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 1-10 of 15 results. Next

A247940 Least integer m > n such that m + n divides L(m) + L(n), where L(k) refers to the Lucas number A000032(k).

Original entry on oeis.org

5, 5, 15, 5, 19, 30, 17, 19, 15, 13, 13, 24, 35, 236, 33, 34, 31, 90, 29, 23, 27, 25, 25, 84, 47, 80, 45, 190, 43, 54, 41, 35, 39, 1216, 37, 72, 59, 212, 57, 43, 55, 66, 53, 86, 51, 76, 49, 60, 71, 53, 69, 55, 67, 222, 65, 122, 63, 112, 61, 264
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 27 2014

Keywords

Comments

Conjecture: Let A be any integer not congruent to 3 modulo 6. Define v(0) = 2, v(1) = A, and v(n+1) = A*v(n) + v(n-1) for n > 0. Then, for any integer n > 0, there are infinitely many positive integers m such that m + n divides v(m) + v(n).
This implies that a(n) exists for any n > 0.

Examples

			 a(3) = 15 since 15 + 3 = 18 divides L(15) + L(3) = 1364 + 4 = 18*76.

Links

Crossrefs

Cf. A000032, A247824, A247937.

Programs

  • Mathematica
    Do[m=n+1;Label[aa];If[Mod[LucasL[m]+LucasL[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A248124 Least integer m > 0 such that gcd(m,n) = 1 and (m+n) | (C(m)+C(n)), where C(k) refers to the k-th Catalan number, binomial(2k,k)/(k+1).

Original entry on oeis.org

1, 19, 1, 20, 1, 95, 1, 4, 1, 242, 241, 478, 1, 23, 1, 5, 7, 109, 1, 17, 1, 227, 467, 37, 1, 209, 1, 330, 2077, 17, 1073, 816, 1, 27, 109, 71, 1, 43, 1, 41, 145, 151, 1, 43, 1, 59, 71, 587, 1, 87, 1775, 344, 1773, 1127, 1, 49, 1
Offset: 4

Views

Author

Zhi-Wei Sun, Oct 01 2014

Keywords

Comments

Conjecture: a(n) exists for all n > 3.

Examples

			a(5) = 19 since 5 is relatively prime to 19 and 5 + 19 = 24 divides C(5) + C(19) = 42 + 1767263190 = 1767263232 = 24*73635968.

Links

Crossrefs

Cf. A000108, A247824, A247937, A247940, A248123, A248125.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[GCD[m,n]==1&&Mod[CatalanNumber[m]+CatalanNumber[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,4,60}]

A248125 Least positive integer m such that m + n divides C(2m,m) + C(2n,n), where C(2k,k) = (2k)!/(k!)^2.

Original entry on oeis.org

1, 2, 5, 16, 3, 6, 2, 22, 101, 6, 21, 86, 43, 16, 15, 4, 3, 6, 21, 20, 11, 8, 49, 48, 7, 22, 29, 28, 27, 26, 25, 49, 11, 29, 133, 20, 19, 22, 71, 70, 7, 18, 13, 46, 11, 14, 25, 24, 23, 93, 45, 80, 43, 67, 29, 286, 171, 102, 97, 38
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 01 2014

Keywords

Comments

Conjecture: a(n) exists for all n > 0. Moreover, for n > 66 we have a(n) < n except for n = 364, 408.
a(n) = n for n = 1, 2, 6, 15, 20, 28, 66, ... The next term, if it exists, is greater than 10^4. - Derek Orr, Oct 01 2014

Examples

			a(3) = 5 since 3 + 5 = 8 divides C(6,3) + C(10,5) = 20 + 252 = 272.

Links

Crossrefs

Cf. A247824, A247937, A247940, A248123, A248124.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[Mod[Binomial[2m,m]+Binomial[2n,n],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((binomial(2*m,m)+binomial(2*n,n))%(m+n),m++);m
vector(100,n,a(n)) \\ Derek Orr, Oct 01 2014

A248133 Least positive integer m such that m + n divides T(m) + T(n), where T(.) is given by A002426.

