A247824 -id:A247824 - cp's OEIS frontend

A247869 (prime(A247824(n)) + prime(n)) / (A247824(n) + n).

2, 2, 2, 2, 2, 2, 4, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 5, 6, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 5, 5, 5, 5

Offset: 1

Reinhard Zumkeller, Sep 27 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A247824, A000040.

a247869 n = (a000040 (a247824 n) + a000040 n) `div` (a247824 n + n)

A247937 Least integer m > n such that m + n divides F(m) + F(n), where F(k) refers to the Fibonacci number A000045(k).

Original entry on oeis.org

5, 22, 9, 8, 8, 18, 10, 16, 21, 14, 35, 24, 17, 34, 21, 32, 20, 30, 31, 28, 87, 26, 47, 36, 28, 46, 63, 32, 80, 42, 151, 40, 75, 38, 38, 60, 113, 39, 51, 56, 109, 49, 307, 52, 63, 50, 50, 72, 101, 70, 57, 68, 97, 66, 58, 64, 93, 62, 191, 84

Offset: 1

Zhi-Wei Sun, Sep 27 2014

nonn

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

This implies that a(n) exists for any n > 0.

			a(2) = 22 since 22 + 2 = 24 divides F(22) + F(2) = 17711 + 1 = 17712 = 24*738.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000045, A247824, A247940, A248032.

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

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

Zhi-Wei Sun, Sep 27 2014

nonn

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.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000032, A247824, A247937.

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}]

A248004 Least positive integer m with prime(m*n) == 1 (mod m+n).

Original entry on oeis.org

3, 4, 1, 2, 2, 15, 1, 1, 5, 10, 2, 3, 4, 18, 6, 27, 4, 7, 35, 4, 45, 2, 47, 9, 5, 10, 16, 11, 3, 3, 9, 61, 1, 52, 3, 60, 53, 74, 8, 47, 7, 60, 128, 5, 21, 12, 2, 29, 15, 127, 53, 28, 17, 21, 303, 80, 72, 8, 61, 36

Offset: 1

Zhi-Wei Sun, Sep 29 2014

nonn

Conjecture: (i) a(n) exists for any n > 0. Moreover, a(n) does not exceed n*(n-1)/2 if n > 2.

(ii) For each positive integer n, there is an integer m > 0 with prime(m*n) == -1 (mod m+n). Moreover, we may require that m does not exceed n*(n-1)/2 if n > 2.

			a(2) = 4 since prime(2*4) = 19 is congruent to 1 modulo 2 + 4 = 6.
a(5146) = 593626 since prime(5146*593626) = prime(3054799396) = 73226821741 is congruent to 1 modulo 5146 + 593626 = 598772.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000040, A247824, A247975.

Do[m=1;Label[aa];If[Mod[Prime[m*n],m+n]==1,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
lpim[n_]:=Module[{m=1},While[Mod[Prime[m*n],m+n]!=1,m++];m]; Array[lpim,60] (* Harvey P. Dale, Oct 01 2017 *)

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

Zhi-Wei Sun, Oct 01 2014

nonn

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 4..10000

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

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}]

A247975 Least positive integer m such that m + n divides prime(m)^2 + prime(n)^2.

Original entry on oeis.org

1, 8, 15479, 30, 29, 68, 51, 2, 73, 15, 39, 13, 12, 36, 10, 25, 33, 8, 15, 38, 40, 108, 42, 1, 16, 39, 31, 57, 5, 4, 27, 2, 17, 51, 30, 14, 36, 20, 11, 21, 32, 23, 39, 689, 29, 4, 27, 1873, 184, 7248, 7, 153, 132, 76, 75, 18, 28, 99, 2, 86

Offset: 1

Zhi-Wei Sun, Sep 28 2014

nonn

Conjecture: a(n) exists for any n > 0. - Zhi-Wei Sun, Sep 28 2014

If a(i) = j, then a(j) <= i. - Derek Orr, Sep 28 2014

			a(2) = 8 since 8 + 2 = 10 divides prime(8)^2 + prime(2)^2 = 19^2 + 3^2 = 370.
a(3) = 15479 since 15479 + 3 = 15482 divides prime(15479)^2 + prime(3)^2 = 169789^2 + 5^2 = 28828304546 = 15482*1862053.
a(4703) = 760027770 since 760027770 + 4703 = 760032473 divides prime(760027770)^2 + prime(4703)^2 = 17111249191^2 + 45329^2 = 292794848878552872722 = 760032473*385239919714.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Zhi-Wei Sun)
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000040, A247824.

Do[m = 1; Label[aa]; If[Mod[Prime[m]^2 + Prime[n]^2, m + n] == 0, Print[n, " ", m]; Goto[bb]]; m = m + 1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

a(n)=m=1;while((prime(m)^2+prime(n)^2)%(m+n),m++);m
vector(75,n,a(n)) \\ Derek Orr, Sep 28 2014

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

Zhi-Wei Sun, Oct 01 2014

nonn

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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

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}]

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

Zhi-Wei Sun, Oct 02 2014

nonn

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.

			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).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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

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

Zhi-Wei Sun, Oct 02 2014

nonn

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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

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

Zhi-Wei Sun, Oct 02 2014

nonn

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..2187

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

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}]

cp's OEIS Frontend

A247869 (prime(A247824(n)) + prime(n)) / (A247824(n) + n).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

A247937 Least integer m > n such that m + n divides F(m) + F(n), where F(k) refers to the Fibonacci number A000045(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A248004 Least positive integer m with prime(m*n) == 1 (mod m+n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A247975 Least positive integer m such that m + n divides prime(m)^2 + prime(n)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs