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 A275055 Irregular triangle read by rows listing divisors d of n in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes. 4
 1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 4, 3, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 4, 8, 3, 6, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(p^e) = A027750(p^e) for e >= 1. The matrix of products that are divisors of n is arranged such that the powers of the prime divisors range across an axis, one axis per prime divisor. Thus a squarefree semiprime has a 2-dimensional matrix, a sphenic number has 3 dimensions, etc. Generally, the number of dimensions for the matrix of divisors = omega(n) = A001221(n). Because of this, tau(n)*(mod omega(n)) = 0 for n > 1. This follows from the formula for tau(n). Prime divisors p of n are considered in numerical order. Product matrix of tensors T = 1,p,p^2,...,p^e that include the powers 1 <= e of the prime divisor p that divide n. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11214 (Rows 1 <= n <= 1500) Eric Weisstein's World of Mathematics, Divisor EXAMPLE Triangle begins: 1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9; 1, 2, 5, 10; 1, 11; 1, 2, 4, 3, 6, 12; 1, 13; 1, 2, 7, 14; 1, 3, 5, 15; 1  2, 4, 8, 16; 1, 17; 1, 2, 3, 6, 9, 18; ... 2 prime divisors: n = 72    1  2  4  8    3  6 12 24    9 18 36 72    thus a(72) = {1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72} 3 prime divisors: n = 60 (the 3 dimensional levels correspond with powers of 5)   level 5^0:        level 5^1:    1  2  4    |     5  10  20    3  6 12    |    15  30  60    thus a(60) = {1, 2, 4, 3, 6, 12, 5, 10, 20, 15, 30, 60} 4 prime divisors: n = 210 (the 3 dimensional levels correspond with powers of 5, the 4 dimensional levels correspond with powers of 7)   level 5^0*7^0:    level 5^1*7^0:      1   2     |     5  10      3   6     |    15  30   level 5^0*7^1:    level 5^1*7^1:      7  14     |    35  70     21  42     |   105 210    thus a(210) = {1,2,3,6,5,10,15,30,7,14,21,42,35,70,105,210} MATHEMATICA {{1}}~Join~Table[TensorProduct @@ Reverse@ Apply[PowerRange[1, #1^#2, #1] &, # &@ FactorInteger@ n, 1], {n, 2, 30}] // Flatten CROSSREFS Cf. A027750, A000005 (row length), A000203 (row sums), A056538. Sequence in context: A162306 A233773 A027750 * A254679 A275280 A087295 Adjacent sequences:  A275052 A275053 A275054 * A275056 A275057 A275058 KEYWORD nonn,easy,tabf AUTHOR Michael De Vlieger, Jul 14 2016 STATUS approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)