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.

User: Greg Martin

Greg Martin's wiki page.

Greg Martin has authored 4 sequences.

A061194 Minimum positive value of lcm{1,...,n}*(s_1/1 + ... + s_n/n), where each s_i equals 1 or -1.

Original entry on oeis.org

1, 1, 1, 1, 7, 3, 11, 13, 11, 11, 23, 23, 607, 251, 251, 125, 97, 97, 3767, 3767, 3767, 2285, 24319, 24319, 71559, 4261, 13703, 13703, 872843, 872843, 17424097, 13828799, 902339, 7850449, 7850449, 7850449, 10683197, 68185267, 37728713, 37728713, 740674333
Offset: 1

Views

Author

Greg Martin (gerg(AT)math.toronto.edu), Apr 19 2001

Keywords

Comments

Nonnegative by considering the largest power of 2 up to n.

Examples

			1/1 - 1/2 - 1/3 - 1/4 - 1/5 + 1/6 + 1/7 = 11/420 = 11/lcm(1,2,3,4,5,6,7), so a(7) <= 11; no other choice of signs yields 1/420, ..., 10/420, so a(7)=11.

Crossrefs

See also A061195.

Extensions

More terms from Naohiro Nomoto, Jun 24 2001
a(23)-a(41) from Giovanni Resta, Jun 12 2016

A061195 Minimum positive numerator of s_1/1 + ... + s_n/n in lowest terms, where each s_i equals 1 or -1.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 11, 9, 11, 11, 23, 23, 607, 251, 59, 25, 97, 97, 2647, 2647, 1337, 457, 8917, 8917, 7951, 4261, 12439, 12439, 587971, 587971, 9687661, 13828799, 505163, 1554793, 1554793, 1554793, 1526171
Offset: 1

Views

Author

Greg Martin (gerg(AT)math.toronto.edu), Apr 19 2001

Keywords

Examples

			1/1 - 1/2 - 1/3 + 1/4 - 1/5 - 1/6 = 1/20, so a(6)=1.

Crossrefs

Cf. A061194.
Cf. A232090 (minimal possible denominator).

Programs

  • Mathematica
    nMax = 19; d = {0}; Table[d = Flatten[{d + 1/n, d - 1/n}]; Min[Abs[Numerator[d]]], {n, nMax}] (* T. D. Noe, Nov 19 2013 *)
  • PARI
    a(n) = {lcmn = 1;for (i=1, n, lcmn = lcm(i, lcmn)); minn = lcmn; for (i=0, 2^(n-1)-1, b = binary(i); while (#b != n, b = concat(0, b);); num = numerator(abs(sum(ii = 1, n, (-1)^b[ii]/ii))); minn = min(minn, num);); return(minn);} \\ Michel Marcus, Jun 15 2013

Extensions

More terms from Naohiro Nomoto, Jun 24 2001
a(22)-a(25) from Zak Seidov, Nov 20 2013
a(26)-a(33) from Zak Seidov, Nov 24 2013
a(34)-a(37) from Giovanni Resta, Jun 12 2016

A033808 All slopes (a(n)-a(m))/(n-m) are distinct; generated from 0 by greedy algorithm.

Original entry on oeis.org

0, 1, 3, 7, 14, 19, 29, 41, 59, 72, 88, 116, 139, 153, 178, 226, 258, 275, 321, 351, 396, 463, 523, 550, 621, 660, 745, 813, 912, 990, 1034, 1136, 1223, 1349, 1406, 1524, 1593, 1714, 1788, 1913, 2032, 2144, 2280, 2387, 2497, 2667, 2787, 2962, 3104, 3298
Offset: 0

Views

Author

Greg Martin (gerg(AT)ias.edu)

Keywords

References

  • A. Granville and J. Jimenez-Urroz, The least common multiple and lattice points on hyperbolas, Q. J. Math. 51 (2000), 343-352.

Links

A033501 Almost-squares: m such that m/p(m) >= k/p(k) for all k

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 28, 30, 35, 36, 40, 42, 48, 49, 54, 56, 60, 63, 64, 70, 72, 77, 80, 81, 88, 90, 96, 99, 100, 108, 110, 117, 120, 121, 130, 132, 140, 143, 144, 150, 154, 156, 165, 168, 169, 176, 180, 182, 192, 195, 196, 204, 208, 210
Offset: 1

Views

Author

Greg Martin; suggested by Jon Grantham

Keywords

Comments

Also integers that can be written in the form k*(k+h), for some integers k>=1 and 0 <= h <= T(k), where T(x) is the number of triangular numbers binomial(x+1,2) not exceeding x. (Corollary 1 in Greg Martin's article) - Hugo Pfoertner, Sep 23 2017

Links

Crossrefs

Cf. A000217.

Programs

  • Mathematica
    chs={1}; For[ n=2, n<=99, n++, chs=Join[ chs, Reverse[ Table[ (n-1-i)(n+i), {i, 0, (Sqrt[ 2n-1 ]-1)/2} ] ], Reverse[ Table[ (n-i)(n+i), {i, 0, n/Sqrt[ 2n-1 ]} ] ] ] ]
(*code uses alternate characterization, lists almost-squares up to 99^2*)

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