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 A256448 a(n) = A250474(n+1) - A250477(n). 7
 -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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 LINKS Antti Karttunen, Table of n, a(n) for n = 1..564 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). EXAMPLE 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. PROG (Scheme) (define (A256448 n) (- (A250474 (+ n 1)) (A250477 n))) CROSSREFS Two less than A256469. Cf. A050216, A250474, A250477, A251723, A256446, A256447, A256449. Sequence in context: A091578 A332363 A258249 * A056756 A120861 A236542 Adjacent sequences:  A256445 A256446 A256447 * A256449 A256450 A256451 KEYWORD sign AUTHOR Antti Karttunen, Mar 29 2015 STATUS approved

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Last modified September 16 08:30 EDT 2021. Contains 347469 sequences. (Running on oeis4.)