cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A108602 Number of distinct prime factors of highly composite numbers (definition 1, A002182).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 1

Views

Author

Jud McCranie, Jun 12 2005

Keywords

Comments

n appears A086334(n) times. - Lekraj Beedassy, Sep 02 2006

Examples

			A002182(8) = 48 = 2^4*3, which has 2 distinct prime factors, so a(8)=2.

Links

Crossrefs

Cf. A002182, A002183.
Cf. A212182, A318490.

Formula

a(n) = A001221(A002182(n)).

Extensions

Edited by Ray Chandler, Nov 11 2005

A318490 Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 11, 2, 3, 5, 7
Offset: 1

Views

Author

Peter J. Marko, Aug 27 2018

Keywords

Comments

The exponents of factors in row n are given by A212182(n).

Examples

			Triangle begins:
  0;
  2;
  2;
  2, 3;
  2, 3;
  2, 3;
  2, 3;
  2, 3;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5, 7;
  ...
1st row: A002182(1) = 1 so T(1,1) = 0;
2nd row: A002182(2) = 2 so T(2,1) = 2;
3rd row: A002182(3) = 4 = 2^2 so T(3,1) = 2;
4th row: A002182(4) = 6 = 2 * 3 so T(4,1) = 2 and T(4,2) = 3;
5th row: A002182(5) = 12 = 2^2 * 3 so T(5,1) = 2 and T(5,2) = 3;
6th row: A002182(6) = 24 = 2^3 * 3 so T(6,1) = 2 and T(6,2) = 3.

Links

Crossrefs

Row n has length A108602(n), n >= 2.
Cf. A000040, A002182, A212182.

A212184 Row n of table gives exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 2, 0, 2, 3, 2, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 3, 2, 4, 4, 5, 2, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6, 3, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6
Offset: 1

Views

Author

Matthew Vandermast, Jul 01 2012

Keywords

Comments

Length of row n equals A212185(n) if A212185(n) is positive, or 1 if A212185(n) = 0.
Row n of table represents second signature of A002182(n) (cf. A212172). The use of 0 in the table to represent squarefree highly composite numbers accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.
No row is repeated an infinite number of times in the table. The contrary to this would imply that at least one integer appeared in A212183 an infinite number of times - something that Ramanujan proved to be false (cf. Ramanujan link). It would be interesting to know if there is an upper bound on the number of times a row can appear.

Examples

			First rows read: 0; 0; 2; 0; 2; 3; 2,2; 4; 2; 3; 2,2; 4;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Only exponents that are 2 or greater appear in a number's second signature; therefore, 12's second signature is {2}.  Since 12 = A002182(5), row 5 represents the second signature {2}.

References

  • Srinivasa 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

Crossrefs

Cf. A002182, A212172, A212182, A212183, A212185.

Programs

  • Mathematica
    With[{v = Import["https://oeis.org/A002182/b002182.txt", "Table"][[;; , 2]]}, exp[n_] := Select[FactorInteger[n][[;; , 2]], # > 1 &]; exp /@ v[[1 ;; 100]] /. {} -> {0} // Flatten] (* Amiram Eldar, Jan 20 2025 *)

Formula

Row n is identical to row A002182(n) of table A212172.

A212185 Number of exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 4, 2
Offset: 1

Views

Author

Matthew Vandermast, Jul 16 2012

Keywords

Comments

Length of row n of A212184 equals a(n) if a(n) is positive, 1 otherwise.

Examples

			The canonical prime factorization of 720 (2^4*3^2*5) has 2 exponents that equal or exceed 2. Since 720 = A002182(14), a(14) = 2.

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

Crossrefs

Cf. A002182, A212172, A212182, A212184.

Programs

Showing 1-4 of 4 results.

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