A087126 -id:A087126 - cp's OEIS frontend

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333, 1641, 1807, 2163, 2757, 3423, 3661, 4423, 4971, 5257, 6163, 6807, 7833, 9313, 10101, 10507, 11343, 11773, 12657, 16003, 17031, 18633, 19183, 22053, 22651, 24493, 26407, 27723, 29757, 31863

Offset: 1

Alexander Adamchuk, Aug 02 2006

nonn

Prime terms belong to A074268, which is a subset of A002383, A087126, A034915, A085104.

In every interval of prime(n)^2 integers, a(n) is the number that are not divisible by prime(n) plus the number that are divisible by prime(n)^2. - Peter Munn, Dec 12 2020

Cf. A002061, A074268, A002383, A087126, A034915, A068468, A085104, A083558.

Mathematica
```
Table[Prime[n]^2-Prime[n]+1,{n,1,100}]
```

a(n) = {my(p = prime(n)); p^2 - p + 1; } \\ Amiram Eldar, Nov 07 2022

a(n) = prime(n)^2 - prime(n) + 1.
a(n) = A036689(n)+1. - R. J. Mathar, Aug 13 2019
Product_{n>=1} (1 - 1/a(n)) = zeta(6)/(zeta(2)*zeta(3)) (A068468). - Amiram Eldar, Nov 07 2022

A087139 Least k>1 such that p^k - p^(k-1) + 1 is prime for p = prime(n).

Original entry on oeis.org

2, 2, 3, 2, 11, 2, 5, 30, 15, 3, 6, 10, 81, 3, 17, 961, 15, 7, 2, 5, 6, 2, 3, 3, 12, 3, 57, 5, 16, 5, 166, 15, 13, 2, 3, 2, 30, 2, 25, 3, 47, 3, 3, 2, 521, 9, 3, 15, 17, 42, 17, 51, 39

Offset: 1

T. D. Noe, Aug 18 2003

The next term in this sequence, a(54) for the prime p=251, is greater than 73000.
Is there a prime p such that p^k - p^(k-1) + 1 is composite for all k > 1? For the related question of Sierpinski numbers (n such that n*2^k+1 is composite for all k ), the answer is yes.
If n=251^k-251^(k-1)+1 is prime then k mod 10 = 1,5,7 or 9 because n mod 3 = 0 iff k is even and n mod 11 = 0 iff k mod 5 = 3. More exponents can be cleared this way. - Bernardo Boncompagni, Oct 23 2005
Note that k cannot be 8, 14, 20, ... (i.e. k == 2 mod 6) because then p^2 - p + 1 divides p^k - p^(k-1) + 1. - T. D. Noe, Aug 31 2006

See A087126.

Cf. A040076 (Sierpinski numbers), A087126 (primes of the form p^k - p^(k-1) + 1).

Cf. A122396.

lst={}; Do[p=Prime[n]; i=2; While[m=p^i-p^(i-1)+1; !PrimeQ[m], i++ ]; AppendTo[lst, i], {n, 53}]; lst

cp's OEIS Frontend

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula