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 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified August 9 10:38 EDT 2020. Contains 336323 sequences. (Running on oeis4.)