A247940 -id:A247940 - cp's OEIS frontend

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

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}]

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}]

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}]

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

Zhi-Wei Sun, Oct 02 2014

nonn

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

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

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

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

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

Zhi-Wei Sun, Oct 02 2014

nonn

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..2853

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

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}]

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

Zhi-Wei Sun, Oct 02 2014

nonn

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

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..765

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

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}]

A248593 Least positive integer m such that m + n divides F(m), where F(m) is the m-th Fibonacci number given by A000045.

Original entry on oeis.org

10, 6, 84, 12, 16, 7, 27, 9, 144, 30, 28, 12, 8, 30, 14, 18, 57, 19, 342, 18, 20, 24, 66, 12, 9, 27, 144, 60, 112, 35, 16, 24, 60, 55, 20, 12, 40, 111, 24, 36, 88, 72, 80, 48, 10, 15, 72, 24, 224, 18, 50, 54, 270, 72, 54, 33, 224, 18, 28, 12

Offset: 1

Zhi-Wei Sun, Oct 09 2014

nonn

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n*(n-1) except for n = 1, 2, 3, 9.
In contrast, it is easy to show that for any integer n > 0, there is a positive integer m such that m + n divides 2^m - 1.
a(n) exists for any n > 0. See Bloom (1998). - Amiram Eldar, Jan 15 2022

			a(1) = 10 since 10 + 1 = 11 divides F(10) = 55.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
David M. Bloom, Offset Entries, Solution to Problem B-830, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 89-90.

Cf. A000045, A247937, A247940, A248588, A248590.

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

cp's OEIS Frontend

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

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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).

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

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PARI

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

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

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

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

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

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