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