cp's OEIS Frontend

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

%I A138511 #64 Aug 18 2025 02:14:17
%S A138511 10,14,22,26,33,34,38,39,46,51,57,58,62,69,74,82,86,87,93,94,106,111,
%T A138511 118,122,123,129,134,141,142,145,146,155,158,159,166,177,178,183,185,
%U A138511 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
%N A138511 Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.
%C A138511 From _Antti Karttunen_, Dec 17 2014, further edited Jan 01 & 04 2014: (Start)
%C A138511 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.
%C A138511 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.
%C A138511 (End)
%C A138511 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.
%H A138511 Reinhard Zumkeller, <a href="/A138511/b138511.txt">Table of n, a(n) for n = 1..10000</a>
%F A138511 Other identities. For all n >= 1 it holds that:
%F A138511 A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.
%e A138511 See A138510.
%t A138511 Select[Range[305],PrimeOmega[#]==2&&Divisors[#][[-2]]>Divisors[#][[-3]]^2&&SquareFreeQ[#]&] (* _James C. McMahon_, Jun 07 2025 *)
%o A138511 (Haskell)
%o A138511 a138511 n = a138511_list !! (n-1)
%o A138511 a138511_list = filter f [1..] where
%o A138511                       f x = p ^ 2 < q && a010051' q == 1
%o A138511                             where q = div x p; p = a020639 x
%o A138511 -- _Reinhard Zumkeller_, Jan 06 2015
%o A138511 (Scheme)
%o A138511 ;; Scheme with _Antti Karttunen_'s IntSeq-library.
%o A138511 (define A138511 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (= (A252375 n) (+ 1 (A006530 n)))))))
%o A138511 (define A138511 (COMPOSE A001358 (MATCHING-POS 1 1 (lambda (n) (= (A138510 n) (+ 1 (A006530 (A001358 n))))))))
%o A138511 ;; _Antti Karttunen_, Dec 16-17 2014
%o A138511 (Scheme)
%o A138511 ;; Scheme with _Antti Karttunen_'s IntSeq-library.
%o A138511 (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
%o A138511 (PARI) isok(s) = my(f=factor(s)); (bigomega(f) == 2) && (#f~ == 2) && (f[1,1]^2 < f[2, 1]); \\ _Michel Marcus_, Sep 15 2020
%Y A138511 Cf. A138510.
%Y A138511 Complement of A251728 in A001358.
%Y A138511 Subsequence of A088381.
%Y A138511 An intersection of A001358 (semiprimes) and A251727.
%Y A138511 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.
%Y A138511 Also an intersection A001358 and A245729.
%Y A138511 Cf. also A006530, A020639, A054272, A083221, A083140, A084127, A138510, A162319, A174956, A251719.
%K A138511 nonn,base,changed
%O A138511 1,1
%A A138511 _Reinhard Zumkeller_, Mar 21 2008
%E A138511 Wrong comment corrected by _Reinhard Zumkeller_, Dec 16 2014
%E A138511 New definition by _Antti Karttunen_, Jan 01 2015; old definition moved to comment.
%E A138511 More terms from _Antti Karttunen_, Jan 09 2015