A243896 -id:A243896 - cp's OEIS frontend

A243895 a(n) = prime(n^2-1).

5, 19, 47, 89, 149, 223, 307, 409, 523, 659, 823, 997, 1187, 1423, 1613, 1877, 2141, 2423, 2731, 3079, 3457, 3797, 4201, 4621, 5039, 5507, 5987, 6473, 6991, 7561, 8147, 8731, 9337, 9929, 10613, 11317, 12043, 12739, 13487, 14323, 15091, 15859, 16741

Offset: 2

Freimut Marschner, Jun 17 2014

nonn

The prime numbers prime(k-1), prime(k) = A001248 and prime(k+1) = A243896 with k = n^2 are building a triple of successive prime numbers. Remark: prime(n^2-1) is not defined for n=1.

			n = 3, n^2 = 9, n^2-1 = 8, prime(8) = 19.

Freimut Marschner, Table of n, a(n) for n = 2..97

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A011757 (prime(n^2)), A055875 (prime(n^3)), A096327 (prime((prime(n)^2))), A096328 (prime(prime(n)^3)), A038580 (prime(prime(prime(n)))).

Table[Prime[n^2-1],{n,2,50}] (* Harvey P. Dale, Jul 16 2025 *)

a(n) = prime(n^2-1) = prime(A000290(n) - 1) = prime(A005563(n-1)).

A326377 For any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; a(n) = g(f(n) o f(n)) (where o denotes function composition).

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 29, 8, 81, 1100, 59, 48, 101, 195478444, 40425, 16, 157, 648, 229, 440000, 64240097649, 1445390468875226977004, 313, 192, 214358881, 44574662297516497591170630280506162081362246142404, 19683, 9921285858330292941824, 421, 72765000, 547, 32

Offset: 1

Rémy Sigrist, Jul 02 2019

nonn

This sequence is the main diagonal of A326376.

			The first terms, alongside the corresponding polynomials, are:
  n   a(n)  f(n)   f(n) o f(n)
  --  ----  -----  -----------
   1     1      0            0
   2     2      1            1
   3     3      x            x
   4     4      2            2
   5    11    x^2          x^4
   6    12    x+1          x+2
   7    29    x^3          x^9
   8     8      3            3
   9    81    2*x          4*x
  10  1100  x^2+1  x^4+2*x^2+2
  11    59    x^4         x^16
  12    48    x+2          x+4

Cf. A060722, A243896, A326376.

g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))
f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i, 2] * v^(primepi(f[i, 1]) - 1))
a(n) = g(f(n, f(n)))

a(n) = A326376(n, n).
a(2^k) = 2^k for any k >= 0.
a(3^k) = A060722(k) for any k >= 0.
a(prime(k)) = A243896(k) for any k >= 1 (where prime(k) denotes the k-th prime number).

cp's OEIS Frontend

A243895 a(n) = prime(n^2-1).

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