Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Feb 10 2020 08:23:21
%S 0,1,2,4,8,3,32,5,128,9,512,6,2048,33,10,65,32768,18,131072,12,34,513,
%T 2097152,7,8388608,2049,130,36,134217728,11,536870912,68,514,32769,40,
%U 19,34359738368,131073,2050,13,549755813888,35,2199023255552,516,136,2097153
%N a(n) is the least k such that A121663(k) = n.
%C The binary representation of a(n) encodes the colexicographically earliest factorization of n into distinct factors greater than 1.
%H Rémy Sigrist, <a href="/A332306/a332306.gp.txt">PARI program for A332306</a>
%F a(n) = 2^(n-2) iff n is a prime number of the square of a prime number (A000430).
%F a(n!) = 2^(n-1)-1 for any n > 0.
%F a(p_1*...*p_k) = 2^(p_1-2)+...+2^(p_k-2) for distinct primes numbers p_1, ..., p_k.
%e The first terms, alongside their binary representations and factorizations, are:
%e n a(n) bin(a(n)) Factorization
%e -- ------ ------------------ -------------
%e 1 0 0
%e 2 1 1 2
%e 3 2 10 3
%e 4 4 100 4
%e 5 8 1000 5
%e 6 3 11 2*3
%e 7 32 100000 7
%e 8 5 101 2*4
%e 9 128 10000000 9
%e 10 9 1001 2*5
%e 11 512 1000000000 11
%e 12 6 110 3*4
%e 13 2048 100000000000 13
%e 14 33 100001 2*7
%e 15 10 1010 3*5
%e 16 65 1000001 2*8
%e 17 32768 1000000000000000 17
%e 18 18 10010 3*6
%e 19 131072 100000000000000000 19
%e 20 12 1100 4*5
%o (PARI) See Links section.
%Y Cf. A000430, A045778, A121663.
%K nonn,base
%O 1,3
%A _Rémy Sigrist_, Feb 09 2020