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.

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

Views

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

A092117 Numbers n such that the concatenation 2n3n5n7n11n13 is prime.

Original entry on oeis.org

10, 43, 51, 55, 58, 60, 136, 171, 204, 213, 214, 222, 270, 288, 309, 334, 339, 364, 366, 376, 414, 423, 460, 477, 492, 501, 502, 507, 513, 519, 565, 585, 586, 597, 621, 649, 726, 729, 787, 852, 861, 870, 903, 906, 915, 933, 946, 981, 988, 1005, 1038, 1071
Offset: 1

Views

Author

Farideh Firoozbakht, Jun 15 2003

Keywords

Comments

This concatenation is fp(6, n) as defined in A083677.

Examples

			10 is in the sequence because 210310510710111013 is prime.

Links

Crossrefs

Cf. A032711, A083677, A083966, A083677, A092115, A083969.

Programs

  • Mathematica
    v={};Do[If[PrimeQ[FromDigits[Join[{2}, IntegerDigits[n], {3}, IntegerDigits[n], {5}, IntegerDigits[n], {7}, IntegerDigits[n], {1, 1}, IntegerDigits[n], {1, 3}]]], v=Append[v, n]], {n, 1400}];v
fp6Q[n_] := PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ Insert[{2, 3, 5, 7, 11, 13}, n, {{2}, {3}, {4}, {5}, {6}}]]]]; Select[ Range[1100], fp6Q[ # ] &] (* Robert G. Wilson v, Dec 11 2004 *)
Select[Range[1100],PrimeQ[FromDigits[Flatten[IntegerDigits/@Riffle[{2,3,5,7,11,13}, #]]]]&] (* Harvey P. Dale, Mar 21 2013 *)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

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

Views

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

A083684 Numbers k such that there is no nonnegative integer m such that m < k*prime(k) and the concatenated decimal number fp(k,m) = prime(1).m.prime(2).m. ... .prime(k-1).m.prime(k) is prime.

Original entry on oeis.org

3, 10, 16, 28, 34, 40, 46, 52, 70, 76, 82, 88, 97, 103, 121, 127, 136, 163, 166, 169, 175, 187, 199, 205, 211, 217, 220, 235, 250, 262, 268
Offset: 1

Views

Author

Farideh Firoozbakht, Jun 15 2003

Keywords

Comments

If k == 1 (mod 3) and 3 divides 2 + 3 + 5 + ... + prime(k) then k
is in the sequence. I conjecture that 3 is the only term of the sequence which is not of this form.

Examples

			For each m, fp(1,m)=2 is prime so 1 is not in the sequence.
fp(2,2) = 2.2.3 = 223 is prime and 2 < 2*prime(2) so 2 isn't in the sequence. Also for each m, 5 divides fp(3,m) = 2.m.3.m.5 so fp(3,m) is composite and we deduce that 3 is in the sequence.

Crossrefs

Cf. A082549, A083677.

Programs

  • PARI
    is(k) = for(m=0, k*prime(k), if(ispseudoprime(eval(concat(concat([""], vector(2*k-1, i, if(i%2, prime(1+i\2), m)))))), return(0))); 1; \\ Jinyuan Wang, Apr 10 2020

Extensions

Corrected and edited by Farideh Firoozbakht, Nov 04 2013

A083750 Numbers n such that 2111131111...p(n-1)1111p(n) is prime.

Original entry on oeis.org

85, 145, 304, 915
Offset: 1

Views

Author

Farideh Firoozbakht, Jun 17 2003

Keywords

Comments

Let fp(n,k) be the decimal concatenation of prime(n), k, and fp(n-1,k) for n > 1, and fp(1,k) = 2. Then a(n) = fp(n, 1111).
No more terms up to 6300.

Examples

			a(1)=85 because fp(85,1111)= 211113111151111...4331111439 is prime and fp(k,1111) is composite for k< 85 (prime(85)=439).
a(3)=304 because fp(304,1111)=211113111151111...199911112003 is a prime related to prime year 2003; this prime number has 2231 digits. fp(915,1111)=211113111151111...712911117151 is a prime with 7119 digits (prime(915)=7151).

Links

Crossrefs

Cf. A083677, A082549.

Extensions

Comment from Charles R Greathouse IV, Oct 12 2009

A084048 Integers m such that the base-10 digit concatenation 2//m//3//m//5//m...//prime(49)//m//prime(50) is prime.

Original entry on oeis.org

96, 359, 546, 1422, 1644, 1980, 2241, 3458, 3606, 4530, 4629, 5018, 5090, 5114, 5166, 7007, 7389, 8534, 9123, 9717, 9771, 10065, 10343, 10355, 10514, 10596, 11307, 11361, 11531, 12401, 12759, 13707, 14810, 15185, 15290, 15614, 15728, 16038
Offset: 1

Views

Author

Farideh Firoozbakht, Jun 19 2003

Keywords

Comments

This is in the family of sequences fp(j,m)=prime(1)//m//prime(2)//m//prime(3)//...//m//prime(j), of which A032711 = fp(2,m) is a simplest type.

Examples

			96 is in the sequence because fp(50,k)=2//k//3//k//5//k//7//k//11//k//13...//k//229 fp(50,96)=296396596...22796229 is prime.
fp(50,96) is prime number with 219 digits.

Crossrefs

Cf. A083677, A032711.

Programs

  • Mathematica
    Select[Range[16100],PrimeQ[FromDigits[Flatten[IntegerDigits/@ Riffle[ Prime[ Range[ 50]],#]]]]&] (* Harvey P. Dale, Jan 07 2021 *)
Showing 1-7 of 7 results.

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

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