A103969 Positions n such that A005941(n) = A005940(n).

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, 112, 128, 144, 160, 192, 224, 256, 288, 320, 384, 448, 512, 576, 640, 768, 896, 1024, 1152, 1280, 1536, 1792, 2048, 2304, 2560, 3072, 3584, 4096, 4608, 5120, 6144, 7168

Offset

1,2

Comments

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

From Antti Karttunen, Aug 01 2023: (Start)

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.

If n is a term then also 2*n is present, and vice versa.

Question: Are 1, 3, 5, 7 and 9 the only odd terms of this sequence?

(End)

Links

R. J. Mathar, Table of n, a(n) for n = 1..60.

Formula

Empirical g.f.: x*(1 +x +x^2 +x^3 +x^4)^2 / (1-2*x^5). - Colin Barker, Nov 18 2016

Example

56 is in the sequence since A005940(56) = A005941(56) = 72.
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

Mathematica

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]]

Prog

(Python)
from math import prod
from itertools import accumulate, count, islice
from sympy import prime, primepi, factorint
from collections import Counter
def A103969_gen(startvalue=1): # generator of terms >= startvalue
    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)))
A103969_list = list(islice(A103969_gen(), 20)) # Chai Wah Wu, Mar 11 2023

Crossrefs

Cf. A005940, A005941, A029747 (subsequence).

Sequence in context: A060472 A130514 A130232 * A271168 A292514 A030141

Adjacent sequences: A103966 A103967 A103968 * A103970 A103971 A103972

Keyword

nonn

Author

Robert G. Wilson v, Feb 22 2005

Extensions

Definition corrected and example updated by R. J. Mathar, Mar 06 2010

Status

approved