OFFSET
1,3
COMMENTS
Row 2 records the primes (A000040). Rows 3 and 4 record the semiprimes (A001358). Rows 5, 6 and 9 record the 3-almost primes (A014612) etc. A058933 is a similar sequence based on k-almost primes.
The graph of this sequence is interesting for large n because it shows multiple curves, one for each prime signature. For example, the six highest curves on the graph of a(n) for n up to 10^4 are for the (1,1), (1,1,1), (1), (2,1,1), (2,1), and (1,1,1,1) prime signatures. The (1) curve dominates until n=58; the (1,1) curve dominates until n=1279786, when the (1,1,1) curve intersects the (1,1) curve. Each (1,1,...,1) curve dominates for a finite number of n.
Ordinal transform of A101296. - Antti Karttunen, May 15 2017
a(n) is the number of positive integers up to n with the same prime signature as n. For example, the a(20) = 3 numbers are {12, 18, 20}. - Gus Wiseman, Jul 08 2019
Ordinal transform of A046523. - Alois P. Heinz, May 31 2020
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
EXAMPLE
The list begins as follows:
1
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ...
4 9 25 49 ...
6 10 14 15 21 22 26 33 34 35 38 39 46 51 ...
8 27 ...
12 18 20 28 44 45 50 52 ...
16 ...
Note: the above array, without the initial 1, is given by A095904 (and its transpose A179216). - Antti Karttunen, May 15 2017
MAPLE
p:= proc() 0 end:
a:= proc(n) option remember; local t; a(n-1);
t:= (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
sort(map(i-> i[2], ifactors(n)[2]), `>`));
p(t):= p(t)+1
end: a(0):=0:
seq(a(n), n=1..100); # Alois P. Heinz, May 31 2020
MATHEMATICA
prisig[n_]:=If[n==1, {}, Sort[Last/@FactorInteger[n]]];
Table[Count[Array[prisig, n], prisig[n]], {n, 30}] (* Gus Wiseman, Jul 08 2019 *)
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
Alford Arnold, Oct 24 2001
EXTENSIONS
More terms from Naohiro Nomoto, Oct 31 2001
STATUS
approved