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A077569
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Irregular triangle of numbers having n divisors. The n-th row ends with 2^(n-1).
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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.
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REFERENCES
| 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.
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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.
M. L. Perez et al., eds., Smarandache Notions Journal
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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;
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;
...
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CROSSREFS
| Cf. A001055, A005179, A077570, A122819.
Sequence in context: A167229 A192333 A068902 * A073935 A073696 A058602
Adjacent sequences: A077566 A077567 A077568 * A077570 A077571 A077572
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KEYWORD
| nonn,tabf
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 11 2002
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EXTENSIONS
| 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
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