A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(x*y) = a(x)*a(y) for x, y > 1.

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101

Offset

1,2

Comments

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.

Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).

Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.

Sequences A072202 and A078613 are closed with respect to this permutation.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).

Other identities. For all n >= 1:

a(A000040(n)) = A267101(n).

a(2*n) = 2*a(n).

a(3*n) = 5*a(n).

a(5*n) = 3*a(n).

a(7*n) = 13*a(n).

a(11*n) = 17*a(n).

etc. See examples in A267101.

A000035(n) = A000035(a(n)). [Preserves the parity of n.]

A005094(a(n)) = -A005094(n).

A079635(a(n)) = -A079635(n).

Prog

(PARI)
up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to), i=0, c=0); forprime(p=2, prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to, m, k) = { my(v=vector(up_to), i=0); forprime(p=2, , if(k==(p%m), i++; v[i] = p; if(i==up_to, return(v)))); };
v002144 = list_primes_of_the_form(2*up_to, 4, 1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to, 4, 3);
A002145(n) = v002145[n];
A267101(n) = if(1==n, 2, if(1==(prime(n)%4), A002145(A267097(n)), A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1] = A267101(primepi(f[k, 1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

Crossrefs

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).

Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.

Cf. A001481, A268377, A004431, A268378, A004613, A004614.

Cf. also A108548.

Sequence in context: A354182 A065652 A235200 * A266416 A266401 A083798

Adjacent sequences: A267096 A267097 A267098 * A267100 A267101 A267102

Keyword

nonn,mult

Author

Antti Karttunen, Feb 01 2016

Extensions

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

Status

approved