login
a(n) = A342767(n, n).
7

%I #18 Sep 13 2021 09:19:53

%S 1,2,3,8,5,12,7,32,27,20,11,48,13,28,45,128,17,108,19,80,63,44,23,192,

%T 125,52,243,112,29,180,31,512,99,68,175,432,37,76,117,320,41,252,43,

%U 176,405,92,47,768,343,500,153,208,53,972,275,448,171,116,59,720

%N a(n) = A342767(n, n).

%C This sequence has similarities with A087019.

%C These are the positions of first appearances of each positive integer in A346701, and also in A346703. - _Gus Wiseman_, Aug 09 2021

%H Rémy Sigrist, <a href="/A342768/b342768.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A342768/a342768.gp.txt">PARI program for A342768</a>

%H <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>

%F a(n) = n iff n = 1 or n is a prime number.

%F a(p^k) = p^(2*k-1) for any k > 0 and any prime number p.

%F A007947(a(n)) = A007947(n).

%F A001222(a(n)) = 2*A001222(n) - 1 for any n > 1.

%F From _Gus Wiseman_, Aug 09 2021: (Start)

%F A001221(a(n)) = A001221(n).

%F If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g.

%F a(n) = A129597(n)/2.

%F (End)

%e For n = 42:

%e - 42 = 2 * 3 * 7, so:

%e 2 3 7

%e x 2 3 7

%e -------

%e 2 3 7

%e 2 3 3

%e + 2 2 2

%e -----------

%e 2 2 3 3 7

%e - hence a(42) = 2 * 2 * 3 * 3 * 7 = 252.

%t Table[n^2/FactorInteger[n][[-1,1]],{n,100}] (* _Gus Wiseman_, Aug 09 2021 *)

%o (PARI) See Links section.

%Y Cf. A007947, A087019, A342767.

%Y The sum of prime indices of a(n) is 2*A056239(n) - A061395(n).

%Y The version for even indices is A129597(n) = 2*a(n) for n > 1.

%Y The sorted version is A346635.

%Y These are the positions of first appearances in A346701 and in A346703.

%Y A001221 counts distinct prime factors.

%Y A001222 counts prime factors with multiplicity.

%Y A027193 counts partitions of odd length, ranked by A026424.

%Y A209281 adds up the odd bisection of standard compositions (even: A346633).

%Y A346697 adds up the odd bisection of prime indices (reverse: A346699).

%Y Cf. A000290, A006530, A033942, A037143, A344653, A345957, A346698, A346700, A346704.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Apr 02 2021