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A064839 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. 11
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
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

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

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

Cf. A000040, A001358, A014612, A025487, A046523, A058933, A101296.

Cf. also arrays A095904, A179216.

Cf. A001222, A085089, A118914, A124010, A325263, A325365, A326438, A326439.

Sequence in context: A057499 A241919 A286469 * A255810 A210256 A332422

Adjacent sequences:  A064836 A064837 A064838 * A064840 A064841 A064842

KEYWORD

easy,nice,nonn

AUTHOR

Alford Arnold, Oct 24 2001

EXTENSIONS

More terms from Naohiro Nomoto, Oct 31 2001

STATUS

approved

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Last modified May 26 22:58 EDT 2020. Contains 334634 sequences. (Running on oeis4.)