A332306 a(n) is the least k such that A121663(k) = n.

0, 1, 2, 4, 8, 3, 32, 5, 128, 9, 512, 6, 2048, 33, 10, 65, 32768, 18, 131072, 12, 34, 513, 2097152, 7, 8388608, 2049, 130, 36, 134217728, 11, 536870912, 68, 514, 32769, 40, 19, 34359738368, 131073, 2050, 13, 549755813888, 35, 2199023255552, 516, 136, 2097153

Offset: 1

Author

Rémy Sigrist, Feb 09 2020

Keywords

• nonn
• base

Comments

The binary representation of a(n) encodes the colexicographically earliest factorization of n into distinct factors greater than 1.

Examples

			The first terms, alongside their binary representations and factorizations, are:
  n   a(n)    bin(a(n))           Factorization
  --  ------  ------------------  -------------
   1       0                   0
   2       1                   1              2
   3       2                  10              3
   4       4                 100              4
   5       8                1000              5
   6       3                  11            2*3
   7      32              100000              7
   8       5                 101            2*4
   9     128            10000000              9
  10       9                1001            2*5
  11     512          1000000000             11
  12       6                 110            3*4
  13    2048        100000000000             13
  14      33              100001            2*7
  15      10                1010            3*5
  16      65             1000001            2*8
  17   32768    1000000000000000             17
  18      18               10010            3*6
  19  131072  100000000000000000             19
  20      12                1100            4*5
		

Links

• Rémy Sigrist, PARI program for A332306

Crossrefs

Cf. A000430, A045778, A121663.

Programs

PARI

See Links section.

Formula

a(n) = 2^(n-2) iff n is a prime number of the square of a prime number (A000430).

a(n!) = 2^(n-1)-1 for any n > 0.

a(p_1*...*p_k) = 2^(p_1-2)+...+2^(p_k-2) for distinct prime numbers p_1, ..., p_k.