A252371 -id:A252371

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A251727

Numbers n > 1 for which gpf(n) > spf(n)^2, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

+10
13

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126, 129, 130, 132, 134, 136, 138, 140, 141, 142, 145, 146, 148, 150, 152 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`Numbers n > 1 for which the smallest r such that r^k <= spf(n) and gpf(n) < r^(k+1) [for some k >= 0] is gpf(n)+1. Here spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n). (The original, equivalent definition of the sequence).` `Numbers n > 1 such that A252375(n) = 1 + A006530(n). Equally, one can substitute A251725 for A252375.` `Numbers n > 1 for which there doesn't exist any r <= gpf(n) such that r^k <= spf(n) and gpf(n) < r^(k+1), for some k >= 0, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10000`
	PROG	`(Scheme with Antti Karttunen's IntSeq-library, three alternative versions)` `(define A251727 (MATCHING-POS 1 2 (lambda (n) (> (A006530 n) (A000290 (A020639 n))))))` `(define A251727 (MATCHING-POS 1 2 (lambda (n) (= (A251725 n) (+ 1 (A006530 n))))))` `(define A251727 (MATCHING-POS 1 2 (lambda (n) (= (A252375 n) (+ 1 (A006530 n))))))`
	CROSSREFS	`Complement: A251726. Subsequence: A138511.` `Gives the positions of zeros in A252374 following its initial term.` `Cf. A252371 (difference between the prime indices of gpf and spf of each a(n)).` `Cf. A006530, A020639, A252375, A251725.` `Related permutations: A252757-A252758.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Dec 17 2014. A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section`
	STATUS	`approved`

A252370

a(n) = A243055(A251726(n)).

+10
3

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 2, 0, 3, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 2, 5, 1, 0, 0, 0, 2, 0, 1, 0, 0, 6, 3, 0, 0, 0, 0, 0, 4, 1, 0, 0, 1, 1, 2, 0, 0, 0, 5, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 1, 0, 0, 0, 6, 3, 0, 1, 7, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 4, 0, 0, 8, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,17`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..10000`
	FORMULA	`a(n) = A243055(A251726(n)).` `a(n) = A243055(A066048(A251726(n))). [Result depends only on the smallest and the largest prime factor of A251726(n).]`
	PROG	`(Scheme) (define (A252370 n) (A243055 (A251726 n)))`
	CROSSREFS	`Cf. A252371, A243055, A251726, A066048.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Dec 17 2014`
	STATUS	`approved`

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