A077569 Irregular triangle read by rows: row n lists smallest numbers in increasing order of all possible prime signatures with n divisors.

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, 360, 420

Offset: 1

Author

Amarnath Murthy, Nov 11 2002

Keywords

• nonn
• tabf

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

The n-th row contains A001055(n) terms. Taking the first term of each row gives A005179.

Examples

			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;
  ...
		

References

• Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..7314 (rows n=1..1000, flattened; rows 1..300 from T. D. Noe)
• 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.

Crossrefs

Cf. A001055, A005179, A077570, A122819, A162247.

Programs

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 *)

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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