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.

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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}]

A248052 Least positive integer m such that m + n divides pi(m^2) + pi(n^2), where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

1, 2, 1, 8, 7, 7, 5, 1, 25, 35, 15, 14, 13, 12, 1, 4, 23, 532, 22, 385, 113, 1, 17, 138, 8, 92, 80, 44, 116, 128, 586, 165, 5, 464, 10, 39, 80, 38, 1, 52, 33, 118, 6, 28, 11, 1239, 47, 92, 517, 3, 145, 40, 8, 184, 104, 104, 16, 73, 53, 52, 5, 145, 172, 68, 11
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 30 2014

Keywords

Comments

Conjecture: (i) a(n) exists for any n > 0.
(ii) For each integer m > 0, there is a positive integer n such that m + n divides prime(m^2) + prime(n^2).

Examples

			a(4) = 8 since 8 + 4 = 12 divides pi(8^2) + pi(4^2) = 18 + 6 = 24.

Links

Crossrefs

Cf. A000040, A000720, A247824, A247975, A248044, A248054.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[Mod[PrimePi[m^2]+PrimePi[n^2],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,70}]
a[ n_] := If[ n<1, 0, Module[ {m=1}, While[ Not @ Divisible[ PrimePi[m^2] + PrimePi[n^2], m + n], m++]; m]]; (* Michael Somos, Sep 30 2014 *)

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}]

A248590 Least positive integer m such that prime(m) == 1 (mod m + n).

Original entry on oeis.org

3, 4, 19, 10, 5, 6, 13, 15, 7, 8, 31, 17, 9, 19, 20, 38, 22, 10, 11, 24, 78, 80, 25, 12, 28, 30, 13, 14, 599, 97, 15, 31, 32, 178, 33, 16, 102, 104, 35, 108, 17, 18, 38, 39, 361, 40, 19, 41, 73, 20, 21, 43, 45, 78, 134, 22, 391, 47, 23, 84
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 09 2014

Keywords

Comments

Conjecture: (i) a(n) exists for any n > 0. Moreover, a(n) < n*(n-1) if n > 3.
(ii) For any n > 0, there is a positive integer m such that prime(m) == -1 (mod m + n). Moreover, we may require m < n*(n-1) if n > 1.

Examples

			a(3) = 19 since prime(19) = 67 == 1 (mod 19 + 3).

Links

Crossrefs

Cf. A000040, A247824, A248004, A248588.

Programs

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

A248044 Least positive integer m such that m + n divides pi(m)^2 + pi(n)^2, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

1, 3, 1, 4, 12, 11, 1, 8, 7, 16, 2, 5, 26, 25, 24, 4, 228, 227, 46, 45, 44, 43, 16, 6, 5, 1, 27, 26, 45, 44, 12526, 12525, 12524, 12523, 2970, 502, 351, 350, 46, 45, 236, 235, 10, 9, 8, 4, 1078, 1077, 576, 575, 574, 198, 63, 62, 61, 176, 16, 10, 362, 70
Offset: 1

Views

Author

Zhi-Wei Sun, Sep 30 2014

Keywords

Comments

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

Examples

			a(5) = 12 since 12 + 5 = 17 divides pi(12)^2 + pi(5)^2 = 5^2 + 3^2 = 34.

Links

Crossrefs

Cf. A000720, A247824, A247975, A248035, A248036.

Programs

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

A248175 Least positive integer m such that m + n divides q(m*n), where q(.) is the strict partition function given by A000009.

Original entry on oeis.org

11, 4, 9, 2, 12, 10, 9, 16, 3, 6, 1, 5, 2, 18, 7, 8, 5, 14, 11, 36, 2, 34, 4, 8, 31, 6, 15, 36, 23, 2, 9, 14, 17, 22, 11, 18, 1, 22, 11, 7, 1, 22, 12, 7, 55, 7, 19, 40, 15, 6, 31, 12, 43, 10, 25, 40, 7, 91, 61, 20
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 03 2014

Keywords

Comments

Conjecture: (i) a(n) exists for any n > 0.
(ii) For each n > 0, there is a positive integer m such that m + n divides q(m) + q(n).

