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 1-6 of 6 results.

A023592 Least odd prime divisor of 2*prime(n)+1.

Original entry on oeis.org

5, 7, 11, 3, 23, 3, 5, 3, 47, 59, 3, 3, 83, 3, 5, 107, 7, 3, 3, 11, 3, 3, 167, 179, 3, 7, 3, 5, 3, 227, 3, 263, 5, 3, 13, 3, 3, 3, 5, 347, 359, 3, 383, 3, 5, 3, 3, 3, 5, 3, 467, 479, 3, 503, 5, 17, 7, 3, 3, 563, 3, 587, 3, 7, 3, 5, 3, 3, 5, 3, 7, 719, 3, 3, 3, 13, 19, 3, 11, 3
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Except for first term, same as A094460.

Programs

  • Mathematica
    Table[FactorInteger[2*Prime[n] + 1][[1, 1]], {n, 1, 80}] (* Amiram Eldar, Mar 05 2020 *)
  • PARI
    a(n) = factor(2*prime(n)+1)[1, 1]; \\ Michel Marcus, Oct 02 2013

A094461 a[n] is the 5th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

13, 13, 331, 13, 7, 6163, 7, 571, 13, 10267, 23, 31, 7, 13, 17, 7, 3, 7, 5227, 43, 7, 2371, 7, 61, 19, 3, 7, 13, 3271, 13, 5, 37, 4111, 43, 3, 13, 47, 7, 5011, 360187, 7, 73, 13, 22003, 23, 7, 8863, 5, 7, 6871, 181, 193, 7, 7, 11, 139, 3, 7, 1297, 73, 7, 7, 31, 3, 7
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p[n], 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is here in A094461;
6th, 7th terms in A094462, A094463;

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=5};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094463 a(n) is the 7th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

5, 5, 199, 5, 433, 1601, 31, 457, 7109609443, 5, 7, 127, 71, 5, 7, 2620003, 4583, 1217, 5, 67, 6729871, 39334891, 5, 53, 461, 449885311, 1511, 197, 7, 22008559, 19, 1249, 7, 7, 3217, 7, 7, 3931, 7, 110663370509047, 375155719, 29, 28529671, 23, 24603331
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p(n), 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is in A094461;
6th-7th terms in A094462, A094463;

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=6};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094462 a(n) is the 6th term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

53, 53, 19, 53, 10627, 7, 3571, 271, 84319, 7, 47059, 7, 47, 53, 23971, 11, 13, 5, 7, 201499, 5, 7, 67, 13, 7, 21211, 5, 29, 10696171, 11, 149, 971, 16896211, 11, 58111, 17, 11, 75307, 25105111, 853, 139, 7, 5, 613, 181, 23, 13, 29, 13, 19, 53, 47, 5, 11, 84811
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p(n), 2nd equals 2;
3rd term is A091460 as largest p-divisor of 2p+1
(occasionally safe primes, A005385);
4th terms listed in A051614; 5th term is in A094461;
6th-7th terms in A094462, A094463;

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308-A051335, A093778-A093782, A094152-A094153, A094460-A094463.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]];ta=Table[0, {168}];a[1]=1; Do[{a[1]=Prime[j], el=6};Print[a[el];ta[[j]]=a[el];j++ ], {j, 1, 168}];ta

A094464 Least initial value for a Euclid/Mullin sequence whose 3rd term (= least prime divisor of 1+2p) equals the n-th prime. prime(1)=2 is never a third term, so offset=2.

Original entry on oeis.org

7, 17, 2, 5, 149, 263, 389, 11, 449, 821, 1091, 881, 1913, 23, 2729, 29, 2531, 6599, 2591, 6971, 3989, 41, 4583, 5189, 7019, 7673, 53, 13679, 7853, 8699, 12641, 9521, 13691, 12143, 19403, 13109, 22901, 83, 18251, 89, 20543, 32183, 23063, 26693
Offset: 2

Views

Author

Labos Elemer, May 10 2004

Keywords

Comments

These primes are congruent to (prime(n)-1)/2 mod prime(n) if n > 4. Presumably all primes occur as 3rd term if initial prime is suitably chosen.

Examples

			n=25: prime(25) = 97 and an Euclid-Mullin sequence started with a(25) = 5189 = 97*53 + 48 is {5189, 2, 97, 101, 3, 7, 167, 356568143863}.
All larger (prime) solutions with 97 as 3rd term have the form 97k + 48 form. However, not all primes of the form 97k + 48 result in Euclid-Mullin (EM) sequences with the property that the 3rd term is 97. For example, 727 = 7*97 + 48 is a prime providing an EM sequence as follows: {727, 2, 3, 4363, 19, 5, 1709, 11, 33988283132431, 7} with 3rd term = 3.
Analogous statements hold for other initial or 3rd primes.

Crossrefs

Cf. A000945, A051308-A051334, A094460.

Programs

  • Mathematica
    a[x_]:=First[Flatten[FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]]; ta=Table[0, {20000}];a[1]=1;Do[{a[1]=Prime[j], el=3}; ta[[j]]=a[el], {j, 1, 20000}] Table[Prime[Min[Flatten[Position[ta, Prime[w]]]]], {w, 1, 100}]

Formula

a(n) = Min[x; A094460(x) = prime(n)].

A094465 Least initial value for an Euclid/Mullin sequence whose 4th term is prime(n). prime(1)=2 is never a fourth term, so offset=2.

Original entry on oeis.org

5, 19, 43, 31, 67, 541, 193, 157, 1213, 811, 487, 2371, 2, 1543, 733, 1319, 1291, 1753, 1621, 2713, 13, 1231, 2833, 2053, 1801, 3313, 5011, 821, 2467, 5101, 3253, 8573, 3637, 1553, 15427, 5521, 3191, 9173, 7237, 10531, 11071, 6271, 9103, 15727, 7993
Offset: 2

Views

Author

Labos Elemer, May 10 2004

Keywords

Examples

			n=14: prime(14) = 43 and an Euclid-Mullin sequence started with a(14) = 2 = prime(1) is {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...} is A000945, the prototype EM-sequence.
n=7: a(7) = prime(100) = 541, with EM sequence as follows: {541, 2, 3, 17, 139, 7, 1871, 100457892907, 19, 11047, ...}, where the 4th term equals prime(n) = prime(7) = 17.
It is characteristic but not so simple congruence relations holds of term(1) mod term(4) form for various first or 4th primes, not necessarily smallest ones. See comment at A094464.

Crossrefs

Cf. A000945, A051308-A051334, A094460, A094464.

Programs

  • Mathematica
    a[x_]:=First[Flatten[FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]]; ta=Table[0, {20000}];a[1]=1;Do[{a[1]=Prime[j], el=4}; ta[[j]]=a[el], {j, 1, 20000}] Table[Prime[Min[Flatten[Position[ta, Prime[w]]]]], {w, 1, 100}]

Formula

a(n) = Min_{k} A051614(k) = prime(n).
Showing 1-6 of 6 results.

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