A240507 -id:A240507 - cp's OEIS frontend

A240503 Numbers k such that 5^k - 4^k - 3^k - 2^k - 1 is prime.

4, 8, 72, 396, 4428, 63152

Offset: 1

Derek Orr, Apr 06 2014

a(7) > 70000. - Giovanni Resta, Apr 09 2014

a(7) > 110000. - Michael S. Branicky, Jul 06 2024

			5^4-4^4-3^4-2^4-1 = 271 is prime. Thus, 4 is a term of this sequence.

Cf. A000043, A240507.

for(n=1,7500,if(ispseudoprime(5^n-4^n-3^n-2^n-1),print(n)))

a(6) from Giovanni Resta, Apr 09 2014

A240747 Least number k > 0 such that n^k - (n-1)^k - ... - 3^k - 2^k - 1 is prime, or 0 if no such k exists.

Original entry on oeis.org

2, 0, 2, 4, 4, 0, 0, 0, 8, 0, 0, 12

Offset: 2

Derek Orr, Apr 11 2014

a(n) is definitely 0 for n == 3 mod 4 or 0 mod 4 (except a(4) = 2). This is the case for n = 3; 7,8; 11,12; 15,16; 19,20; ...
Further, if n is not squarefree, then a(n) = 0. Thus, if n is in A013929, then a(n) = 0. This is the case for n = 8, 9, 12, 16, 18, 20, 24, ...
Let S = n^k - (n-1)^k - ... - 3^k - 2^k - 1. Then S is divisible by gpf(n) when k is not a multiple of gpf(n)-1, where gpf(x) denotes the greatest prime factor of x. This means that if a(n) is not 0, then a(n) must be a multiple of gpf(n)-1. Note that this holds with the previous findings.
For n <= 200, if n = {21, 26, 29, 34, 38, 58, 61, 62, 65, 70, 74, 86, 89, 97, 101, 102, 106, 110, 133, 137, 142, 146, 157, 173, 178, 181, 182, 185, 190}, there is a pattern with the factorization of S when k is a multiple of gpf(n)-1. Thus, a(n) = 0 is definite for these n-values.
For other n-values <= 200, there is not a concrete pattern when k is a multiple of gpf(n)-1. If n = 14, a(n) > 30000, if n = {33, 37, 42, 46, 53, 57, 69, 73, 77, 78, 82, 85, 93, 94}, a(n) > 8000, and if n = {105, 109, 113, 114, 118, 122, 129, 130, 134, 138, 141, 145, 149, 154, 158, 161, 165, 166, 170, 174, 177, 186, 193, 197}, a(n) > 6000. Here, a(n) could still be nonzero.
For n <= 200, it is known that a(17) = 16, a(22) = 3190, a(30) = 124, a(41) = 520, a(66) = 260, and a(194) = 288.

			5^1 - 4^1 - 3^1 - 2^1 - 1 = -5 is not prime;
5^2 - 4^2 - 3^2 - 2^2 - 1 = -5 is not prime;
5^3 - 4^3 - 3^3 - 2^3 - 1 = 25 is not prime;
5^4 - 4^4 - 3^4 - 2^4 - 1 = 271 is prime. Thus, a(5) = 4.

Cf. A240748, A240503, A240507.

s(n)=for(k=1,6000,if(ispseudoprime(n^k-sum(i=1,n-1,i^k)),return(k)))
n=1; while(n<200,print(s(n));n+=1)

A240748 Numbers n such that n^k - (n-1)^k - ... - 3^k - 2^k - 1 is prime for some k.

Original entry on oeis.org

2, 4, 5, 6, 10, 13

Offset: 1

Derek Orr, Apr 11 2014

a(7) > 13. See A240747 for more information.
a(n) is also the n-values such that A240747(n) is nonzero.
It is known that a(n) == 1 mod 4 or 2 mod 4 (except a(2) = 4).
If n is not squarefree, then n is not a member of this sequence.
It is known that 17, 22, 30, 41, 66, and 194 are members of this sequence.

			There are primes of the form 2^k-1 (A000043) so 2 is a member of this sequence.

Cf. A000043, A240747, A240507, A240503.

s(n) = for(k=1,6000,if(ispseudoprime(n^k-sum(i=1,n-1,i^k)),return(k)))
n=1; while(n<200,if(s(n),print(n));n+=1)

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A240503 Numbers k such that 5^k - 4^k - 3^k - 2^k - 1 is prime.

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A240747 Least number k > 0 such that n^k - (n-1)^k - ... - 3^k - 2^k - 1 is prime, or 0 if no such k exists.

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A240748 Numbers n such that n^k - (n-1)^k - ... - 3^k - 2^k - 1 is prime for some k.

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