A241912 Fixed points of A241916.

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, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 98, 101, 103, 105, 107, 108, 109, 113, 119, 127, 128, 131, 135, 137, 139, 149, 150, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Offset

1,2

Comments

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.]

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..4931

Formula

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

Example

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.
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.

Mathematica

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 *)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A241912 (FIXED-POINTS 1 1 A241916))
;; Alternatively:
(define (A241912 n) (/ (A242418 (+ n 1)) 2))

Crossrefs

Complement: A241913.

A079704 is a subsequence.

Cf. A088902, A241916, A242418.

Sequence in context: A090467 A177202 A053868 * A269870 A307824 A081730

Adjacent sequences: A241909 A241910 A241911 * A241913 A241914 A241915

Keyword

nonn

Author

Antti Karttunen, May 03 2014

Status

approved