1,2

There are SFP(n) different prime signatures which yield numbers with n divisors, where SFP (n) is the Smarandache Factor Partition of n (the number of factorizations as a product of 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.

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

Amarnath Murthy, Generalization of partition function introducing Smarandache factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

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

T. D. Noe, Rows n=1..300, flattened

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.

M. L. Perez et al., eds., Smarandache Notions Journal

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;

360,420,480,864,1152,6144,8388608;

1296,16777216;

12288,33554432;

900,2304,67108864;

960,1728,24576,134217728;

268435456;

720,2592,4608,49152,536870912;

1073741824;

840,1080,1920,2310,3456,98304,2147483648;

...

Cf. A001055, A005179, A077570, A122819.

Sequence in context: A000937 A167229 A068902 this_sequence A073935 A073696 A058602

Adjacent sequences: A077566 A077567 A077568 this_sequence A077570 A077571 A077572

nonn,tabf

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 11 2002

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Aug 12 2003

Improved definition from T. D. Noe, Aug 31 2008

Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 05 2008