A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Author

Antti Karttunen, Nov 17 2014

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A250245.

Other similar permutations: A250243, A250248, A250250, A163511, A252756.

Cf. A003961, A005843, A020639, A055396, A078898, A246278, A250470, A268674, A278524, A302042, A302046.

Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

Differs from A250250 for the first time at n=73, where a(73) = 73, while A250250(73) = 103.

Programs

PARI

up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k))); \\ Antti Karttunen, Apr 01 2018
(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library, three alternative definitions)
(definec (A250246 n) (cond ((<= n 1) n) (else (A246278bi (A055396 n) (A250246 (A078898 n)))))) ;; Code for A246278bi given in A246278
(definec (A250246 n) (cond ((<= n 1) n) ((even? n) (* 2 (A250246 (/ n 2)))) (else (A003961 (A250246 (A250470 n))))))
(define (A250246 n) (A163511 (A252756 n)))

Formula

a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A250470(2n+1))). - Antti Karttunen, Jan 18 2015 - Instead of A250470, one may use A268674 in above formula. - Antti Karttunen, Apr 01 2018

As a composition of related permutations:

a(n) = A163511(A252756(n)).

Other identities. For all n >= 1:

a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]

A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A001221(a(n)) = A302041(n).

A001222(a(n)) = A253557(n).

A008683(a(n)) = A302050(n).

A000005(a(n)) = A302051(n)

A010052(a(n)) = A302052(n), for n >= 1.

A056239(a(n)) = A302039(n).