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| A000031
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Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.
(history;
published version)
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#213 by R. J. Mathar at Sun Jan 28 08:03:05 EST 2024
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#212 by R. J. Mathar at Sun Jan 28 08:02:43 EST 2024
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| FORMULA
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a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d), ) = A053635(n)/n, where phi is A000010.
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| STATUS
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approved
editing
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#211 by Joerg Arndt at Thu Nov 17 05:11:17 EST 2022
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#210 by Michel Marcus at Thu Nov 17 01:25:57 EST 2022
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#209 by Jon E. Schoenfield at Wed Nov 16 22:53:53 EST 2022
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#208 by Jon E. Schoenfield at Wed Nov 16 22:53:50 EST 2022
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| EXAMPLE
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The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111... }....}.
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| STATUS
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proposed
editing
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#207 by Chai Wah Wu at Wed Nov 16 19:59:18 EST 2022
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#206 by Chai Wah Wu at Wed Nov 16 19:59:04 EST 2022
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| PROG
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(Python)
from sympy import totient, divisors
def A000031(n): return sum(totient(d)*(1<<n//d) for d in divisors(n, generator=True))//n if n else 1 # Chai Wah Wu, Nov 16 2022
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| STATUS
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approved
editing
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#205 by Michael De Vlieger at Sat Apr 30 10:36:52 EDT 2022
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#204 by Jon E. Schoenfield at Sat Apr 30 10:36:05 EDT 2022
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