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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.

%I A317945 #20 Nov 23 2018 21:00:41
%S A317945 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,
%T A317945 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U A317945 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
%N 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.
%C A317945 Differs from A000027(n) = n (positive integers) from a(193) = 191 on.
%C A317945 For all i, j: a(i) = a(j) => A317838(i) = A317838(j).
%C A317945 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?
%H A317945 Antti Karttunen, <a href="/A317945/b317945.txt">Table of n, a(n) for n = 1..65537</a>
%H A317945 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%o A317945 (PARI) \\ Needs also code from A286378.
%o A317945 up_to = 65537;
%o A317945 A317944(n) = { my(m=1); fordiv(n,d, if(d>1, m *= prime(A286378(d)-1))); (m); };
%o A317945 v317945 = rgs_transform(vector(up_to, n, A317944(n)));
%o A317945 A317945(n) = v317945[n];
%Y A317945 Cf. A002487, A125184, A260443, A286378, A317943, A317944.
%Y A317945 Cf. also A305795, A317838.
%K A317945 nonn
%O A317945 1,2
%A A317945 _Antti Karttunen_, Aug 12 2018