login
A212182
Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists exponents of distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).
5
0, 1, 2, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 2, 1, 4, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 4, 2, 1, 1, 3, 3, 1, 1, 5, 2, 1, 1, 4, 3, 1, 1, 6, 2, 1, 1, 4, 2, 2, 1, 3, 2, 1, 1, 1, 4, 4, 1, 1, 5, 2, 2, 1, 4, 2, 1
OFFSET
1,3
COMMENTS
Length of row n = A108602(n).
For n > 1, row n of table gives the "nonincreasing order" version of the prime signature of A002182(n) (cf. A212171). This order is also the natural order of the exponents in the prime factorization of any highly composite number.
The distinct prime factors corresponding to exponents in row n are given by A318490(n, k), where k = 1, 2, 3, ..., A108602(n).
REFERENCES
S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
LINKS
Peter J. Marko, Table of i, a(i) for i = 1..10022 (corresponding to first n = 584 rows of irregular triangle; using data from Flammenkamp)
A. Flammenkamp, Highly composite numbers
Peter J. Marko, Table of n, T(n, k) by rows for n = 1..10000 (using data from Flammenkamp)
FORMULA
Row n equals row A002182(n) of table A124010. For n > 1, row n equals row A002182(n) of table A212171.
EXAMPLE
First rows read:
0;
1;
2;
1, 1;
2, 1;
3, 1;
2, 2;
4, 1;
2, 1, 1;
3, 1, 1;
2, 2, 1;
4, 1, 1;
...
1st row: A002182(1) = 1 so T(1, 1) = 0;
2nd row: A002182(2) = 2^1 so T(2, 1) = 1;
3rd row: A002182(3) = 4 = 2^2 so T(3, 1) = 2;
4th row: A002182(4) = 6 = 2^1 * 3^1 so T(4, 1) = 1 and T(4, 2) = 1;
5th row: A002182(5) = 12 = 2^2 * 3^1 so T(5, 1) = 2 and T(5, 2) = 1;
6th row: A002182(6) = 24 = 2^3 * 3^1 so T(6, 1) = 3 and T(6, 2) = 1.
CROSSREFS
Row n has length A108602(n), n >= 2.
Sequence in context: A220465 A050305 A117164 * A236567 A360564 A247299
KEYWORD
nonn,tabf
AUTHOR
Matthew Vandermast, Jun 08 2012
EXTENSIONS
Edited by Peter J. Marko, Aug 30 2018
STATUS
approved