A143205 -id:A143205 - cp's OEIS frontend

A143201 Product of distances between prime factors in factorization of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 3, 1, 1, 2, 1, 4, 5, 10, 1, 2, 1, 12, 1, 6, 1, 6, 1, 1, 9, 16, 3, 2, 1, 18, 11, 4, 1, 10, 1, 10, 3, 22, 1, 2, 1, 4, 15, 12, 1, 2, 7, 6, 17, 28, 1, 6, 1, 30, 5, 1, 9, 18, 1, 16, 21, 12, 1, 2, 1, 36, 3, 18, 5, 22, 1, 4, 1, 40, 1, 10, 13, 42, 27, 10, 1, 6, 7, 22

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

a(n) is the product of the sum of 1 and first differences of prime factors of n with multiplicity, with a(n) = 1 for n = 1 or prime n. - Michael De Vlieger, Nov 12 2023.
a(A007947(n)) = a(n);
A006093 and A001747 give record values and where they occur:
A006093(n)=a(A001747(n+1)) for n>1.
a(n) = 1 iff n is a prime power: a(A000961(n))=1;
a(n) = 2 iff n has exactly 2 and 3 as prime factors:
a(A033845(n))=2;
a(n) = 3 iff n is in A143202;
a(n) = 4 iff n has exactly 2 and 5 as prime factors:
a(A033846(n))=4;
a(n) = 5 iff n is in A143203;
a(n) = 6 iff n is in A143204;
a(n) = 7 iff n is in A143205;
a(n) <> A006512(k)+1 for k>1.
a(A033849(n))=3; a(A033851(n))=3; a(A033850(n))=5; a(A033847(n))=6; a(A033848(n))=10. [Reinhard Zumkeller, Sep 19 2011]

			a(86) = a(43*2) = 43-2+1 = 42;
a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A000961, A001747, A006093, A020639, A033845, A033846.
Cf. A033847, A033848, A033849, A033850.
Cf. A143203, A143204, A143205.
Cf. A027746.

a143201 1 = 1
a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
   where pfs = a027748_row n
-- Reinhard Zumkeller, Sep 13 2011

Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* Harvey P. Dale, Dec 07 2011 *)

a(n) = f(n,1,1) where f(n,q,y) = if n=1 then y else if q=1 then f(n/p,p,1)) else f(n/p,p,y*(p-q+1)) with p = A020639(n) = smallest prime factor of n.

A111192 Product of the n-th sexy prime pair.

Original entry on oeis.org

55, 91, 187, 247, 391, 667, 1147, 1591, 1927, 2491, 3127, 4087, 4891, 5767, 7387, 9991, 10807, 11227, 12091, 17947, 23707, 25591, 28891, 30967, 37627, 38407, 51067, 52891, 55687, 64507, 67591, 70747, 75067, 78391, 96091, 98587, 111547, 122491, 126727, 136891

Offset: 1

Shawn M Moore (sartak(AT)gmail.com), Oct 23 2005

nonn

Semiprime of the form 4*m^2-9 = (2*m-3)*(2*m+3). - Vincenzo Librandi, Jan 26 2016

			a(2)=91 because the second sexy prime pair is (7, 13) and 7*13=91.

Reinhard Zumkeller, 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].

Cf. A023201, A104229.

Cf. A037074, A143206, A195118; intersection of A143205 and A001358.

a111192 n = a111192_list !! (n-1)
a111192_list = f a000040_list where
   f (p:ps@(q:r:_)) | q - p == 6 = (p*q) : f ps
                    | r - p == 6 = (p*r) : f ps
                    | otherwise  = f ps
-- Reinhard Zumkeller, Sep 13 2011

IsSemiprime:=func; [s: n in [1..300] | IsSemiprime(s) where s is 4*n^2-9]; // Vincenzo Librandi, Jan 26 2016

#(#+6)&/@Select[Prime[Range[100]], PrimeQ[#+6]&] (* Harvey P. Dale, Dec 17 2010 *)
(* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is not in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {6}]; Select[ Range[125000], fQ] (* Robert G. Wilson v, Feb 08 2012 *)
Select[Table[4 n^2 - 9, {n, 300}], PrimeOmega[#] == 2 &] (* Vincenzo Librandi, Jan 26 2016 *)

a(n) = A023201(n) * A046117(n). - Reinhard Zumkeller, Sep 13 2011

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 *)

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A143201 Product of distances between prime factors in factorization of n.

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A111192 Product of the n-th sexy prime pair.

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

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Haskell

Mathematica