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

%I A143201 #24 Nov 12 2023 13:08:40
%S A143201 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,
%T A143201 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,
%U A143201 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
%N A143201 Product of distances between prime factors in factorization of n.
%C A143201 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.
%C A143201 a(A007947(n)) = a(n);
%C A143201 A006093 and A001747 give record values and where they occur:
%C A143201 A006093(n)=a(A001747(n+1)) for n>1.
%C A143201 a(n) = 1 iff n is a prime power: a(A000961(n))=1;
%C A143201 a(n) = 2 iff n has exactly 2 and 3 as prime factors:
%C A143201 a(A033845(n))=2;
%C A143201 a(n) = 3 iff n is in A143202;
%C A143201 a(n) = 4 iff n has exactly 2 and 5 as prime factors:
%C A143201 a(A033846(n))=4;
%C A143201 a(n) = 5 iff n is in A143203;
%C A143201 a(n) = 6 iff n is in A143204;
%C A143201 a(n) = 7 iff n is in A143205;
%C A143201 a(n) <> A006512(k)+1 for k>1.
%C A143201 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]
%H A143201 Reinhard Zumkeller, <a href="/A143201/b143201.txt">Table of n, a(n) for n = 1..10000</a>
%H A143201 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F A143201 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.
%e A143201 a(86) = a(43*2) = 43-2+1 = 42;
%e A143201 a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
%e A143201 a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
%e A143201 a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
%e A143201 a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
%e A143201 a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.
%t A143201 Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* _Harvey P. Dale_, Dec 07 2011 *)
%o A143201 (Haskell)
%o A143201 a143201 1 = 1
%o A143201 a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
%o A143201    where pfs = a027748_row n
%o A143201 -- _Reinhard Zumkeller_, Sep 13 2011
%Y A143201 Cf. A000961, A001747, A006093, A020639, A033845, A033846.
%Y A143201 Cf. A033847, A033848, A033849, A033850.
%Y A143201 Cf. A143203, A143204, A143205.
%Y A143201 Cf. A027746.
%K A143201 nonn
%O A143201 1,6
%A A143201 _Reinhard Zumkeller_, Aug 12 2008