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

1,2

Comments

The divisors of n counted in A038548(n) are sorted, each divisor is represented by a digit of 1 to 4, and these digits are concatenated to form the decimals of a(n).

The parity digits are 1,2,3,4 and are mapped as follows:

1: odd factor of an odd number

2: even factor of an even number, paired with an even factor

3: odd factor of an even number

4: even factor of an even number, paired with an odd factor

a(n) gives the significant or first half of the parity of n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

G. R. Bryant, Divisor 4 Parity

Formula

a(odd prime) = 1. - Michel Marcus, Jul 05 2018

Example

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.)

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 *)

Prog

(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

Crossrefs

Cf. A247795.

Sequence in context: A270086 A141411 A016481 * A047815 A095844 A113110

Adjacent sequences: A303815 A303816 A303817 * A303819 A303820 A303821

Keyword

nonn,base

Author

Gregory Bryant, Apr 30 2018

Status

approved