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Revision History for A243061

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Showing entries 1-10 | older changes
A243061
Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).
(history; published version)
#12 by N. J. A. Sloane at Sun Dec 12 16:08:13 EST 2021
STATUS

reviewed

approved

#11 by Michel Marcus at Sat Dec 11 04:40:05 EST 2021
STATUS

proposed

reviewed

#10 by Antti Karttunen at Fri Dec 10 18:41:19 EST 2021
STATUS

editing

proposed

#9 by Antti Karttunen at Fri Dec 10 18:40:22 EST 2021
LINKS

Antti Karttunen, <a href="/A243061/b243061.txt">Table of n, a(n) for n = 1..1603</a>

<a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

#8 by Antti Karttunen at Fri Dec 10 18:34:32 EST 2021
PROG

(Scheme) (define (A243061 n) (A064216 (A241909 n)))

(PARI)

A064216(n) = A064989(n+n-1);

A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };

A241909(n) = if(1==n||isprime(n), 2^primepi(n), my(f=factor(n), h=1, i, m=1, p=1, k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k, 1]); m *= p^(i-h); h = i; if(f[k, 2]>1, f[k, 2]--, k++)); (p*m));

A243061(n) = A064216(A241909(n)); \\ Antti Karttunen, Dec 10 2021

STATUS

approved

editing

#7 by N. J. A. Sloane at Mon Jun 02 22:25:06 EDT 2014
STATUS

proposed

approved

#6 by Antti Karttunen at Mon Jun 02 16:26:10 EDT 2014
STATUS

editing

proposed

#5 by Antti Karttunen at Mon Jun 02 16:24:39 EDT 2014
COMMENTS

This is A241909-conjugate of A243065. Please see the comments at the latter sequence.

FORMULA

Equally, a A241909-conjugate of A243065, a(n) = A241909(A243065(A241909(n))). [A243065 has more interesting properties.]

a(n) = A241909(A243065(A241909(n))).

#4 by Antti Karttunen at Mon Jun 02 10:39:19 EDT 2014
FORMULA

Equally, a A241909-conjugate of A243065, a(n) = A241909(A243065(A241909(n))). [A243065 has more interesting properties.]

CROSSREFS

Cf. A064216, A241909, A243065, -A243066,.

#3 by Antti Karttunen at Mon Jun 02 08:08:32 EDT 2014
DATA

1, 2, 5, 3, 6, 13, 29, 4, 7, 47, 20, 25, 113, 95, 15, 11, 78, 23, 355, 158, 103, 267, 406, 89, 19, 1247, 17, 1237, 1577, 139, 660, 10, 221, 4363, 67, 38, 8179, 13109, 967, 393, 9266, 515, 21605, 4162, 28, 23601, 19578, 239, 43, 83, 987, 31247

LINKS

<a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

Discussion
Mon Jun 02
09:39
Antti Karttunen: If n is even, 2k, then A241909(n) is odd and
by the recurrence A241909(n)  = A000040(A001511(n)) * A242378(A007814(n), A241909(n)),
a(n) is A064216 (p_{A001511(n)} * A242378(A007814(n), A241909(n)).
09:41
Antti Karttunen: Above in cases where n = 2 mod 4 (i.e. k is odd)
results that a(n) = A064216 (3 * A003961(A241909(n))) = A064989((6 * A003961(A241909(n))) - 1).
09:53
Antti Karttunen: <Bad editing facilities, mistake in cut-and-paste!, Correction to above:>
A241909(n) = A000040(A001511(n)) * A242378(A007814(n), A241909(A064989(n))).
///
Then, if n is even, 2k, A241909(n) is odd and a(n) = A064216 (p_{A001511(n)} * A242378(A007814(n), A241909(A064989(n))))
= A064216(3 * A003961(A241909(A064989(n)))).
09:57
Antti Karttunen: (i.e. above when n = 2 mod 4, of the form 2(2t+1)).
So, checking for n=6, we have A064989(6) = 2, A241909(2) = 2, A003961(2) = 3, A064216(3*3) = A064989((2*9)-1) = A064989(17) = 13, thus a(6) = 13.)
///
However, this doesn't give any nice reduced form yet for those cases.
10:07
Antti Karttunen: Neither it is any better with odd numbers. Then the result of A241909 will be even, of the form 2 * A241909(A064989(n)) = 2*A243065(n), and a(n) = A064216(2*A243065(n)) = A064989((4*A243065(n)) - 1).
10:18
Antti Karttunen: Correction to above, when n is odd, A241909(n) = 2 * A241909(A064989(n)) = 2 * A241909(A064216((n+1)/2)) = 2 *A243065((n+1)/2).
10:20
Antti Karttunen: For n=3 we get 2*A243065(2) = 2*2 = 4. Then A064216(4) = 5, thus a(4) = 5.
10:23
Antti Karttunen: Summa summarum so far: The pair A243065 - A243066 (A241909-conjugates of these, with opposite composition order) seem more interesting and amenable to further discoveries. This pair left for the reference's sake.

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