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A303818
Representation of the divisor set of n based on parities of divisor and complementary divisor.
1
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
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
KEYWORD
nonn,base
AUTHOR
Gregory Bryant, Apr 30 2018
STATUS
approved