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A354091 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the next larger such prime, and primes of the form 3k and 3k+1 stay as they are.

%I A354091 #22 May 18 2022 10:06:57
%S A354091 1,5,3,25,11,15,7,125,9,55,17,75,13,35,33,625,23,45,19,275,21,85,29,
%T A354091 375,121,65,27,175,41,165,31,3125,51,115,77,225,37,95,39,1375,47,105,
%U A354091 43,425,99,145,53,1875,49,605,69,325,59,135,187,875,57,205,71,825,61,155,63,15625,143,255,67,575,87,385,83,1125
%N A354091 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the next larger such prime, and primes of the form 3k and 3k+1 stay as they are.
%C A354091 Permutation of odd numbers. Preserves the prime signature.
%H A354091 Antti Karttunen, <a href="/A354091/b354091.txt">Table of n, a(n) for n = 1..16383</a>
%H A354091 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A354091 Fully multiplicative with a(A003627(n)) = A003627(1+n), a(A007645(n)) = A007645(n).
%F A354091 For all n >= 1, A354092(a(n)) = n.
%F A354091 For all n >= 1, A046523(a(n)) = A046523(n) and A074941(a(n)) = A074941(n).
%e A354091 The primes in A003627 are replaced by the next prime in that sequence, as: 2 -> 5 -> 11 -> 17 -> 23 -> 29 -> 41 -> ..., while other kinds of primes (A002476) stay intact, thus for 60 = 2^2 * 3^1 * 5^1, we have a(60) = 5^2 * 3^1 * 11^1 = 825.
%o A354091 (PARI) A354091(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); factorback(f); };
%Y A354091 Cf. A003627, A007645, A074941, A353815.
%Y A354091 Cf. A354092 (left inverse), A354093 (inverse Möbius transform), A354094 (Möbius transform), A354095, A354096.
%Y A354091 Cf. also A003961, A332818, A348746 for similar constructions.
%K A354091 nonn,mult
%O A354091 1,2
%A A354091 _Antti Karttunen_, May 17 2022