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 1-5 of 5 results.

A084036 Duplicate of A092117.

Original entry on oeis.org

10, 43, 51, 55, 58, 60, 136, 171, 204, 213, 214, 222, 270, 288, 309, 334, 339, 364, 366
Offset: 1

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Author

Keywords

A032711 Numbers k such that k prefixed by '2' and followed by '3' is prime.

Original entry on oeis.org

2, 3, 6, 8, 9, 11, 14, 15, 20, 21, 24, 27, 29, 33, 38, 39, 42, 47, 50, 54, 59, 63, 66, 68, 69, 71, 75, 80, 83, 84, 90, 95, 96, 101, 102, 114, 116, 119, 128, 131, 132, 138, 143, 149, 150, 152, 156, 161, 167, 168, 171, 177, 180, 186, 189, 194, 200, 201, 206, 207, 209
Offset: 1

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Author

Patrick De Geest, May 15 1998

Keywords

Comments

This sequence is infinite, a consequence of the Prime Number Theorem in arithmetic progressions. - Charles R Greathouse IV, Sep 26 2012

Examples

			8 and 21 are in the sequence because 283 and 2213 are primes.

Links

Crossrefs

Cf. A032702-A032733.
Cf. A083677, A083966, A083969, A092117.

Programs

  • Mathematica
    v={};Do[If[PrimeQ[FromDigits[Join[{2},IntegerDigits[n],{3}]]], v=Append[v,n]],{n, 260}];v (* Farideh Firoozbakht, Jun 15 2003 *)
Select[Range[210],PrimeQ[FromDigits[Join[{2},IntegerDigits[#],{3}]]]&] (* Harvey P. Dale, May 02 2012 *)
  • PARI
    for( n=1,300, isprime(eval(Str(2,n,3))) & print1(n",")) \\ M. F. Hasler, Mar 18 2008

Extensions

Merged with data from duplicate entry A092114. - M. F. Hasler, Mar 18 2008

A083677 Define f(n, k) to be the concatenation of the first n primes, with n-1 k's inserted between the primes. Then a(n) is the smallest k >= 0 such that f(n, k) is prime, or -1 if no such prime exists.

Original entry on oeis.org

0, 2, -1, 1, 4, 10, 38, 20, 0, -1, 163, 46, 8, 53, 0, -1, 74, 5, 8, 5, 180, 4, 280, 191, 0, 337, 191, -1, 105, 88, 19, 28, 111, -1, 525, 13, 24, 102, 159, -1, 288, 142, 31, 743, 81, -1, 183, 202, 100, 96, 380, -1, 1227, 5, 113, 123, 20, 23, 0, 48, 148, 438, 52, 144, 128, 297, 206
Offset: 1

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Author

Farideh Firoozbakht, Jun 15 2003

Keywords

Comments

a(3) = -1 because f(3, k) is always a multiple of 5. For any n such that n = 1 (mod 3) and A007504(n) = 0 (mod 3), a(n) = -1 because f(n, k) is always a multiple of 3. It is my conjecture that for all other n, -1 < a(n) < n*p(n). I've checked for all n < 270.

Examples

			a(4) = 1 because 2030507 is composite and 2131517 is prime.

Links

Crossrefs

A082549 gives the n such that a(n) = 0. A083684 gives the n such that a(n)=-1.
Cf. A082549, A032711, A083966, A083969, A092117, A090529.

Programs

  • Mathematica
    fpkQ[k_, n_] := PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ Insert[ Table[ Prime[i], {i, k}], n, Table[{i}, {i, 2, k}]]]]]; a[1] = 0; a[3] = a[10] = a[16] = a[28] = a[34] = a[40] = a[46] = a[52] = a[70] = a[76] = a[82] = a[88] = a[97] = -1; a[n_] := Block[{k = 0}, While[ fpkQ[n, k] != True, k++ ]; k]; Table[ a[n], {n, 70}] (* Robert G. Wilson v, Dec 11 2004 *)

Extensions

Edited and extended by Robert G. Wilson v, Dec 11 2004

A083966 Numbers n such that the concatenation 2n3n5n7 is prime.

