cp's OEIS Frontend

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.

%I A267099 #41 May 20 2022 08:58:09
%S A267099 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,
%T A267099 125,52,19,30,41,32,85,22,39,100,23,58,35,24,31,130,53,68,75,74,61,80,
%U A267099 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
%N 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.
%C A267099 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.
%C A267099 Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
%C A267099 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.
%C A267099 Sequences A072202 and A078613 are closed with respect to this permutation.
%H A267099 Antti Karttunen, <a href="/A267099/b267099.txt">Table of n, a(n) for n = 1..10000</a>
%H A267099 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A267099 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)).
%F A267099 Other identities. For all n >= 1:
%F A267099 a(A000040(n)) = A267101(n).
%F A267099 a(2*n) = 2*a(n).
%F A267099 a(3*n) = 5*a(n).
%F A267099 a(5*n) = 3*a(n).
%F A267099 a(7*n) = 13*a(n).
%F A267099 a(11*n) = 17*a(n).
%F A267099 etc. See examples in A267101.
%F A267099 A000035(n) = A000035(a(n)). [Preserves the parity of n.]
%F A267099 A005094(a(n)) = -A005094(n).
%F A267099 A079635(a(n)) = -A079635(n).
%o A267099 (PARI)
%o A267099 up_to = 2^16;
%o A267099 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); };
%o A267099 v267097 = A267097list(up_to);
%o A267099 A267097(n) = v267097[n];
%o A267099 A267098(n) = ((n-1)-A267097(n));
%o A267099 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)))); };
%o A267099 v002144 = list_primes_of_the_form(2*up_to,4,1);
%o A267099 A002144(n) = v002144[n];
%o A267099 v002145 = list_primes_of_the_form(2*up_to,4,3);
%o A267099 A002145(n) = v002145[n];
%o A267099 A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
%o A267099 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
%o A267099 (Scheme, with memoization-macro definec)
%o A267099 (definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))
%Y A267099 Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
%Y A267099 Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
%Y A267099 Cf. A001481, A268377, A004431, A268378, A004613, A004614.
%Y A267099 Cf. also A108548.
%K A267099 nonn,mult
%O A267099 1,2
%A A267099 _Antti Karttunen_, Feb 01 2016
%E A267099 Verbal description prefixed to the name by _Antti Karttunen_, May 19 2022