Original entry on oeis.org

1, 3, 1, 1, 7, 2, 2, 2, 1, 1, 7, 4, 37, 145, 35, 1, 25, 16, 5, 16, 1, 1, 18, 19, 3, 11, 41, 1, 7, 2, 48, 415, 1, 2, 15, 7, 13, 34, 97, 1, 27, 18, 56, 22, 1, 1, 5, 26, 22, 36, 18, 1, 117, 52, 376, 11, 1, 1, 23, 26
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 02 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n^2 - n + 1 except for n = 274.
Note that a(274) = 188847 > 2*274^2.

Examples

			a(5) = 7 since 5 + 7 divides T(5) + T(7) = 51 + 393 = 444 = 12*37.
a(2539) = 643425 since 2539 + 643425 = 645964 divides T(2539) + T(643425).

Links

Crossrefs

Cf. A002426, A247824, A247937, A247940, A248124, A248125, A248136, A248137, A248139, A248142.

Programs

  • Mathematica
    T[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,n/2}]
Do[m=1;Label[aa];If[Mod[T[m]+T[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A248136 Least positive integer m such that m + n divides D(m) + D(n), where D(.) is given by A001850.

Original entry on oeis.org

1, 20, 3, 6, 1, 4, 200, 299, 5, 29, 4, 119, 5, 61, 3, 3, 6, 64, 31, 2, 21, 35, 6, 2974, 17, 1052, 27, 109, 10, 4, 3, 50, 65, 177, 22, 29, 5, 25, 15, 29, 29, 584, 83, 163, 9, 152, 19, 19, 29, 32, 15, 35, 4, 25, 239, 1122, 185, 76, 35, 97
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 02 2014

Keywords

Comments

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

Examples

			a(2) = 20 since 2 + 20 = 22 divides D(2) + D(20) = 13 + 260543813797441 = 260543813797454 = 22*11842900627157.

Links

Crossrefs

Cf. A001850, A247824, A247937, A247940, A248124, A248125, A248133, A248137, A248139, A248142.

Programs

  • Mathematica
    d[n_]:=Sum[Binomial[n,k]Binomial[n+k,k],{k,0,n}]
Do[m=1;Label[aa];If[Mod[d[m]+d[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A248137 Least positive integer m such that m + n divides M(m) + M(n), where M(.) is given by A001006.

Original entry on oeis.org

1, 1, 244, 1, 23, 4, 1, 1, 3494, 1, 68058, 4, 20, 18, 35, 1, 4, 14, 32, 13, 21, 1, 5, 22, 172, 7, 8, 1, 1, 28, 14, 19, 2, 178, 15, 227, 2, 6, 109, 1, 22, 122, 47, 22, 126, 1, 43, 60, 41, 18, 24, 1, 13, 23, 21, 24, 126, 1, 152, 6
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 02 2014

Keywords

Comments

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

Examples

			a(5) = 23 since 5 + 23 = 28 divides M(5) + M(23) = 21 + 1129760415 = 1129760436 = 28*40348587.

Links

Crossrefs

Cf. A001006, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248139, A248142.

Programs

  • Mathematica
    M[n_]:=Sum[Binomial[n,2k]Binomial[2k,k]/(k+1),{k,0,n/2}]
Do[m=1;Label[aa];If[Mod[M[m]+M[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A248139 Least positive integer m such that m + n divides f(m) + f(n), where f(.) is given by A000172.

