A209934 a(n) is the first value to occur consecutively in the sequence b_n defined by p_2k(b_n(k)) = p_k(n)^2, k=1,2,3,..., where p_k(n) is the n-th k-almost prime.

1, 3, 8, 12, 23, 26, 32, 66, 68, 78, 83, 106, 116, 169, 181, 201, 210, 216, 234, 273, 282, 296, 427, 436, 501, 504, 513, 538, 547, 583, 655, 688, 711, 738, 751, 851, 866, 947, 1065, 1088, 1155, 1274, 1277, 1285, 1350, 1369, 1389, 1456, 1594, 1615, 1702, 1734

Offset: 1

Author

Daniel Tisdale, Mar 15 2012

Keywords

• nonn

Comments

A k-almost prime has exactly k prime factors, repetitions included.

Conjecture: Each sequence b_n repeats indefinitely. (Example: for n=3, b_n = 9, 8, 8, 8, 8, 8, .... It looks like b_3(k) is 8 for all k > 1.)

The conjecture follows from the formula that uses A078843 below (and the strict monotonicity of A078843). However the first repeated value is not for every n the value that repeats indefinitely. For example a(8) = b_8(2) = b_8(3) = 66, but b_8(k) = 64 for k >= 4. - Peter Munn, Aug 05 2019

Examples

			for k = 1, 2, 3, 4, 5, 6, ...:
p_k(3) = 5, 9, 18, 36, 72, 144, ... (the 3rd k-almost prime);
p_k(3)^2 = 25, 81, 324, 1296, 5184, 20736, ...;
b_3(k) = 9, 8, 8, 8, 8, 8, ... (index in the 2k-almost primes);
so since b_3(3) = b_3(2) = 8, a(3) = 8.
		

Crossrefs

Cf. A058933, A078840, A078843, A091538.

Programs

PARI

get_p(m,k) = {local(i,n);i=0;n=1;while(iA209934(n) = {local(m,k,k_old);m=3;k_old=get_k(2,get_p(1,n)^2);k=get_k(4,get_p(2,n)^2);while(kMichael B. Porter, Mar 20 2012

Formula

From Peter Munn, Aug 05 2019: (Start)

b_n(k) = A058933(A078840(k,n)^2).

a(n) = b_n(min {k : b_n(k) = b_n(k+1)}).

If n < A078843(k+1) and b_n(k) < A078843(2k+1) then b_n(i) = b_n(k) for i >= k.

(End)

Extensions

Edited, correcting the subscripting, by Peter Munn, Aug 04 2019