Original entry on oeis.org

1, 6, 8, 9, 16, 17, 18, 21, 23, 24, 29, 32, 39, 64, 70, 78, 84, 85, 98, 1000, 1005, 1013, 1033, 1038, 1041, 1047, 1056, 1065, 1066, 1076, 1087, 1091, 1102, 1107, 1109, 1115, 1118, 1121, 1137, 1139, 1152, 1156, 1164, 1167, 1171, 1173, 1185, 1199, 1220, 1241
Offset: 1

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Author

Farideh Firoozbakht, Jun 15 2003, Jun 19 2003

Keywords

Comments

Numbers n such that the concatenation of 2, n, 3, n, 5, n and 7 is prime.
This concatenation is fp(4, n) as defined in A083677.
For any 3-digit number n, fp(4, n) is divisible by 7, so there are no 3-digit numbers in the sequence.
More generally, there are no (3+6*k)-digit numbers in the sequence for any k. - Robert Israel, Nov 12 2019

Examples

			8 and 21 are in the sequence because 2838587 and 2213215217 are primes.
16 is in the sequence because 2163165167 is prime.

Links

Crossrefs

Cf. A032711, A083677, A083969, A092117.

Programs

  • Maple
    filter:= proc(n) local m;
  m:= ilog10(n)+1;
isprime(n*(10 + 10^(m+2)+ 10^(2*m+3))+7+5*10^(m+1)+3*10^(2*m+2)+2*10^(3*m+3))
end proc:
select(filter, [$1..2000]); # Robert Israel, Nov 12 2019
  • Mathematica
    v={};Do[If[PrimeQ[FromDigits[Join[{2},IntegerDigits[n],{3}, IntegerDigits[n],{5},IntegerDigits[n],{7}]]],v=Append[v,n]], {n,1300}];v
Select[Range[1300],PrimeQ[FromDigits[Flatten[IntegerDigits/@ Riffle[ {2,3,5,7}, Table[#,{3}]]]]]&](* Harvey P. Dale, Nov 24 2015 *)
  • PARI
    is(n)=isprime(eval(Str(2,n,3,n,5,n,7))) \\ Charles R Greathouse IV, May 15 2013

Extensions

Edited and extended by David Wasserman, Dec 06 2004
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007

A083969 Numbers n such that 2.n.3.n.5.n.7.n.11 is prime (dot means concatenation).

Original entry on oeis.org

4, 18, 33, 42, 43, 57, 73, 76, 78, 87, 91, 93, 97, 102, 112, 114, 120, 141, 151, 177, 186, 193, 196, 219, 261, 267, 276, 280, 300, 307, 318, 322, 342, 352, 364, 366, 402, 435, 438, 445, 457, 462, 468, 484, 511, 580, 582, 633, 646, 651, 679, 706, 745, 774, 783
Offset: 1

Views

Author

Farideh Firoozbakht, Jun 19 2003

Keywords

Examples

			2.4.3.4.5.4.7.4.11 = 2434547411, which is prime. Hence 4 is in the sequence.

Links

Crossrefs

Cf. A083677, A032711, A092115, A092117.

Programs

  • Mathematica
    v={};Do[If[PrimeQ[FromDigits[Join[{2}, IntegerDigits[n], {3}, IntegerDigits[n], {5}, IntegerDigits[n], {7}, IntegerDigits[n], {1, 1}]]], v=Append[v, n]], {n, 1000}];v
Select[Range[660], PrimeQ[FromDigits[Join[{2}, IntegerDigits[ # ], {3}, IntegerDigits[ # ], {5}, IntegerDigits[ # ], {7}, IntegerDigits[ # ], {1, 1}]]] &] (* Stefan Steinerberger, Jun 28 2007 *)
  • Python
    from sympy import isprime
def aupton(terms):
  n, alst = 1, []
  while len(alst) < terms:
    s = str(n)
    t = int('2'+s+'3'+s+'5'+s+'7'+s+'11')
    if isprime(t): alst.append(n)
    n += 1
  return alst
print(aupton(55)) # Michael S. Branicky, Apr 18 2021

Extensions

Edited by Stefan Steinerberger, Jun 28 2007
Edited by N. J. A. Sloane, Sep 18 2008 at the suggestion of R. J. Mathar
Showing 1-5 of 5 results.

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