A143203 -id:A143203 - 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.

A033850 Numbers whose prime factors are 3 and 7.

Original entry on oeis.org

21, 63, 147, 189, 441, 567, 1029, 1323, 1701, 3087, 3969, 5103, 7203, 9261, 11907, 15309, 21609, 27783, 35721, 45927, 50421, 64827, 83349, 107163, 137781, 151263, 194481, 250047, 321489, 352947, 413343, 453789, 583443, 750141, 964467

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k)/k = 4/7, where phi is the Euler totient function A000010. - Lekraj Beedassy, Jul 18 2008

Subsequence of A143203. - Reinhard Zumkeller, Sep 13 2011

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 189, p. 57, Ellipses, Paris 2008.

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

Cf. A000010, A033845, A033846, A033847, A033848, A033849, A033851, A143203.

import Data.Set (singleton, deleteFindMin, insert)
a033850 n = a033850_list !! (n-1)
a033850_list = f (singleton (3*7)) where
   f s = m : f (insert (3*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

Select[Range[10^6],Union[FactorInteger[#][[;;,1]]]=={3,7}&] (* Harvey P. Dale, Mar 01 2023 *)

A143201(a(n)) = 5. - Reinhard Zumkeller, Sep 13 2011

Sum_{n>=1} 1/a(n) = 1/12. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A143206 Product of the n-th cousin prime pair.

Original entry on oeis.org

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).

A195106 Numbers with largest and smallest prime factors differing by 4.

Original entry on oeis.org

21, 63, 77, 105, 147, 189, 221, 315, 437, 441, 525, 539, 567, 735, 847, 945, 1029, 1323, 1517, 1575, 1701, 2021, 2205, 2625, 2835, 2873, 3087, 3675, 3757, 3773, 3969, 4725, 4757, 5103, 5145, 5929, 6557, 6615, 7203, 7875, 8303, 8505, 9261, 9317, 9797, 10051

Offset: 1

Reinhard Zumkeller, Sep 13 2011

nonn

			a(10) = 441 = 3^2 * 7^2.
a(11) = 525 = 3 * 5^2 * 7.
a(12) = 539 = 7^2 * 11.

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

Cf. A023200, A143206, A195118.

A143203 is a subsequence.

a195106 n = a195106_list !! (n-1)
a195106_list = filter (\x -> a006530 x - a020639 x == 4) [1,3..]

pf4Q[n_]:=Module[{pfs=Transpose[FactorInteger[n]][[1]]}, Max[pfs]- Min[pfs]==4]; Select[Range[11000],pf4Q] (* Harvey P. Dale, Sep 24 2011 *)

Sum_{n>=1} 1/a(n) = 1/48 + Sum_{n>=1} 1/A143203(n) = 1/48 + Sum_{n>=1} 1/((A023200(n)+1)^2-4) = 0.130715767205... . - Amiram Eldar, Oct 26 2024

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

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A033850 Numbers whose prime factors are 3 and 7.

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A143206 Product of the n-th cousin prime pair.

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A195106 Numbers with largest and smallest prime factors differing by 4.

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