A195106 -id:A195106 - cp's OEIS frontend

A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.

21, 63, 77, 147, 189, 221, 437, 441, 539, 567, 847, 1029, 1323, 1517, 1701, 2021, 2873, 3087, 3757, 3773, 3969, 4757, 5103, 5929, 6557, 7203, 8303, 9261, 9317, 9797, 10051, 11021, 11907, 12317, 15309, 16637, 21609

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Subsequence of A007774.
A033850 is a subsequence.
Subsequence of A195106. - Reinhard Zumkeller, Sep 13 2011

			a(1) = 21 = 3 * 7 = A023200(1) * A046132(1).
a(2) = 63 = 3^2 * 7 = A023200(1)^2 * A046132(1).
a(3) = 77 = 7 * 11 = A023200(2) * A046132(2).
a(4) = 147 = 3 * 7^2 = A023200(1) * A046132(1)^2.
a(5) = 189 = 3*3 * 7 = A023200(1)^3 * A046132(1).
a(6) = 221 = 13 * 17 = A023200(3) * A046132(3).
a(7) = 437 = 19 * 23 = A023200(4) * A046132(4).
a(8) = 441 = 3^2 * 7^2 = A023200(1)^2 * A046132(1)^2.
a(9) = 539 = 7^2 * 11 = A023200(2)^2 * A046132(2).
a(10) = 567 = 3^4 * 7 = A023200(1)^4 * A046132(1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Cousin Primes.
Index entries for primes, gaps between.

Cf. A001221, A006530, A007774, A020639, A023200, A027748, A033850, A046132, A143201, A195106.

a143203 n = a143203_list !! (n-1)
a143203_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 4 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

dpf2Q[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==4]; Select[Range[22000],dpf2Q] (* Harvey P. Dale, Mar 18 2023 *)

A143201(a(n)) = 5.
A020639(a(n)) in A023200 and A006530(a(n)) in A046132.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023200(n)+1)^2-4) = 0.109882433872... . - Amiram Eldar, Oct 26 2024

A195118 Numbers with largest and smallest prime factors differing by 6.

Original entry on oeis.org

55, 91, 187, 247, 275, 385, 391, 605, 637, 667, 1001, 1147, 1183, 1375, 1591, 1925, 1927, 2057, 2431, 2491, 2695, 3025, 3127, 3179, 3211, 4087, 4199, 4235, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7007, 7387, 7429, 8281, 8993, 9625, 9991, 10807, 11011

Offset: 1

Reinhard Zumkeller, Sep 13 2011

nonn

			a(10) = 667 = 23 * 29;
a(11) = 1001 = 7 * 11 * 13;
a(12) = 1147 = 31 * 37;
a(13) = 1183 = 7 * 13^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]

Cf. A195106, A111192; A143205 is a subsequence.

a195118 n = a195118_list !! (n-1)
a195118_list = filter f [3,5..] where
   f x = last pfs - head pfs == 6 where pfs = a027748_row x

spf6Q[n_]:=With[{fi=FactorInteger[n]},fi[[-1,1]]-fi[[1,1]]==6]; Select[Range[12000],spf6Q] (* Harvey P. Dale, May 14 2024 *)

cp's OEIS Frontend

A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.

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