A256448 a(n) = A250474(n+1) - A250477(n).

-1, 1, 2, 7, 3, 12, 4, 13, 23, 6, 28, 21, 7, 21, 40, 40, 14, 45, 33, 13, 52, 37, 60, 86, 42, 18, 43, 21, 50, 192, 50, 82, 25, 156, 30, 90, 95, 61, 94, 97, 34, 174, 35, 69, 35, 234, 250, 81, 36, 79, 139, 45, 220, 140, 132, 153, 44, 143, 92, 51, 250, 379, 103, 53, 105, 396, 174, 294, 59, 121, 181, 245, 182, 184, 129, 203, 261, 136, 265, 339, 72

Offset: 1

Author

Antti Karttunen, Mar 29 2015

Keywords

• sign

Comments

a(n) tells how many more positive integers there are <= prime(n+1)^2 whose smallest prime factor is at least prime(n+1), as compared to how many positive integers there are <= (prime(n) * prime(n+1)) whose smallest prime factor is at least prime(n).

Conjecture 1: for n >= 2, a(n) > 0.

Conjecture 2: ratio a(n)/A256447 converges towards 1. See the associated plots in A256447 and A256449 and comments in A050216.

As what comes to the second conjecture, it's not necessarily true. See the plots linked into A256468. - Antti Karttunen, Mar 30 2015

Examples

			For n=1, the respective primes are prime(1) = 2 and prime(2) = 3, and the ranges in question are [1, 9] and [1, 6]. The former range contains 4 such numbers whose lpf (A020639) is at least 3, namely {3, 5, 7, 9}, while the latter range contains 5 such numbers whose lpf is at least 2, namely {2, 3, 4, 5, 6}, thus a(1) = 4 - 5 = -1.
For n=2, the respective primes are prime(2) = 3 and prime(3) = 5, and the ranges in question are [1, 25] and [1, 15]. The former range contains 8 such numbers whose lpf is at least 5, namely {5, 7, 11, 13, 17, 19, 23, 25}, while the latter range contains 7 such numbers whose lpf is at least 3, namely {3, 5, 7, 9, 11, 13, 15}, thus a(2) = 8 - 7 = 1.
For n=3, the respective primes are prime(3) = 5 and prime(4) = 7, and the ranges in question are [1, 49] and [1, 35]. The former range contains 13 such numbers whose lpf is at least 7, namely {7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49}, while the latter range contains 11 such numbers whose lpf is at least 5, namely {5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35}, thus a(3) = 13 - 11 = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..564

Crossrefs

Two less than A256469.

Cf. A050216, A250474, A250477, A251723, A256446, A256447, A256449.

Programs

Scheme

(define (A256448 n) (- (A250474 (+ n 1)) (A250477 n)))

Formula

a(n) = A256469(n) - 2.

a(n) = A250474(n+1) - A250477(n).

a(n) = A251723(n) - A256447(n).

a(n) = A256446(n) - A256447(n+1).

a(n) = A256447(n) - A256449(n).