 Numbers k such that there are no primes of the form 2^(k-m)3^m + 1 or 2^(k-m)3^m - 1 for 0 <= m <= k.

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Maple

cp's OEIS Frontend

A167505 Number of primes of the form 2^(n-m) 3^m + 1, 0 <= m <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A167506 Number of m >= 0, m <= n such that 2^(n-m)*3^m + 1 or 2^(n-m)*3^m - 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A381970 Numbers k such that there are no primes of the form 2^(k-m)*3^m + 1 or 2^(k-m)*3^m - 1 for 0 <= m <= k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A167506 Number of m >= 0, m <= n such that 2^(n-m)3^m + 1 or 2^(n-m)3^m - 1 is prime.

A381970 Numbers k such that there are no primes of the form 2^(k-m)3^m + 1 or 2^(k-m)3^m - 1 for 0 <= m <= k.