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

1, 3, 5, 17, 25, 45, 49, 133, 257, 65537

Offset: 1

Author

Antti Karttunen, Sep 02 2023

Keywords

• nonn
• hard
• more

Comments

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.

Examples

			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.
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.
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).
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.)
		

Crossrefs

Cf. A005940, A019434 (subsequence), A348717, A364961, A364962.

Cf. also A364297.

Programs

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
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));
isA364959(n) = ((n%2)&&(A348717(n)==A348717(A005940(n))));