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 1-7 of 7 results.

A116602 Values n such that prime(2*n) mod 12 == 5.

Original entry on oeis.org

5, 8, 12, 13, 15, 20, 30, 31, 33, 49, 51, 52, 54, 58, 63, 70, 71, 74, 76, 79, 80, 81, 85, 88, 89, 92, 93, 97, 98, 103, 106, 109, 110, 117, 119, 126, 134, 135, 144, 145, 148, 149, 151, 156, 160, 165, 166, 169, 171, 175, 178, 181, 191, 192, 196, 199, 200, 201, 202, 204
Offset: 1

Views

Author

Roger L. Bagula, Mar 29 2006

Keywords

Comments

Eight sets of integers (this, A116610 and A116612-A116617), dealing with the remainder classes {1,5,7,11} modulo 12 are inspired by the Taylor series of sin(x) = x-x^3/3!+x^5/5!-... and cos(x)=1-x^2/2!+x^4/4!-x^6/6!+... which might be constructed via partitioning of the series terms according to factorial denominators of 2n+1 = 1,3,5,7,.. and 2n=1,2,4,6,...

Examples

			33 is in the sequence because the 66th prime is 317 and 317 mod 12=5.

Links

Programs

  • Maple
    a:=proc(n) if ithprime(2*n) mod 12 = 5 then n fi end: seq(a(n),n=1..220);
  • Mathematica
    a[1] = Flatten[Table[If[Mod[Prime[2*n], 12] == 5, n, {}], {n, 1, 100}]]
Select[Range[210],Mod[Prime[2#],12]==5&] (* Harvey P. Dale, Apr 12 2018 *)
  • PARI
    for(n=1,999, prime(2*n)%12==5 & print1(n",")) \\ M. F. Hasler, May 22 2009

Formula

A116602 = 1/2 * {even terms in A160591}. - M. F. Hasler, May 22 2009

Extensions

Edited by N. J. A. Sloane, Apr 05 2006, May 22 2009

A116612 Values of n such that p(2n+1) mod 12 = 1, where p(j) denotes the j-th prime.

Original entry on oeis.org

10, 12, 14, 18, 26, 29, 32, 60, 63, 72, 73, 75, 79, 84, 91, 93, 94, 95, 98, 101, 105, 110, 115, 118, 125, 128, 131, 135, 136, 137, 139, 143, 147, 150, 159, 165, 168, 169, 170, 173, 177, 180, 184, 187, 195, 200, 203, 205, 207, 212, 215, 217, 218, 221, 222, 224
Offset: 1

Views

Author

Roger L. Bagula, Mar 29 2006

Keywords

Examples

			29 is in the sequence because the 59th prime is 277 and 277 mod 12=1.

Links

Crossrefs

Cf. A116602, A116610, A116613-A116617, A160591-A160594. [M. F. Hasler, May 22 2009]

Programs

  • Maple
    a:=proc(n) if ithprime(2*n+1) mod 12 = 1 then n else fi end: seq(a(n),n=0..250);
  • Mathematica
    Select[Range[0, 2000], Mod[Prime[2*# + 1], 12] == 1 &] (* G. C. Greubel, Oct 27 2017 *)
  • PARI
    for(n=1,999, prime(2*n+1)%12==1 & print1(n",")) \\ M. F. Hasler, May 22 2009

Formula

A116612 = integer part of 1/2 * { odd terms in A160594 = A000720(A068228) }. - M. F. Hasler, May 22 2009

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

A116613 Values of n such that prime(2n+1) mod 12 = 5.

Original entry on oeis.org

1, 3, 6, 16, 17, 22, 25, 27, 28, 35, 38, 39, 43, 44, 48, 56, 59, 61, 67, 68, 69, 82, 99, 100, 104, 111, 113, 122, 127, 129, 132, 133, 145, 146, 156, 161, 162, 171, 172, 176, 179, 183, 186, 189, 190, 202, 209, 210, 234, 238, 250, 251, 258, 261, 272, 275, 280, 284
Offset: 1

Views

Author

Roger L. Bagula, Mar 29 2006

Keywords

Examples

			25 is in the sequence because the 51st prime is 233 and 233 mod 12 = 5.

Links

Crossrefs

Cf. A116602, A116610, A116612-A116617, A160591-A160594. - M. F. Hasler, May 22 2009

Programs

  • Maple
    a:=proc(n) if ithprime(2*n+1) mod 12 = 5 then n else fi end: seq(a(n),n=0..300);
  • Mathematica
    Select[Range[0, 500], Mod[Prime[2*# + 1], 12] == 5 &] (* G. C. Greubel, Nov 19 2017 *)
  • PARI
    for(n=1,999, prime(2*n+1)%12==5 & print1(n",")) \\ M. F. Hasler, May 22 2009

Formula

Equals the integer part of { odd terms in A160591 = A000720(A040117) } / 2. - M. F. Hasler, May 22 2009

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

A116614 Values of n such that prime(2n) mod 12 = 1.

Original entry on oeis.org

3, 6, 9, 21, 22, 25, 34, 35, 37, 39, 40, 41, 42, 44, 50, 53, 55, 56, 61, 65, 67, 68, 84, 86, 87, 90, 99, 102, 108, 113, 122, 127, 130, 132, 139, 142, 153, 154, 155, 158, 161, 163, 176, 183, 189, 190, 194, 198, 211, 219, 225, 232, 233, 237, 238, 248, 258, 259, 278
Offset: 1

Views

Author

Roger L. Bagula, Mar 29 2006

Keywords

Examples

			53 is in the sequence because the 106th prime is 577 and 577 mod 12 = 1.

