A241918 -id:A241918 - cp's OEIS frontend

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

Antti Karttunen, May 03 2014

nonn

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.

			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.

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

Complement: A241913.
A079704 is a subsequence.
Cf. A088902, A241916, A242418.

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 *)

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

A241914 After a(1)=0, numbers 0 .. A061395(n)-1, followed by numbers 0 .. A061395(n+1)-1, etc.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0

Offset: 1

Antti Karttunen, May 01 2014

			Viewed as an irregular table, the sequence is constructed as:
"Row"
  [1] 0; (by convention, a(1)=0)
  [2] 0; (because A061395(2)=1 (the index of the largest prime factor), we have here terms from 0 to 1-1)
  [3] 0, 1; (because A061395(3)=2, we have terms from 0 to 2-1)
  [4] 0;
  [5] 0, 1, 2; (because A061395(5)=3, we have terms from 0 to 3-1)
  [6] 0, 1;    (because A061395(6)=2, we have terms from 0 to 2-1)
  [7] 0, 1, 2, 3; (because A061395(7)=4, we have terms from 0 to 4-1)
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10082

One less than A241915.

Cf. A203623, A241920, A241910, A241918.

(define (A241914 n) (if (= n 1) 0 (- n (+ 2 (A203623 (- (A241920 n) 1))))))

a(1)=0, a(n) = n - A203623(A241920(n)-1) - 2.

A241920 After a(1)=1, each n appears A061395(n) times, where A061395 gives the index of the largest prime factor of n.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22

Offset: 1

Antti Karttunen, May 01 2014

nonn

Only numbers that occur just once are the powers of two (A000079).

Antti Karttunen, Table of n, a(n) for n = 1..10081

Cf. A082288, A061395, A203623, A067255, A241918, A241914, A241915.

A253890 a(n) = A253560(A253883(n)) = A122111((2*A122111(n)) - 1).

Original entry on oeis.org

1, 4, 16, 8, 18, 32, 2048, 9, 128, 512, 100, 256, 2147483648, 32768, 54, 64, 1200, 1024, 10616832, 144, 1048576, 864, 43200, 25, 65536, 8796093022208, 81, 4194304, 644972544, 131072, 7260, 36, 486, 75557863725914323419136, 268435456, 8192

Offset: 1

Antti Karttunen, Jan 17 2015

nonn

Conjugate the partition defined by the prime factorization of n (see, e.g., table A112798 or A241918), resulting k = A122111(n), then take the k-th odd number (2k-1), and conjugate again, giving a(n) = A122111(2k-1).
Thus after a(1)=1, this is a permutation of A070003 (numbers divisible by the square of their largest prime factor).
When A122111 is represented as a binary tree, then node A122111(t > 1) = n has as its left child A122111(2t-1) = a(n).

Cf. A070003 (same sequence without 1, sorted into ascending order).
Cf. A005408, A122111, A253560, A253883.
Cf. also A112798 and A241918.

a(n) = A122111((2*A122111(n)) - 1) = A122111(A005408(A122111(n) - 1)).

a(n) = A253560(A253883(n)).

cp's OEIS Frontend

A241912 Fixed points of A241916.

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A241914 After a(1)=0, numbers 0 .. A061395(n)-1, followed by numbers 0 .. A061395(n+1)-1, etc.

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A241920 After a(1)=1, each n appears A061395(n) times, where A061395 gives the index of the largest prime factor of n.

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A253890 a(n) = A253560(A253883(n)) = A122111((2*A122111(n)) - 1).

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