A303818 Representation of the divisor set of n based on parities of divisor and complementary divisor.
1, 3, 1, 32, 1, 34, 1, 32, 11, 34, 1, 323, 1, 34, 11, 322, 1, 343, 1, 324, 11, 34, 1, 3232, 11, 34, 11, 324, 1, 3433, 1, 322, 11, 34, 11, 32342, 1, 34, 11, 3223, 1, 3434, 1, 324, 111, 34, 1, 32322, 11, 343, 11, 324, 1, 3434, 11, 3223, 11, 34, 1, 323432, 1, 34, 111
Offset: 1
Examples
For n=24, 24 has the following divisors: {1, 2, 3, 4, 6, 8, 12, 24} with the following divisor pairings {{1,24}, {2,12}, {3,8}, {4,6}}.
The first divisor is 1, odd, and paired with an even, so we have: 3;
the second divisor is 2, even, and paired with an even, so we have: 2;
the third divisor is 3, odd, and paired with an even, so we have: 3;
the fourth divisor is 4, even, and paired with an even, so we have: 2.
That gives us the significant portion of the parity as 3232. (The full parity would include the complement and be 32322424.)
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- G. R. Bryant, Divisor 4 Parity
Crossrefs
Cf. A247795.
Programs
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Mathematica
Table[FromDigits[Map[Boole[OddQ@ #] & /@ {#, n/#} &, Take[#, Ceiling[Length[#]/2]] &@ Divisors@ n] /. {{1, 1} -> 1, {0, 0} -> 2, {1, 0} -> 3, {0, 1} -> 4}], {n, 100}] (* Michael De Vlieger, May 03 2018 *) -
PARI
par(d, nd) = if (d % 2, if (nd % 2, 1, 3), if (nd % 2, 4, 2)); a(n) = my(s=""); fordiv (n, d, if (d <= n/d, s = concat(s, par(d, n/d)))); eval(s); \\ Michel Marcus, Jul 05 2018
Formula
a(odd prime) = 1. - Michel Marcus, Jul 05 2018
Comments