Examples

			a(3) = 9 since 9 + 3 = 12 divides q(9*3) = 192 = 12*16.

Links

Crossrefs

Cf. A000009, A247824, A248004, A248143, A248144.

Programs

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

A248366 Least positive integer m such that prime(m+n) - prime(m) divides m + n.

Original entry on oeis.org

1, 1, 5, 10, 175, 22, 23, 34, 35, 102, 57, 54, 63, 70, 345, 74, 279, 198, 225, 124, 127, 294, 145, 130, 149, 334, 831, 164, 191, 720, 183, 520, 209, 486, 259, 990, 231, 226, 227, 268, 663, 294, 701, 326, 301, 308, 335, 310, 311, 790
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 05 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n^2 except for n = 5, 10, 15, 27.
See also A248369 for a similar conjecture.

Examples

			a(5) = 175 since prime(175+5) - prime(175) = 1069 - 1039 = 30 divides 175 + 5 = 180.

Links

Crossrefs

Cf. A000040, A247824, A248369.

Programs

  • Mathematica
    q[n_]:=q[n]=PartitionsQ[n]
Do[m=1;Label[aa];If[Mod[m+n,Prime[m+n]-Prime[m]]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,50}]
  • PARI
    a(n)=m=1;while((m+n)%(prime(m+n)-prime(m)),m++);m
vector(100,n,a(n)) \\ Derek Orr, Oct 05 2014

A248369 Least positive integer m such that prime(m+n) - prime(m) divides m.

Original entry on oeis.org

1, 6, 54, 18, 26, 24, 80, 120, 180, 48, 70, 160, 92, 82, 220, 98, 228, 102, 378, 130, 348, 152, 158, 172, 202, 372, 204, 720, 206, 448, 218, 560, 236, 228, 222, 1480, 282, 1656, 636, 300, 312, 322, 764, 350, 356, 352, 362, 420, 434, 860
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 05 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0. Moreover, for n > 9 we have a(n) < n^2 except for n = 12, 19, 36, 38.
Note that for each n > 1 the term a(n) should be even and at least 2*n.

Examples

			a(7) = 80 since prime(80+7) - prime(80) = 449 - 409 = 40 divides 80.

Links

Crossrefs

Cf. A000040, A247824, A248366.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[Mod[m,Prime[m+n]-Prime[m]]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,50}]
lpi[n_]:=Module[{m=1},While[Mod[m,Prime[n+m]-Prime[m]]!=0,m++];m]; Array[ lpi,50] (* Harvey P. Dale, Jan 17 2022 *)
  • PARI
    a(n)=m=1;while(m%(prime(m+n)-prime(m)),m++);m
vector(100,n,a(n)) \\ Derek Orr, Oct 05 2014

A248197 Least positive integer m such that m + n divides prime(prime(m)) + prime(prime(n)).

Original entry on oeis.org

1, 9, 4, 1, 17, 12, 3, 4, 2, 4, 15, 6, 1, 20, 4, 74, 4, 3, 2, 8, 9, 5, 3, 17, 5, 9, 8, 26, 8, 1, 14, 4, 17, 35, 33, 52, 29, 46, 35, 95, 4, 4, 23, 24, 23, 38, 135, 64, 11, 62, 222, 36, 92, 41, 1, 39, 6, 37, 3, 18
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 03 2014

Keywords

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) < n*(n-1) if n > 2.

Examples

			a(3) = 4 since 3 + 4 = 7 divides prime(prime(3)) + prime(prime(4)) = prime(5) + prime(7) = 11 + 17 = 28.
a(2479) = 3386154 since 2479 + 3386154 = 3388633 divides prime(prime(2479)) + prime(prime(3386154)) = prime(22111) + prime(56851657) = 250963 + 1124775193 = 1125026156 = 332*3388633.

Links

Crossrefs

Cf. A000040, A247824, A247975.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[Mod[Prime[Prime[m]]+Prime[Prime[n]],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
Previous Showing 11-20 of 22 results. Next

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