A317945 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each divisor d of n. Restricted growth sequence transform of A317944.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Author

Antti Karttunen, Aug 12 2018

Keywords

• nonn

Comments

Differs from A000027(n) = n (positive integers) from a(193) = 191 on.

For all i, j: a(i) = a(j) => A317838(i) = A317838(j).

There are certain prime pairs p, q for which the Stern polynomials B(p,t) and B(q,t) (see table A125184) have equal multisets of nonzero coefficients. For example, for primes 191 and 193 these coefficients are {1, 2, 2, 2, 2, 3, 1} and {1, 2, 2, 2, 3, 2, 1} (from which follows that A278243(191) = A278243(193), A286378(191) = A286378(193) and thus => a(191) = a(193) => A002487(191) = A002487(193) as well). Other such prime pairs currently known are {419, 461}, {2083, 2143} and {11777, 12799}. Whenever a(p) = a(q) for such a prime pair, then also a(2^k * p) = a(2^k * q) for all k >= 0. It would be nice to know whether there could exist any other cases of a(i) = a(j), i != j, but for example both i and j being odd semiprimes?

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to Stern's sequences

Crossrefs

Cf. A002487, A125184, A260443, A286378, A317943, A317944.

Cf. also A305795, A317838.

Programs

PARI

\\ Needs also code from A286378.
up_to = 65537;
A317944(n) = { my(m=1); fordiv(n,d, if(d>1, m *= prime(A286378(d)-1))); (m); };
v317945 = rgs_transform(vector(up_to, n, A317944(n)));
A317945(n) = v317945[n];