A077569 Irregular triangle read by rows: row n lists numbers in the range 1 to 2^(n-1) (inclusive) that have exactly 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

1,2

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.

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

References

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

Links

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.

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;
...

Crossrefs

Cf. A001055, A005179, A077570, A122819.

Sequence in context: A192333 A068902 A269332 * A270140 A333020 A325792

Adjacent sequences: A077566 A077567 A077568 * A077570 A077571 A077572

Keyword

nonn,tabf

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

Status

approved