

A095904


Triangular array of natural numbers (greater than 1) arranged by prime signature.


15



2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, 14641, 56, 66, 243, 36, 31, 841, 34, 6859, 50, 28561, 88, 70
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OFFSET

0,1


COMMENTS

The unit, 1, has the empty prime signature { } (thus not in triangle).
Downwards diagonals:
* Rightmost diagonal: smallest numbers of a given prime signature in increasing order (A025487). This defines the order of signatures used.
This special ordering of prime signatures (by increasing smallest numbers of a given prime signature, A181087) is unrelated to any of the 8 variants of graded lexicographic or colexicographic orderings (based on the exponents only) since it depends on the magnitudes of the prime numbers. It is not even graded by Omega(n).
* Second rightmost diagonal: second smallest numbers of a given prime signature (A077560). (They are not increasing anymore.)
Upwards diagonals:
* Leftmost diagonal: primes. {1} (A000040)
* 2nd leftmost diagonal: squares of primes. {2} (A001248)
* 3rd leftmost diagonal: squarefree biprimes. {1,1} (A006881)
* 4th leftmost diagonal: cubes of primes. {3} (A030078)
* 5th leftmost diagonal: signature (Achilles numbers) {1,2} (A054753)
* 6th leftmost diagonal: fourth powers of primes. {4} (A030514)
* 7th leftmost diagonal: signature (Achilles numbers) {1,3} (A065036)
* 8th leftmost diagonal: squarefree triprimes. {1,1,1} (A007304)
The Achilles numbers are nonsquarefree while not perfect powers.
Prime signatures are often expressed in increasing order of exponents. The decreasing order of exponents (as on the Wiki page, see links) has the advantage of listing the exponents in the same order (with the canonical factorization convention) as the smallest number of a given prime signature.


LINKS

Table of n, a(n) for n=0..62.
OEIS Wiki, Prime signature.
OEIS Wiki, Orderings.


EXAMPLE

343 is in the 4th left and 4th rightmost diagonal, because it is the 4th value with the 4th prime signature {3}.
First 8 rows of triangular array (Cf. table link for this sequence):
2
3 4
5 9 6
7 25 10 8
11 49 14 27 12
13 121 15 125 18 16
17 169 21 343 20 81 24
19 289 22 1331 28 625 40 30


CROSSREFS

Cf. A083140, A064839, A181087.
Sequence in context: A240062 A255129 A119586 * A096153 A083140 A246279
Adjacent sequences: A095901 A095902 A095903 * A095905 A095906 A095907


KEYWORD

nonn,tabl


AUTHOR

Alford Arnold, Jul 10 2004


EXTENSIONS

Extended by Ray Chandler, Jul 31 2004
Corrected (minor) by Daniel Forgues, Jan 21 2011
Example, comments by Daniel Forgues, Jan 21 2011
Edited by Alois P. Heinz, Jan 23 2011
Edited by Daniel Forgues, Jan 23 2011


STATUS

approved



