cp's OEIS Frontend

A364959 Odd numbers k such that A348717(k) = A348717(A005940(k)).

%I A364959 #14 Sep 02 2023 21:37:44
%S A364959 1,3,5,17,25,45,49,133,257,65537
%N A364959 Odd numbers k such that A348717(k) = A348717(A005940(k)).
%C A364959 In contrast to condition A348717(n) = A348717(A163511(n)), which seems to admit only Mersenne primes for odd n (see A364297), here we also have some extra terms in addition to Fermat primes, A019434.
%e A364959 For n = 17, A005940(17) = 11, and A348717(11) = A348717(17) = 2, therefore 17 is included in this sequence. Moreover, any prime in this sequence must be one of the Fermat primes (A019434), because the primes are located at positions 2^k + 1 in the offset-1 variant of Doudna-tree, A005940.
%e A364959 For n = 25 and n = 49, A005940(25) = 49 and A005940(49) = 121, which are all squares of primes, and thus have the same A348717-value (4), therefore both 25 and 49 are terms.
%e A364959 For n = 45 = 3^2 * 5, A005940(45) = 175 = 5^2 * 7 [= A003961(45)], with A348717(45) = A348717(175) = 12, therefore 45 is included as a term. (See also examples in A364961).
%e A364959 For n = 133 = 7*19, A005940(133) = 85 = 5*17 [= A064989(133)], with A348717(133) = A348717(85) = 22, therefore 133 is included as a term. (See also example in A364962.)
%o A364959 (PARI)
%o A364959 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
%o A364959 A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
%o A364959 isA364959(n) = ((n%2)&&(A348717(n)==A348717(A005940(n))));
%Y A364959 Cf. A005940, A019434 (subsequence), A348717, A364961, A364962.
%Y A364959 Cf. also A364297.
%K A364959 nonn,hard,more
%O A364959 1,2
%A A364959 _Antti Karttunen_, Sep 02 2023