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%I #27 Aug 02 2023 10:02:03
%S 1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,24,28,32,36,40,48,56,64,72,80,96,
%T 112,128,144,160,192,224,256,288,320,384,448,512,576,640,768,896,1024,
%U 1152,1280,1536,1792,2048,2304,2560,3072,3584,4096,4608,5120,6144,7168
%N Positions n such that A005941(n) = A005940(n).
%C Sequence with n>=5 appears to be quintisected with the quintisections multiples of A000079 (powers of two): a(5m) = 5,10,20,40... = 5*2^(m-1) for m>0; a(5m+1) = 6,12,24,48,... = 6*2^(m-1); likewise a(5m+2) = 7*2^(m-1); a(5m+3) = 8*2^(m-1); a(5m+4) = 9*2^(m-1). - _Ralf Stephan_, Nov 13 2010
%C From _Antti Karttunen_, Aug 01 2023: (Start)
%C Numbers k for which A005940(A005940(k)) = k, or equally, for which A005941(A005941(k)) = k, i.e., numbers that are either fixed points of permutation A005940/A005941, or elements of its 2-cycles.
%C If n is a term then also 2*n is present, and vice versa.
%C Question: Are 1, 3, 5, 7 and 9 the only odd terms of this sequence?
%C (End)
%H R. J. Mathar, <a href="/A103969/b103969.txt">Table of n, a(n) for n = 1..60</a>.
%F Empirical g.f.: x*(1 +x +x^2 +x^3 +x^4)^2 / (1-2*x^5). - _Colin Barker_, Nov 18 2016
%e 56 is in the sequence since A005940(56) = A005941(56) = 72.
%e 7 is in the sequence since A005940(7) = 9, and A005940(9) = 7, thus also A005941(7) = 9, and A005941(9) = 7. - _Antti Karttunen_, Aug 01 2023
%t f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; t = Table[ f[n], {n, 10^4}]; u = Flatten[ Table[ Position[t, n, 1, 1], {n, 10^4}]]; Do[ If[ u[[n]] == {}, u[[n]] = {0}], {n, 10^4}]; Flatten[ Position[ Take[t, 10^4] - Flatten[u], 0]]
%o (Python)
%o from math import prod
%o from itertools import accumulate, count, islice
%o from sympy import prime, primepi, factorint
%o from collections import Counter
%o def A103969_gen(startvalue=1): # generator of terms >= startvalue
%o return filter(lambda n:sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n, multiple=True)))+1==prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(n-1)[2:].split('1')[:0:-1])).items()),count(max(startvalue,1)))
%o A103969_list = list(islice(A103969_gen(),20)) # _Chai Wah Wu_, Mar 11 2023
%Y Cf. A005940, A005941, A029747 (subsequence).
%K nonn
%O 1,2
%A _Robert G. Wilson v_, Feb 22 2005
%E Definition corrected and example updated by _R. J. Mathar_, Mar 06 2010
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