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A129597 Central diagonal of array A129595.

%I A129597 #20 Jul 09 2025 04:26:46
%S A129597 1,4,6,16,10,24,14,64,54,40,22,96,26,56,90,256,34,216,38,160,126,88,
%T A129597 46,384,250,104,486,224,58,360,62,1024,198,136,350,864,74,152,234,640,
%U A129597 82,504,86,352,810,184,94,1536,686,1000,306,416,106,1944,550,896,342
%N A129597 Central diagonal of array A129595.
%C A129597 These are the positions of first appearances of each positive integer in A346704. - _Gus Wiseman_, Oct 16 2021
%H A129597 Antti Karttunen, <a href="/A129597/b129597.txt">Table of n, a(n) for n = 1..10000</a>
%F A129597 From _Gus Wiseman_, Aug 10 2021: (Start)
%F A129597 For n > 1, A001221(a(n)) = A099812(n).
%F A129597 If g = A006530(n) is the greatest prime factor of n > 1, then a(n) = 2n^2/g.
%F A129597 a(n) = A100484(A000720(n)) = 2n iff n is prime.
%F A129597 a(n > 1) = 2*A342768(n).
%F A129597 (End)
%t A129597 Table[If[n==1,1,2*n^2/FactorInteger[n][[-1,1]]],{n,100}] (* _Gus Wiseman_, Aug 10 2021 *)
%o A129597 (PARI) A129597(n) = if(1==n, n, my(f=factor(n)); (2*n*n)/f[#f~, 1]); \\ _Antti Karttunen_, Oct 16 2021
%Y A129597 a(n) = A129595(n,n).
%Y A129597 The sum of prime indices of a(n) is 2*A056239(n) - A061395(n) + 1 for n > 1.
%Y A129597 The version for odd indices is A342768(n) = a(n)/2 for n > 1.
%Y A129597 Except the first term, the sorted version is 2*A346635.
%Y A129597 These are the positions of first appearances in A346704.
%Y A129597 A001221 counts distinct prime factors.
%Y A129597 A001222 counts prime factors with multiplicity.
%Y A129597 A027187 counts partitions of even length, ranked by A028260.
%Y A129597 A346633 adds up the even bisection of standard compositions (odd: A209281).
%Y A129597 A346698 adds up the even bisection of prime indices (reverse: A346699).
%Y A129597 Cf. A000290, A006530, A037143, A329888, A344606, A345957, A346697, A346700, A346701.
%K A129597 nonn
%O A129597 1,2
%A A129597 _Antti Karttunen_, May 01 2007, based on _Marc LeBrun_'s Jan 11 2006 message on SeqFan mailing list