Original entry on oeis.org

1, 1, 25, 6, 14, 4, 13, 49, 19, 10, 2, 56, 2, 5, 6, 5, 27, 61, 9, 33, 23, 53, 21, 15, 3, 24, 11, 58, 39, 118, 3, 1598, 20, 40, 4, 2, 58, 26, 29, 17, 47, 34, 4, 31, 43, 163, 41, 25, 8, 26, 67, 40, 21, 214, 535, 12, 7, 22, 164, 74
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 02 2014

Keywords

Comments

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

Examples

			a(5) = 14 since 5 + 14 = 19 divides f(5) + f(14) = 2252 + 112738423360 = 112738425612 = 19*5933601348.

Links

Crossrefs

Cf. A000172, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248137, A248142.

Programs

  • Mathematica
    f[n_]:=Sum[Binomial[n,k]^3,{k,0,n}]
Do[m=1; Label[aa]; If[Mod[f[m]+f[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

A248142 Least positive integer m such that m + n divides A(m) + A(n), where A(.) is given by A005259.

Original entry on oeis.org

1, 1, 7, 2238, 5, 9, 3, 3, 1, 2484, 2, 2, 26, 12, 24, 5, 41, 32, 14, 3, 29, 29, 6, 15, 30, 7, 8, 37, 21, 5, 44, 18, 5, 16, 39, 34, 8, 1, 6, 5, 17, 8, 26, 6, 865, 39, 8, 13, 16, 781, 356, 35, 184, 65, 30, 139, 18, 25, 16, 123
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 02 2014

Keywords

Comments

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

Examples

			 a(3) = 7 since 3 + 7 = 10 divides A(3) + A(7) = 1445 + 584307365 = 584308810.

Links

Crossrefs

Cf. A005259, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248137, A248139.

Programs

  • Mathematica
    A[0]=1;A[1]=5
A[n_]:=((2n-1)(17*n(n-1)+5)*A[n-1]-(n-1)^3*A[n-2])/n^3
Do[m=1; Label[aa]; If[Mod[A[m]+A[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

A248032 Least positive number k such that n + k divides F(n) + F(k), where F(n) denotes the n-th Fibonacci number.

Original entry on oeis.org

1, 22, 9, 8, 1, 18, 10, 4, 3, 5, 35, 12, 17, 10, 3, 8, 13, 6, 31, 4, 9, 2, 47, 12, 5, 8, 9, 20, 1, 18, 19, 16, 3, 5, 10, 12, 13, 10, 38, 8, 109, 6, 7, 4, 9, 2, 14, 12, 1, 22, 39, 8, 97, 18, 58, 4, 32, 14, 191, 12, 89, 1, 27, 8, 85, 6, 70, 4, 39, 2, 11, 24, 77, 22, 15, 13, 73, 18, 10
Offset: 1

Views

Author

Derek Orr, Sep 29 2014

Keywords

Comments

I believe a(n) exists for all n > 0.

Links

Crossrefs

Cf. A000045, A247937.

Programs

  • PARI
    a(n)=m=1;while((fibonacci(n)+fibonacci(m))%(m+n),m++);m
vector(100,n,a(n))

Extensions

Definition clarified by Chai Wah Wu, Jan 02 2018

A248143 Least integer m > 0 such that m + n divides p(m) + p(n), where p(.) is the partition function given by A000041.

Original entry on oeis.org

1, 1, 1, 61, 13, 7, 1, 25, 109, 41, 60, 1, 5, 24, 18, 6, 3, 7, 38, 12, 86, 31, 18, 14, 8, 96, 470, 2, 37, 245, 8, 6, 37, 2, 20, 137, 3, 19, 24, 63, 10, 99, 52, 32, 16, 638, 15, 20, 61, 45, 288, 43, 52, 12, 371, 123, 94, 8, 483, 11
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 02 2014

Keywords

Comments

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

Examples

			a(5) = 13 since 5 + 13 = 18 divides p(5) + p(13) = 7 + 101 = 108.

Links

Crossrefs

Cf. A000041, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248137, A248139, A248142, A248144.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[Mod[PartitionsP[m]+PartitionsP[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
Showing 1-10 of 15 results. Next

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