A241912 - OEIS

%I #23 Aug 27 2016 09:56:45

%S 1,2,3,4,5,7,8,11,13,15,16,17,18,19,23,29,31,32,37,41,43,45,47,50,53,

%T 55,59,61,64,67,71,73,79,83,89,97,98,101,103,105,107,108,109,113,119,

%U 127,128,131,135,137,139,149,150,151,157,163,167,173,179,181,191,193,197,199

%N Fixed points of A241916.

%C A natural number n occurs here if and only if it is either a power of 2, or satisfies A001511(n) = A071178(n) [the exponent of highest power of 2 dividing n is one less than the exponent of the largest prime factor of n], and all the intermediate exponents form a palindrome. [Please see the definition of A241916.]

%C Numbers for which the corresponding rows in A112798 and A241918 are the conjugate partitions of each other.

%H Antti Karttunen, <a href="/A241912/b241912.txt">Table of n, a(n) for n = 1..4931</a>

%F a(n) = A242418(n+1)/2.

%e 98 = 2*7*7 = p_1^1 * p_2^0 * p_3^0 * p_4^2 is included because 2 occurs once, the highest prime factor 7 occurs twice (thus A001511(150) = A071178(150) = 2), and the intermediate exponents (in this case {0,0}) form a palindrome.

%e 150 = 2*3*5*5 = p_1^1 * p_2^1 * p_3^2 is included because 2 occurs once, the highest prime factor 5 occurs twice (thus A001511(150) = A071178(150) = 2), and the intermediate exponents (in this case 1) form a palindrome.

%t f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[#[[n + 1]]/2, {n, Length@ # - 1}] &@ Select[Range@ 400, g@ f@ # == g@ Reverse@ f@ # &] (* _Michael De Vlieger_, Aug 27 2016 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A241912 (FIXED-POINTS 1 1 A241916))

%o ;; Alternatively:

%o (define (A241912 n) (/ (A242418 (+ n 1)) 2))

%Y Complement: A241913.

%Y A079704 is a subsequence.

%Y Cf. A088902, A241916, A242418.

%K nonn

%O 1,2

%A _Antti Karttunen_, May 03 2014