A032711 -id:A032711 - cp's OEIS frontend

A092114 Duplicate of A032711.

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

Offset: 1

dead

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.

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

Farideh Firoozbakht, Jun 15 2003

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.

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

C. Rivera, On Solution Of Puzzle 208 .

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.

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 *)

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

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

Farideh Firoozbakht, Jun 15 2003

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

			10 is in the sequence because 210310510710111013 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

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 *)

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

Farideh Firoozbakht, Jun 15 2003, Jun 19 2003

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

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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A032711, A083677, A083969, A092117.

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

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 *)

is(n)=isprime(eval(Str(2,n,3,n,5,n,7))) \\ Charles R Greathouse IV, May 15 2013

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

Farideh Firoozbakht, Jun 19 2003

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

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A083677, A032711, A092115, A092117.

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 *)

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

Edited by Stefan Steinerberger, Jun 28 2007

Edited by N. J. A. Sloane, Sep 18 2008 at the suggestion of R. J. Mathar

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

Farideh Firoozbakht, Jun 19 2003

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.

			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.

Cf. A083677, A032711.

Select[Range[16100],PrimeQ[FromDigits[Flatten[IntegerDigits/@ Riffle[ Prime[ Range[ 50]],#]]]]&] (* Harvey P. Dale, Jan 07 2021 *)

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A092114 Duplicate of A032711.

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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.

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

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

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

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

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