A111192 -id:A111192 - cp's OEIS frontend

A143206 Product of the n-th cousin prime pair.

21, 77, 221, 437, 1517, 2021, 4757, 6557, 9797, 11021, 12317, 16637, 27221, 38021, 50621, 53357, 77837, 95477, 99221, 123197, 145157, 159197, 194477, 210677, 216221, 239117, 250997, 378221, 416021, 455621, 549077, 576077, 594437, 680621

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Intersection of A143203 and A001358.

Sum_{n>=2} 1/a(n) > 0.02187310784. - R. J. Mathar, Jan 23 2013

			a(1) = 3*7 = 3*(3+4) = 21;
a(2) = 7*11 = 7*(7+4) = 77;
a(3) = 13*17 = 13*(13+4) = 221;
a(4) = 19*23 = 19*(19+4) = 437.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A037074, A111192.

a143206 n = a143206_list !! (n-1)
a143206_list = (3*7) : f a000040_list where
   f (p:ps@(p':_)) | p'-p == 4 = (p*p') : f ps
                   | otherwise = f ps
-- Reinhard Zumkeller, Sep 13 2011

[(p*(p+4)): p in PrimesUpTo(1000)| IsPrime(p+4)]; // Vincenzo Librandi, Jan 04 2018

fQ[n_] := Block[{fi = FactorInteger@ n}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {4}]; Select[ Range@ 700000, fQ] (* Robert G. Wilson v, Feb 08 2012 *)

lista(nn) = forprime(p=2, nn, if (isprime(q=p+4), print1(p*q, ", "))); \\ Michel Marcus, Jan 04 2018

a(n) = A023200(n)*A046132(n).

A104229 Primes equal to the product of two successive sexy primes plus 6.

Original entry on oeis.org

61, 97, 193, 397, 673, 1153, 1597, 1933, 4093, 7393, 12097, 37633, 64513, 70753, 96097, 122497, 126733, 136897, 190093, 211597, 256033, 313597, 329473, 348097, 430333, 541693, 781453, 891133, 988033, 1267873, 1416097, 1674433, 2102497

Offset: 1

Giovanni Teofilatto, Apr 02 2005

Primes of the form 6 + A111192(i). - R. J. Mathar, Nov 26 2008

All numbers in this sequence are of the form 12n + 1. Also, as one would expect from a random distribution of sexy prime pairs, with the exception of 61, in decimal two thirds of these numbers end in 3, and the other third end in 7. - Daniel Mondot, Apr 29 2024

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A023201, A046117, A104227, A104228, A111192.

Extended by R. J. Mathar, Nov 26 2008

A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

Original entry on oeis.org

55, 91, 187, 247, 275, 391, 605, 637, 667, 1147, 1183, 1375, 1591, 1927, 2057, 2491, 3025, 3127, 3179, 3211, 4087, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7387, 8281, 8993, 9991, 10807, 11227, 12091, 15125, 15341, 15379, 17947, 19343, 22627, 23707

Offset: 1

Reinhard Zumkeller, Jul 30 2008

nonn

Subsequence of A007774.
A111192 is a subsequence.
Subsequence of A195118. - Reinhard Zumkeller, Sep 13 2011

			a(1) = 55 = 5 * 11 = A023201(1) * A046117(1).
a(2) = 91 = 7 * 13 = A023201(2) * A046117(2).
a(3) = 187 = 11 * 17 = A023201(3) * A046117(3).
a(4) = 247 = 13 * 19 = A023201(4) * A046117(4).
a(5) = 275 = 5^2 * 11 = A023201(1)^2 * A046117(1).
a(6) = 391 = 17 * 23 = A023201(5) * A046117(5).
a(7) = 605 = 5 * 11^2 = A023201(1) * A046117(1)^2.
a(8) = 637 = 7^2 * 13 = A023201(2)^2 * A046117(2).
a(9) = 667 = 23 * 29 = A023201(6) * A046117(6).
a(10) = 1147 = 31 * 37 = A023201(7) * A046117(7).

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]
Index entries for primes, gaps between.

Cf. A001221, A006530, A007774, A020639, A023201, A027748, A046117, A111192, A143201, A195118.

a143205 n = a143205_list !! (n-1)
a143205_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 6 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

okQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]},Length[fi]==2 && Last[fi]-First[fi]==6]; Select[Range[25000],okQ]  (* Harvey P. Dale, Apr 18 2011 *)

A143201(a(n)) = 7.
A020639(a(n)) in A023201 and A006530(a(n)) in A046117.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023201(n)+2)^2-9) = 0.058842810164... . - 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 *)

A210477 Product of adjacent primes with a gap of 6.

Original entry on oeis.org

667, 1147, 2491, 3127, 4087, 5767, 7387, 17947, 23707, 25591, 28891, 30967, 55687, 64507, 67591, 70747, 75067, 111547, 126727, 136891, 141367, 148987, 190087, 198907, 256027, 295927, 313591, 320347, 329467, 348091, 355207, 364807, 372091, 422491, 430327, 462391, 532891

Offset: 1

R. J. Mathar, Jan 23 2013

nonn

Subsequence of A111192.

Sum_{n>=1} 1/a(n) > 0.00405067912.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A210477 := proc(n)
    A031924(n)*(A031924(n)+6) ;
end proc:
seq(A210477(n),n=1..100) ; # R. J. Mathar, Jan 23 2013

Times@@@Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]]==6&] (* Harvey P. Dale, Aug 22 2019 *)

a(n) = A031924(n)*(A031924(n)+6).

