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 1-4 of 4 results.

A279097 Numbers k such that prime(k) divides primorial(j) + 1 for some j.

Original entry on oeis.org

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 59, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 177, 179, 180, 186, 205, 207, 208, 210, 211, 218, 221, 233, 243, 249, 256, 260, 261, 262
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			1 is in the sequence because primorial(0) + 1 = 1 + 1 = 2 is divisible by prime(1) = 2.
4 is in the sequence because primorial(2) + 1 = 2*3 + 1 = 7 is divisible by prime(4) = 7.
8 is in the sequence because primorial(7) + 1 = 2*3*5*7*11*13*17 + 1 = 510511 is divisible by prime(8) = 19.
59 is in the sequence because primorial(7) + 1 = 510511 is divisible by prime(59) = 277 (and primorial(17) + 1 = 1922760350154212639071 is divisible by prime(59) as well).
5 is not in the sequence because there is no number j such that primorial(j) + 1 is divisible by prime(5) = 11:
    primorial(1) + 1 = 2       + 1 =   3 == 3 (mod 11)
    primorial(2) + 1 = 2*3     + 1 =   7 == 7 (mod 11)
    primorial(3) + 1 = 2*3*5   + 1 =  31 == 9 (mod 11)
    primorial(4) + 1 = 2*3*5*7 + 1 = 211 == 2 (mod 11)
and primorial(j) + 1 = 2*...*11*... + 1  == 1 (mod 11) for all j >= 5.

Links

Crossrefs

Cf. A000040, A002110, A006862, A113165, A279098, A279099, A283928.

Programs

  • Mathematica
    np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] > 0 &] (* Giovanni Resta, Mar 29 2017 *)

A279098 Numbers k such that prime(k) divides primorial(j) + 1 for exactly one integer j.

Original entry on oeis.org

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 179, 180, 186, 205, 207, 208, 210, 211, 218, 233, 243, 249, 256, 261, 262, 297, 308, 316, 318
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			59 is not in this sequence because both primorial(7) + 1 = 510511 and primorial(17) + 1 = 1922760350154212639071 are divisible by prime(59) = 277.

Links

Crossrefs

Subsequence of A279097 (which also includes numbers k such that prime(k) divides primorial(j) + 1 for more than one integer j).
Cf. A000040, A002110, A006862, A113165, A279099, A283928.

Programs

  • Mathematica
    np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] == 1 &] (* Giovanni Resta, Mar 29 2017 *)

A279099 Numbers k such that prime(k) divides primorial(j) + 1 for exactly two integers j.

Original entry on oeis.org

59, 177, 221, 260, 285, 431, 476, 489, 625, 653, 686, 860, 957, 1320, 1334, 1734, 1987, 2140, 2215, 2854, 2991, 3051, 3341, 3455, 3535, 3591, 3645, 3695, 3798, 4020, 4032, 4079, 4612, 4614, 4856, 4904, 5019, 5234, 5263, 5842, 6178, 6184, 6491, 6639, 6745, 7151
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			59 is in this sequence because prime(59) = 277 divides primorial(j) + 1 for exactly two integers j: 7 and 17.
436 is not in this sequence because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j: 206, 263, and 409.

Links

Crossrefs

Subsequence of A279097 (which includes all numbers k such that prime(k) divides primorial(j) + 1 for one or more integers j); cf. A279098 (exactly one integer j).
Cf. A000040, A002110, A006862, A113165, A279098, A283928.

Programs

  • Mathematica
    np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[1000], np[#] == 2 &] (* Giovanni Resta, Mar 29 2017 *)

A284754 a(n) is the smallest number k such that prime(k) divides primorial(j) + 1 for exactly n integers j.

Original entry on oeis.org

1, 59, 436, 995752, 180707
Offset: 1

Views

Author

Jon E. Schoenfield, Apr 01 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).
a(n) > 10^7 for n > 5. - Giovanni Resta, Apr 03 2017

Examples

			a(1) = 1 because the first prime, prime(1) = 2, divides primorial(j) + 1 for exactly one integer j, namely, j = 0 (since primorial(0) = 1).
a(2) = 59 because prime(59) = 277 divides primorial(j) + 1 for exactly two integers j (i.e., 7 and 17), and 59 is the smallest integer for which this is the case.
a(3) = 436 because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j (i.e., 206, 263, and 409), and 436 is the smallest integer for which this is the case.
a(5) = 180707 because prime(180707) = 2464853 divides primorial(j) + 1 for exactly five integers j (i.e., 75366, 79914, 139731, 139990, and 175013), and 180707 is the smallest integer for which this is the case.

Crossrefs

Cf. A000040, A002110, A006862, A113165, A279097, A279098, A279099, A283928.

Extensions

a(4) from Giovanni Resta, Apr 02 2017
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