Links

Crossrefs

Cf. A116602, A116610, A116612-A116617, A160591-A160594. [M. F. Hasler, May 22 2009]

Programs

  • Maple
    a:=proc(n) if ithprime(2*n) mod 12 = 1 then n else fi end: seq(a(n),n=1..300);
  • Mathematica
    Select[Range[1, 500], Mod[Prime[2*#], 12] == 1 &] (* G. C. Greubel, Nov 19 2017 *)
  • PARI
    for(n=1,999, prime(2*n)%12==1 & print1(n",")) \\ M. F. Hasler, May 22 2009

Formula

Equals (1/2) * { even terms in A160594 = A000720(A068228) }. - M. F. Hasler, May 22 2009

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

A116615 Values of n such that prime(2n) mod 12 = 7.

Original entry on oeis.org

2, 4, 7, 11, 17, 18, 19, 23, 24, 29, 45, 57, 69, 94, 101, 105, 111, 112, 116, 121, 129, 133, 136, 137, 138, 141, 150, 157, 162, 164, 170, 172, 174, 177, 184, 187, 197, 203, 207, 209, 220, 231, 235, 239, 245, 250, 251, 252, 254, 255, 260, 261, 270, 273, 276, 283
Offset: 1

Views

Author

Roger L. Bagula, Mar 29 2006

Keywords

Examples

			23 is in the sequence because the 46th prime is 199 and 199 mod 12=7.

Links

Crossrefs

Cf. A116602, A116610, A116612-A116617, A160591-A160594. [M. F. Hasler, May 22 2009]

Programs

  • Maple
    a:=proc(n) if ithprime(2*n) mod 12 = 7 then n else fi end: seq(a(n),n=1..300);
  • Mathematica
    Select[Range[1, 500], Mod[Prime[2*#], 12] == 7 &] (* G. C. Greubel, Nov 19 2017 *)
  • PARI
    for( n=1,999, prime(2*n)%12==7 & print1(n",")) \\ M. F. Hasler, May 22 2009

Formula

Equals (1/2) * { even terms in A160592 = A000720(A068229) }. - M. F. Hasler, May 22 2009

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

A160593 Indices of primes congruent to 11 modulo 12.

Original entry on oeis.org

5, 9, 15, 17, 20, 23, 28, 32, 39, 41, 43, 49, 52, 54, 56, 64, 69, 72, 76, 81, 83, 86, 91, 92, 94, 96, 103, 107, 109, 118, 120, 124, 128, 132, 144, 146, 150, 154, 156, 161, 164, 166, 171, 173, 182, 185, 190, 192, 195, 200, 205, 208, 214, 215, 219, 225, 228, 230, 236
Offset: 1

Views

Author

M. F. Hasler, May 22 2009

Keywords

Comments

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 02 2021

Examples

			a(1) = 5 since the 5th prime, A000040(5) = 11, is the first one to be equal to 11 (mod 12).
a(2) = 9 since the 9th prime, A000040(9) = 23, is the second one to be equal to 11 (mod 12).

Links

Crossrefs

A116610 lists the even terms of this sequence, divided by 2.
Cf. A000720, A068231.

Programs

  • Mathematica
    Position[Mod[Prime[Range[250]],12],11]//Flatten (* Harvey P. Dale, Apr 13 2022 *)
  • PARI
    for( n=1,999, prime(n)%12==11 & print1(n","))

Formula

a(n) = A000720(A068231(n)).

A116616 Values of n such that prime(2n+1) mod 12 = 7.

Original entry on oeis.org

5, 9, 13, 15, 23, 30, 31, 33, 36, 37, 42, 46, 47, 49, 50, 52, 55, 57, 58, 62, 64, 65, 66, 70, 71, 74, 76, 77, 78, 81, 83, 87, 88, 89, 90, 96, 103, 106, 108, 114, 116, 117, 121, 123, 124, 130, 134, 142, 144, 148, 151, 152, 160, 163, 166, 167, 175, 182, 185, 191, 192
Offset: 1

Views

Author

Roger L. Bagula, Mar 29 2006

Keywords

Examples

			33 is in the sequence because the 67th prime is 331 and 331 mod 12=7.

Links

Crossrefs

Cf. A116602, A116610, A116612-A116617, A160591-A160594. [M. F. Hasler, May 22 2009]

Programs

  • Maple
    a:=proc(n) if ithprime(2*n+1) mod 12 = 7 then n else fi end: seq(a(n),n=0..215);
  • Mathematica
    Select[Range[220], Mod[Prime[2# + 1], 12] == 7 &] (* Stefan Steinerberger, Apr 08 2006 *)
  • PARI
    for(n=1,999, prime(2*n+1)%12==7 & print1(n",")) \\ M. F. Hasler, May 22 2009

Formula

Equals { odd terms in A160592 = A000720(A068229) } / 2, rounded towards zero. - M. F. Hasler, May 22 2009

Extensions

Edited by N. J. A. Sloane, Apr 05 2006
Showing 1-7 of 7 results.

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

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