cp's OEIS Frontend

There are primes p for which there exist a positive integer k and another prime q such that p=q*(2k+1)-2k. See A136020, A091180, A136061 and the subsequent sequences. Such k is called an "order" of the prime p. Note that q is necessarily larger than 2 and that 4*k is necessarily smaller than p-1. A prime may belong to more than one order, but the primes listed in the present sequence do not belong to any order.

As soon as they are larger than 8, all members minus 1 are multiples of 8.

Terms belong to A172186 but not to A344378. Even though a(n)*(a(n)+1)*(2*a(n)+1) is squarefree, Sum_{j=1..a(n)} j^(2k) always has a prime divisor which is smaller than 2*a(n)+3, whatever k. For the integers m such that m*(m+1)*(2*m+1) is nonsquarefree, Sum_{j=1..m} j^(2k) always has a prime divisor which is smaller than 2*m+3, whatever k, because it is divisible by any prime p such that p^2 divides m*(m+1)*(2*m+1).

a(n)*(a(n)+1)*(2a(n)+1) must be squarefree, so this is a subsequence of A172186. It is the complement of A344380 in A172186.

Let L= LCM_j[(p_j-1)/2], where p_j run through the set of the prime divisors of a(n)*(a(n)+1)*(2a(n)+1). For a given member a(n) any admissible k must be a multiple of L, and for any prime p smaller than a(n) such that (p-1)/2 divides L, it holds that p does not divide a(n)-Floor[a(n)/p]. But the converse is not true: 397 is squarefree and satisfies the former condition, but Sum_{j=1..397} j^(2k) is always divisible either by 17 or by 73. 397 is the smallest "false positive" with the above test. Other "false positives" are rather scarce: 397,469,478,561,885,1002,1554,1658,1702,1977,... - René Gy, Apr 15 2025

|s(n,k)| is the unsigned Stirling number of first kind (see A008275), B_k is the Bernoulli number and i^2=-1. All even-indexed terms are positive integers, and the odd-indexed terms are zero. A generating function would be welcomed.

This is the sequence defined by the third-order non-linear recurrence a(n+1) = a(n)*a(n-1) - a(n-2) and a(0)=2, a(1)=3, a(2)=3.

For n>0, k>=0, the k-sequence q_k=lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)) is a periodic integer sequence with period a(n).

a(n) is a divisor of A003418(n-1) and a multiple of A003418(n)/n.

a(n) = A003418(n-1) if n is a member of A027854 (a mutinous number), otherwise a(n) = A003418(n)/q^v where q^v is the highest prime power which divides n.

a(n) = A003418(n-1) iff n is a mutinous number or n is a prime number.

a(n) = A003418(n) iff n is a mutinous number.

lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)) is a divisor of lcm(1,2,...,n)/n, therefore a(n) is also the period of the periodic k-sequence r_k= binomial(k+n,n)*lcm(1,2,...,n)/lcm(k+1,k+2,...,k+n).

Let g be the smallest multiple of A003418(n)/n such that r_g=r_0=1 and r_{g+1}=r_1=gcd(m+1,A003418(n)), then a(n)=g.

a(n+j) is a multiple of binomial(n+j-1,j).

All these statements require proofs.

|s(n,k)| is the unsigned Stirling number of first kind (see A008275), B_k is the Bernoulli number and i^2=-1. All even-indexed terms are positive integers, and the odd-indexed terms are zero. A generating function would be welcomed.

No new term less than 2000000. This sequence is included in A274994 because it can be shown that Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1) == p*((2^(p-1)-1)/p)*(2*((p-1)!+1)/p +(2^(p-1)-1)/p) (mod p^2).

Any prime p divides Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1). But a restricted list of primes p are such that p^2 divides Sum_{k=1..(p-1)/2}(k^(p-2))*(k^(p-1)-1).

Also primes p such that (2^(p-1)-1)/p == 0 (mod p) or 2*((p-1)!+1)/p +(2^(p-1)-1)/p == 0 (mod p), because it can be shown that Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1) == p*((2^(p-1)-1)/p)*(2*((p-1)!+1)/p +(2^(p-1)-1)/p) (mod p^2).

The Wieferich primes (A001220) belong to the sequence.

No more terms up to 2000000, because A280300 has no more terms up to 2000000, and A001220 has no other terms below 4.97*10^17 (see the comments in these sequences). - René Gy, Jan 01 2017

The above congruence is true modulo p for all odd primes. See A089043. But like for Wilson congruence, it is true modulo p^2, for a restricted number of primes. After 53, the next one (if any) seems very far away (>500000).

The fact that the congruence is true modulo p for all odd primes was proved by Lagrange in 1771. Using a theorem of Mathews (1892) and Eisenstein's logarithmetic rule for the Fermat quotient, the condition stated in the definition can be restated as W_p == -2q_p(2) (mod p), where W_p is the Wilson quotient of p (A007619) and q_p(2) is the Fermat quotient of p, base 2 (A007663). - John Blythe Dobson, Jul 31 2017