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 1-3 of 3 results.

A248123 Least integer m > 0 such that gcd(m,n) = 1 and m*n | C(m+n), where C(k) refers to the k-th Catalan number binomial(2k,k)/(k+1).

Original entry on oeis.org

1, 3, 2, 21, 9, 11, 11, 77, 5, 13, 6, 85, 10, 5, 1, 77, 11, 5, 11, 1, 4, 7, 13, 29, 18, 7, 14, 1, 15, 11, 17, 189, 19, 9, 6, 5, 23, 15, 7, 49, 23, 1, 22, 17, 1, 13, 25, 13, 26, 19, 11, 9, 28, 71, 18, 29, 10, 15, 31, 13, 34, 17, 5, 381, 9, 1, 35, 9, 19, 9
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 01 2014

Keywords

Comments

Conjecture: a(n) exists for all n > 0.

Examples

			a(4) = 21 since 4*21 divides C(4+21) = 4861946401452.

Links

Crossrefs

Cf. A000108, A248058, A248124.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[GCD[m,n]==1&&Mod[CatalanNumber[m+n],m*n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,70}]

A248189 Least positive integer m such that m*n divides sigma(m^2+n^2), where sigma(k) is the sum of all positive divisors of k.

Original entry on oeis.org

1, 1, 1, 7, 2, 38, 4, 81, 1, 102, 868, 1, 9, 3, 702, 26505, 1554, 14, 3, 243, 1, 650, 108, 1833, 34542, 18, 68, 186, 7252, 39, 58, 736839, 1, 3108, 72, 778, 210, 6, 3, 4830, 267, 2, 567, 5859, 6640, 6363, 3178412, 155771, 4964, 9
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 03 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0.
See also the comments in A248058.

Examples

			a(6) = 38 since 6*38 = 228 divides sigma(6^2+38^2) = sigma(1480) = 3420 = 15*228.

Links

Crossrefs

Cf. A000203, A248054, A248058, A248123.

Programs

  • Mathematica
    Do[m=1; Label[aa]; If[Mod[DivisorSigma[1,m^2+n^2], m*n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 50}]
  • PARI
    a(n)=m=1;while(sigma(n^2+m^2)%(m*n),m++);m
n=1;while(n<50,print1(a(n),", ");n++) \\ Derek Orr, Oct 03 2014

A248568 Least positive integer m such that phi(m+n) divides m, where phi(.) is Euler's totient function.

Original entry on oeis.org

1, 2, 6, 2, 20, 4, 8, 4, 12, 40, 24, 6, 20, 12, 24, 8, 48, 16, 32, 18, 24, 8, 72, 12, 44, 40, 36, 24, 132, 12, 56, 16, 60, 96, 40, 18, 180, 36, 60, 36, 144, 40, 80, 16, 72, 20, 168, 24, 92, 88, 184, 80, 276, 24, 104, 42, 48, 264, 312, 24
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 08 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n*(n-1) for all n > 1.
Note that for n > 1 the term a(n) should be even.

Examples

			a(10) = 40 since phi(40+10) = 20 divides 40.

Links

Crossrefs

Cf. A000010, A248007, A248058.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[Mod[m,phi[m+n]]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
  • PARI
    a(n)=m=1;while(m%eulerphi(m+n),m++);m
vector(100,n,a(n)) \\ Derek Orr, Oct 08 2014
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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