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.

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A377488 Irregular triangle r ead by rows where row n lists powers p^k for primes p | n such that k = floor(log n/log p).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 4, 7, 8, 9, 5, 8, 11, 8, 9, 13, 7, 8, 5, 9, 16, 17, 9, 16, 19, 5, 16, 7, 9, 11, 16, 23, 9, 16, 25, 13, 16, 27, 7, 16, 29, 16, 25, 27, 31, 32, 11, 27, 17, 32, 7, 25, 27, 32, 37, 19, 32, 13, 27, 25, 32, 41, 7, 27, 32, 43, 11, 32, 25, 27, 23, 32
Offset: 1

Views

Author

Michael De Vlieger, Nov 01 2024

Keywords

Comments

Row 1 is {1} by convention, since 1 is the empty product.

Examples

			Table of the first 12 rows:
   1:  1;
   2:  2;
   3:  3:
   4:  4;
   5:  5;
   6:  3, 4;
   7:  7;
   8:  8;
   9:  9;
  10:  5, 8;
  11: 11;
  12:  8, 9.

Links

Crossrefs

Cf. A001221, A064446, A339378, A377485.

Programs

  • Mathematica
    {{1}}~Join~Table[Union[Join @@ Map[#^Range[Floor@ Log[#, n]] &, FactorInteger[n][[All, 1]] ] ], {n, 2, 30}]

Formula

Row n is { p^k : p | n, k = floor(log n/log p) }.
Row p^k = { p^k } for prime p and k > 0.
A001221(n) = length of row n for n > 1.
A064446(n) = product of row n.
A339378(n) = sum of row n for n > 1.

A339377 Number of triples (x, y, z) of natural numbers satisfying x+y = n and 2*x*y = z^2.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 4, 2, 2, 4, 2, 4, 6, 4, 2, 4, 4, 2, 4, 2, 2, 8, 2, 2, 4, 2, 2, 10, 4, 2, 6, 2, 4, 4, 2, 4, 4, 4, 4, 6, 2, 2, 4, 2, 2, 10, 2, 2, 8, 4, 2, 10, 2, 4, 4, 2, 2, 6, 2, 2, 10, 4, 4, 4, 2, 2, 6, 4, 2, 4, 4, 4, 4, 2, 2, 10, 4, 4, 4, 4, 4, 4
Offset: 0

Views

Author

Bernard Schott, Dec 02 2020

Keywords

Comments

This sequence is inspired by the 4th problem proposed during the second day of the final round of the 18th Austrian Mathematical Olympiad in 1987. The problem asked to find all triples solutions (x, y, z) only for n = 1987 (see Link, Reference and last example).
Some properties:
-> Inequalities, 0 <= x, y <= n; 0 <= z <= floor(n*sqrt(2)/2)
-> z is even and (x,y) are not together even.
-> a(n) = 1 iff n = 0, and the only solution is (0,0,0).
-> for n >= 1, a(n) >= 2 because (0,n,0) and (n,0,0) are always solutions.
-> a(n) is even for n >= 1.
-> If n = 3k, then (k,2k,2k) and (2k,k,2k) are solutions.
-> If 2*(n-1) = m^2, then (1,n-1,m) and (n-1,1,m) are solutions (with n in A058331).
-> The formula for n>0 comes from (x+y=n and 2*x*y=z^2) <==> n^2 = |x-y|^2 + 2*z^2.

Examples

			a(9) = 6 and these 6 solutions are: (0, 9, 0), (1, 8, 4), (3, 6, 6), (6, 3, 6), (8, 1, 4), (9, 0, 0).
a(1987) = 4 and these 4 solutions are: (0, 1987, 0), (529, 1458, 1242), (1458, 529, 1242), (1987, 0, 0); this is the answer to the Olympiad problem in link.

References

  • Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 4 of Austrian Mathematical Olympiad 1987, page 29 [Warning: solution proposed in this book has a mistake with (x, y, z) = ([0, 1987], 1987-x, sqrt(2xy))].

Links

Crossrefs

Cf. A058331, A218799, A339378 (variant with x+y = n and x*y = z^2).

Formula

a(0)=A218799(0); then for n>=1, a(n)=2*A218799(n) (remark from Hugo Pfoertner, Dec 02 2020).
Showing 1-2 of 2 results.

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