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A064839
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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.
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12
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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
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OFFSET
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1,3
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COMMENTS
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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.
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
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LINKS
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EXAMPLE
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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 ...
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MAPLE
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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:
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MATHEMATICA
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prisig[n_]:=If[n==1, {}, Sort[Last/@FactorInteger[n]]];
Table[Count[Array[prisig, n], prisig[n]], {n, 30}] (* Gus Wiseman, Jul 08 2019 *)
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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