A249744 a(n) = 0 if n is 1 or a prime, otherwise, when n = A020639(n) * A032742(n), a(n) = the largest m < n such that A020639(m) = A020639(n), where A020639(n) and A032742(n) are the smallest prime and the largest proper divisor dividing n.

0, 0, 0, 2, 0, 4, 0, 6, 3, 8, 0, 10, 0, 12, 9, 14, 0, 16, 0, 18, 15, 20, 0, 22, 5, 24, 21, 26, 0, 28, 0, 30, 27, 32, 25, 34, 0, 36, 33, 38, 0, 40, 0, 42, 39, 44, 0, 46, 7, 48, 45, 50, 0, 52, 35, 54, 51, 56, 0, 58, 0, 60, 57, 62, 55, 64, 0, 66, 63, 68, 0, 70, 0, 72, 69, 74, 49, 76, 0, 78, 75, 80, 0, 82, 65, 84, 81, 86, 0, 88, 77, 90, 87, 92, 85, 94, 0, 96, 93, 98, 0, 100

Offset

1,4

Comments

For all composite numbers, a(n) tells what is the previous number processed by the sieve of Eratosthenes, i.e., number which is immediately left of n on the same row where n is in arrays like A083140, A083221.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A020639(n) * A249738(n).

Other identities. For all n >= 1 it holds:

a(2n) = 2n-2.

a(A001248(n)) = A000040(n). [I.e., a(p^2) = p for primes p.]

Mathematica

a[1] = 0; a[_?PrimeQ] = 0; a[n_] := For[p = FactorInteger[n][[1, 1]]; m = n - p, True, m = m - p, If[FactorInteger[m][[1, 1]] == p, Return[m]]]; Array[a, 102] (* Jean-François Alcover, Mar 08 2016 *)

Prog

(Scheme) (define (A249744 n) (* (A020639 n) (A249738 n)))

Crossrefs

Can be used to compute A078898.

Cf. A000040, A001248, A020639, A083140, A083221, A249738, A250469, A250470.

Sequence in context: A130891 A091371 A144775 * A245075 A046666 A276093

Adjacent sequences: A249741 A249742 A249743 * A249745 A249746 A249747

Keyword

nonn

Author

Antti Karttunen, Dec 06 2014

Status

approved