A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

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

Offset: 0

Author

Alford Arnold, Jul 10 2004

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• OEIS Wiki, Prime signature.
• OEIS Wiki, Orderings.

Crossrefs

Cf. A083140, A064839, A181087.

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