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

1,2

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

Links

Table of n, a(n) for n=1..52.

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)

Example

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.

Prog

(PARI) get_p(m, k) = {local(i, n); i=0; n=1; while(i<k, n=n+1; if(bigomega(n)==m, i=i+1)); n}
get_k(m, p) = {local(i, n); i=0; n=1; while(n<p, n=n+1; if(bigomega(n)==m, i=i+1)); i};
A209934(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(k<k_old, k_old=k; k=get_k(2*m, get_p(m, n)^2); m=m+1); k} \\ Michael B. Porter, Mar 20 2012

Crossrefs

Cf. A058933, A078840, A078843, A091538.

Sequence in context: A330897 A169923 A158022 * A007434 A379717 A128303

Adjacent sequences: A209931 A209932 A209933 * A209935 A209936 A209937

Keyword

nonn

Author

Daniel Tisdale, Mar 15 2012

Extensions

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

Status

approved