A391072 Numbers of twin prime candidates in primorial interval of prime(n)# after elimination by prime factor prime(n-1).

0, 2, 5, 21, 165, 1755, 25245, 423225, 8709525, 230613075, 6655970475, 230382461925, 8935103590875, 365467529802375, 16378044882073875, 831098660505238125, 47184450367174745625, 2780683625875704924375, 180197415952240353541875, 12412105591337152710384375, 880560223430496312707128125

Offset

1,2

Comments

The number of twin prime candidates remaining in the primorial(n)# interval after sieving with prime(n-1).

Conjecture: This number goes to infinity when n approaches infinity.

The interval in which to count twin prime candidates is to start at prime(n)+1 and to have the length of one primorial, that is to end at the number (n)# + prime(n).

The prime candidates occur in the process of identifying primes with the sieve of Eratosthenes. For each prime p used to identify composite numbers, a number of twin prime candidates are eliminated within the interval by one of its components being identified as composite. A twin prime candidate not eliminated when the process has advanced beyond the moment where its components are smaller than p^2 cannot be eliminated and is identified as a twin prime number.

The number of eliminations of twin prime candidates within the primorial interval that can be attributed to sieving with the prime(n) is given by A121406.

Conjecture: The number of twin prime candidates remaining in the primorial(n)# interval after sieving with the prime(n) is given by A059861.This number goes to infinity when (n) approaches infinity.

When a number of prime twin candidates given by this sequence is eliminated from a prime(n)# interval, by the eliminators given by A121406, the corresponding numbers of remaining candidates are given by A059861.

Links

Paolo Xausa, Table of n, a(n) for n = 1..300

Dennis R. Martin, On the Infinite Series Characterizing the Elimination of Twin Prime Candidates [Cached copy, with permission of the author]

Formula

For n>=3, a(n) = prime(n)*(Product_{i=2..n-1} (prime(i) - 2)), where prime(n) is the n-th prime.

For n>3, a(n) = prime(n)*A059861(n-1) = A059861(n) + A121406(n).

Example

First formula, for n=4, a(4) = 7*1*(3-2)*(5-2) = 21.
Second formula, for n=4, a(4) = (1*(3-2)*(5-2)*(7-2)) + (3*2) = 21.

Mathematica

Join[{0, 2}, Rest[#]*FoldList[Times, Most[#] - 2] & [Prime[Range[2, 20]]]] (* Paolo Xausa, Dec 18 2025 *)

Prog

(PARI) a(n) = if (n==1, 0, if (n==2, 2, prime(n)*prod(i=2, n-1, prime(i) - 2))); \\ Michel Marcus, Dec 10 2025

Crossrefs

Cf. A002110 (primorial), A059861, A121406.

Sequence in context: A144271 A193189 A358316 * A176527 A117261 A108021

Adjacent sequences: A391069 A391070 A391071 * A391073 A391074 A391075

Keyword

nonn

Author

Leif Nordqvist, Nov 27 2025

Status

approved