A254690 Number of decompositions of 2n into a sum of two primes p1 < p2 such that p2-p1 is between a pair of sexy primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 3, 3, 2, 3, 5, 4, 2, 5, 2, 3, 5, 2, 4, 6, 2, 5, 6, 3, 4, 6, 4, 3, 7, 2, 3, 8, 3, 4, 6, 2, 5, 7, 3, 3, 7, 5, 5, 8, 4, 3, 9, 2, 4, 8, 2, 5, 7, 2, 2, 4, 6, 5, 7, 4, 2, 10, 2, 4, 7, 1, 6, 7, 1, 4, 10, 7, 3, 8

Offset: 1

Lei Zhou, Feb 05 2015

"A pair of sexy primes" is defined as two primes p_a < p_b such that p_b = p_a + 6, with p_a from A023201. See the Weisstein link.
The restriction is therefore p_a < p2 - p1 < p_a + 6 for p_a from A023201.
Conjecture: when n>=7, a(n)>0.
The products of sexy prime pairs are listed in A111192.

			n=7, 2n=14=3+11. 11-3=8, 5<8<11 where {5, 11} is a pair of sexy primes. So a(7)=1.
n=8, 2n=16=3+13=5+11. 13-3=10, 5<10<11; 11-5=6, 5<6<11, where {5, 11} is a pair of sexy primes: two cases found, so a(8)=2.
n=17, 2n=34=3+31=5+29=11+23. 31-3=28, 23<28<29; 29-5=24, 23<24<29; 23-11=12, 7<12<13; where {23,29} and {7,13} are sexy prime pairs: three cases found, so a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000
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].
Lei Zhou, Plot of a(n) for n <= 20000.

Cf. A023201, A045917, A111192.

Table[e = 2 n; ct = 0; p1 = 1; While[p1 = NextPrime[p1]; p1 < n, p2 = e - p1; If[PrimeQ[p2], c = p2 - p1; If[c >= 6, found = 0; Do[If[PrimeQ[c - i] && PrimeQ[c + 6 - i], found = 1], {i, 1, 5, 2}]; If[found == 1, ct++]]]]; ct, {n, 1, 100}]

Edited by Wolfdieter Lang, Feb 20 2015

A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.

Original entry on oeis.org

110, 182, 374, 494, 782, 1334, 2294, 3182, 3854, 4982, 6254, 7905, 7917, 8174, 9782, 11534, 12765, 14774, 15810, 15834, 18705, 19982, 20757, 21614, 22330, 22454, 24182, 25530, 27265, 28210, 30381, 30597, 32637, 35894, 37410, 40205, 41181, 41514, 43005, 47414, 49210

Offset: 1

David A. Corneth, Apr 09 2019

nonn

This sequence is a primitive subsequence of A200070. If p|a(n) for some prime p then p*a(n) is in A200070.
From Robert Israel, Apr 09 2019: (Start)
All terms have at least three prime factors.
The number of prime factors is odd if and only if the term is even.
The terms with three prime factors are 2*A111192. (End)

			110 = 2 * 5 * 11 is squarefree. The minimal and maximal prime divisors of 110 are 2 and 11 respectively. Twice their difference is 2 * (11-2) = 18 which is also the sum of the distinct prime divisors of 110; 2 + 5 + 11 = 18.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A200070, A111192.

filter:= proc(n) local P;
if not numtheory:-issqrfree(n) then return false fi;
P:= numtheory:-factorset(n);
  convert(P, `+`) = 2*(max(P)-min(P))
end proc:
select(filter, [$1..50000]);# Robert Israel, Apr 09 2019

Select[Select[Range[2, 5*10^4], SquareFreeQ], Total@ # == 2 (Last@ # - First@ #) &@ FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Apr 11 2019 *)

is(n) = if(!issquarefree(n), return(0)); my(f=factor(n)[, 1]~); sum(i=1, #f, f[i])==2*(f[#f]-f[1])
forcomposite(c=1, 50000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Apr 11 2019

A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)p(p + 6).

Original entry on oeis.org

935, 1729, 4301, 11339, 49321, 102131, 146969, 298351, 386389, 1089019, 1221191, 3864241, 5171489, 12640949, 16965341, 18181979, 21243961, 43974269, 51881689, 178433279, 208506509, 223626691, 230324329, 270816731, 278421569, 393806449, 849244031, 932539661

Offset: 1

Matthew Goers, Oct 25 2023

nonn

			5, 11, and 17 are primes p, p+6, p+12, called a sexy prime triple. 5*11*17 = 935, so 935 is a term.
7, 13, and 19 are the second set of sexy prime triples. 7*13*19=1729, so 1729 is the second term.

Matthew Goers, Table of n, a(n) for n = 1..1000 [a(535) corrected by _Georg Fischer_, Sep 21 2024]

Cf. A006489, A111192. Subsequence of A007304.

(#*(#^2 - 36)) & /@ Select[Prime[Range[180]], PrimeQ[# - 6] && PrimeQ[# + 6] &] (* Amiram Eldar, Oct 27 2023 *)

apply(x->x*(x-6)*(x+6), select(x->(isprime(x-6) && isprime(x) && isprime(x+6)), [1..1000])) \\ Michel Marcus, Oct 27 2023

a(n) = (A006489(n) - 6)*A006489(n)*(A006489(n) + 6).

cp's OEIS Frontend

A143206 Product of the n-th cousin prime pair.

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A104229 Primes equal to the product of two successive sexy primes plus 6.

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A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

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

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A210477 Product of adjacent primes with a gap of 6.

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A254690 Number of decompositions of 2n into a sum of two primes p1 < p2 such that p2-p1 is between a pair of sexy primes.

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A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.

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Maple

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A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)*p*(p + 6).

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A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)p(p + 6).