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).

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

A. Flammenkamp, List of the first 1200 highly composite numbers

A. Flammenkamp, List of the first 779,674 highly composite numbers

Peter J. Marko, Table of n, T(n, k) by rows for n = 1..10000 (using data from Flammenkamp)

S. Ramanujan, Highly Composite Numbers

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.

Cf. A000040, A002182, A124010, A212171, A318490.

Sequence in context: A220465 A050305 A117164 * A236567 A360564 A247299

Adjacent sequences: A212179 A212180 A212181 * A212183 A212184 A212185

Keyword

nonn,tabf

Author

Matthew Vandermast, Jun 08 2012

Extensions

Edited by Peter J. Marko, Aug 30 2018

Status

approved