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%I #29 Dec 21 2021 23:33:12
%S 1,1,2,1,3,1,4,1,2,2,5,1,6,3,4,1,7,2,8,3,5,6,9,1,3,7,2,4,10,1,11,1,8,
%T 9,10,1,12,11,12,2,13,2,14,5,6,13,15,1,4,7,14,8,16,3,15,4,16,17,17,1,
%U 18,18,9,1,19,3,19,10,20,4,20,1,21,21,11,12,22,5,22,2,2,23,23,2,24,25,26
%N List the natural numbers starting a new row only with each new least prime signature (A025487). a(n) is the column position associated with n.
%C 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.
%C 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.
%C Ordinal transform of A101296. - _Antti Karttunen_, May 15 2017
%C 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
%C Ordinal transform of A046523. - _Alois P. Heinz_, May 31 2020
%H T. D. Noe, <a href="/A064839/b064839.txt">Table of n, a(n) for n = 1..10000</a>
%e The list begins as follows:
%e 1
%e 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ...
%e 4 9 25 49 ...
%e 6 10 14 15 21 22 26 33 34 35 38 39 46 51 ...
%e 8 27 ...
%e 12 18 20 28 44 45 50 52 ...
%e 16 ...
%e Note: the above array, without the initial 1, is given by A095904 (and its transpose A179216). - _Antti Karttunen_, May 15 2017
%p p:= proc() 0 end:
%p a:= proc(n) option remember; local t; a(n-1);
%p t:= (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
%p sort(map(i-> i[2], ifactors(n)[2]), `>`));
%p p(t):= p(t)+1
%p end: a(0):=0:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 31 2020
%t prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
%t Table[Count[Array[prisig,n],prisig[n]],{n,30}] (* _Gus Wiseman_, Jul 08 2019 *)
%Y Cf. A000040, A001358, A014612, A025487, A046523, A058933, A101296.
%Y Cf. also arrays A095904, A179216.
%Y Cf. A001222, A085089, A118914, A124010, A325263, A325365, A326438, A326439.
%K easy,nice,nonn
%O 1,3
%A _Alford Arnold_, Oct 24 2001
%E More terms from _Naohiro Nomoto_, Oct 31 2001
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