OFFSET
1,2
COMMENTS
There are A001055(n) different prime signatures with n divisors.
If a*b*c... is a factorization of n then the corresponding prime signature is p^(a-1)*q^(b-1)*r^(c-1)... etc.
The corresponding term of the n-th array is obtained by arranging a>b>c>... and p<q<r<... i.e. p = 2, q = 3 and r = 5 etc.
REFERENCES
Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10003 (first 1275 rows; rows 1..300 from T. D. Noe, rows 301..1000 from Amiram Eldar)
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.
EXAMPLE
The row for n = 12 contains 60,72,96 and 2048, each having 12 divisors, with prime signature p^2qr, p^3q^2, p^5q, p^11.
The triangle begins:
1;
2;
4;
6, 8;
16;
12, 32;
64;
24, 30, 128;
36, 256;
48, 512;
1024;
60, 72, 96, 2048;
4096;
192, 8192;
144, 16384;
120, 210, 216, 384, 32768;
65536;
180, 288, 768, 131072;
262144;
240, 432, 1536, 524288;
576, 1048576;
3072, 2097152;
4194304;
...
MATHEMATICA
row[n_] := Module[{e = f[n] - 1}, Sort[Times @@ (Prime[Range[Length[#]]]^Reverse[#]) & /@ e]]; Table[row[n], {n, 1, 25}] // Flatten (* Amiram Eldar, Jun 28 2025 using the function f by T. D. Noe at A162247 *)
CROSSREFS
KEYWORD
nonn,tabf,changed
AUTHOR
Amarnath Murthy, Nov 11 2002
EXTENSIONS
More terms from Ray Chandler, Aug 12 2003
Improved definition from T. D. Noe, Aug 31 2008
Edited by N. J. A. Sloane, Sep 05 2008
Name corrected by Amiram Eldar, Jun 28 2025
STATUS
approved