A138511 Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.

10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 291, 295, 298, 302, 303, 305

Offset: 1

Author

Reinhard Zumkeller, Mar 21 2008

Keywords

• nonn
• base
• changed

Comments

From Antti Karttunen, Dec 17 2014, further edited Jan 01 & 04 2014: (Start)

Semiprimes p*q, p < q, such that the smallest r for which r^k <= p and q < r^(k+1) [for some k >= 0] is q+1, and thus k = 0. In other words, semiprimes whose both prime factors do not fit (simultaneously) between any two consecutive powers of any natural number r less than or equal to the larger prime factor. This condition forces the larger prime factor q to be greater than the square of the smaller prime factor because otherwise the opposite condition given in A251728 would hold.

Assuming that A054272(n), the number of primes in interval [prime(n), prime(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true), these are also "unsettled" semiprimes that occur in a square array A083221 constructed from the sieve of Eratosthenes, "above the line A251719", meaning that if and only if row < A251719(col) then a semiprime occurring at A083221(row, col) is in this sequence, and conversely, all the semiprimes that occur at any position A083221(row, col) where row >= A251719(col) are in the complementary sequence A251728.

(End)

Semiprimes p*q, p < q, such that b = q+1 is the minimal base with the property that p and q have equal length representations in base b. This was the original definition, which is based primarily on A138510: A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

Examples

			See A138510.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A138510.

Complement of A251728 in A001358.

Subsequence of A088381.

An intersection of A001358 (semiprimes) and A251727.

Also an intersection of A001358 and A253569, from the latter which this sequence differs for the first time at n=60, where A253569(60) = 290, while here a(60) = 291.

Also an intersection A001358 and A245729.

Cf. also A006530, A020639, A054272, A083221, A083140, A084127, A138510, A162319, A174956, A251719.

Programs

Haskell

a138511 n = a138511_list !! (n-1)
a138511_list = filter f [1..] where
                      f x = p ^ 2 < q && a010051' q == 1
                            where q = div x p; p = a020639 x
-- Reinhard Zumkeller, Jan 06 2015

Mathematica

Select[Range[305],PrimeOmega[#]==2&&Divisors[#][[-2]]>Divisors[#][[-3]]^2&&SquareFreeQ[#]&] (* James C. McMahon, Jun 07 2025 *)

PARI

isok(s) = my(f=factor(s)); (bigomega(f) == 2) && (#f~ == 2) && (f[1,1]^2 < f[2, 1]); \\ Michel Marcus, Sep 15 2020

Scheme

;; Scheme with Antti Karttunen's IntSeq-library.
(define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (= (A252375 n) (+ 1 (A006530 n)))))))
(define A138511 (COMPOSE A001358 (MATCHING-POS 1 1 (lambda (n) (= (A138510 n) (+ 1 (A006530 (A001358 n))))))))
;; Antti Karttunen, Dec 16-17 2014

Scheme

;; Scheme with Antti Karttunen's IntSeq-library.
(define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (> (A006530 n) (A000290 (A020639 n))))))) ;; Based on the new alternative definition - Antti Karttunen, Jan 01 2015

Formula

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

A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

Extensions

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014

New definition by Antti Karttunen, Jan 01 2015; old definition moved to comment.

More terms from Antti Karttunen, Jan 09 2015