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.

Previous Showing 11-17 of 17 results.

A244057 Semiprimes which are concatenation of two consecutive primes.

Original entry on oeis.org

35, 57, 1317, 1923, 2329, 2931, 4143, 5359, 5961, 6167, 7379, 8997, 103107, 131137, 181191, 193197, 211223, 227229, 281283, 307311, 347349, 367373, 379383, 383389, 421431, 443449, 503509, 547557, 557563, 577587, 587593, 593599, 607613, 619631, 641643, 691701, 709719
Offset: 1

Views

Author

K. D. Bajpai, Jun 18 2014

Keywords

Comments

The semiprimes in A045533.

Examples

			35 is in the sequence because the concatenation of [3, 5] = 35 = 5 * 7, which is semiprime.
1923 is in the sequence because concatenation of [19, 23] = 1923 = 3 * 641, which is semiprime.
1113 is not in the sequence because, though 1113 is concatenation of two consecutive primes [11, 13], 1113 = 3 * 7 * 53, which is not semiprime.

Links

Crossrefs

Cf. A000040, A001358, A030469, A045533, A244007.

Programs

  • Maple
    with(numtheory):with(StringTools):A244057:= proc() local a,b,k; a:=ithprime(n); b:=ithprime(n+1); k:=parse(cat(a,b)); if bigomega(k)=2 then RETURN (k); fi; end: seq(A244057 (), n=1..200);
  • Mathematica
    A244057 = {}; Do[t = FromDigits[Flatten[IntegerDigits /@ {Prime[n], Prime[n + 1]}]]; If [PrimeOmega[t] == 2, AppendTo[A244057, t]], {n, 100}]; A244057

A244163 Primes which are the concatenation of three consecutive primes p, q, r while the sum (p + q + r) yields another prime.

Original entry on oeis.org

5711, 111317, 171923, 313741, 414347, 229233239, 389397401, 401409419, 409419421, 449457461, 701709719, 773787797, 787797809, 797809811, 140914231427, 157915831597, 163716571663, 202920392053, 212921312137, 252125312539, 259125932609, 263326472657, 268926932699
Offset: 1

Views

Author

K. D. Bajpai, Jun 21 2014

Keywords

Comments

Subsequence of A030469.
The first five terms of this sequence resemble exactly those of A030469.

Examples

			5711 is in the sequence because the concatenation of [5, 7, 11] = 5711 which is prime. The sum [5 + 7 + 11] = 23 is another prime.
111317 is in the sequence because the concatenation of [11, 13, 17] = 111317 which is prime. The sum [11 + 13 + 17] = 41 is another prime.

Links

Crossrefs

Cf. A000040, A030469, A034961, A034962, A132903.

Programs

  • Maple
    A244163:= proc() local a,b,c,k,m; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); m:=a+b+c; k:=parse(cat(a,b,c)); if isprime(k) and isprime(m) then RETURN (k); fi; end: seq(A244163 (), n=1..500);
  • Mathematica
    prQ[{a_,b_,c_}]:=Module[{p=FromDigits[Flatten[IntegerDigits/@ {a,b,c}]]}, If[ AllTrue[ {p,a+b+c},PrimeQ],p,Nothing]]; prQ/@Partition[ Prime[ Range[ 500]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 05 2021 *)

A384953 First of three consecutive primes whose concatenations, both forward and backward, are primes.

Original entry on oeis.org

313, 359, 383, 449, 619, 787, 827, 907, 1697, 2503, 2521, 2857, 3673, 3853, 4139, 4363, 4993, 5281, 5527, 5563, 5641, 5851, 6037, 6043, 6719, 7019, 7477, 9281, 10177, 10459, 13799, 14009, 15013, 15511, 17167, 17209, 19183, 19423, 20483, 20743, 21397, 21407, 25111
Offset: 1

Views

Author

Robert Israel, Jun 13 2025

Keywords

Examples

			a(3) = 383 is a term because 383, 389 and 397 are consecutive primes and both 383389397 and 397389383 are prime.

Links

Crossrefs

Cf. A030469, A104328.

Programs

  • Maple
    rcat:= proc(L) local x,i;
  x:= L[1];
  for i from 2 to nops(L) do
    x:= 10^(1+ilog10(x))*L[i] + x
  od;
  x
end proc:
fcat:= proc(L) local x,i;
  x:= L[1];
  for i from 2 to nops(L) do
    x:= 10^(1+ilog10(L[i]))*x + L[i]
  od;
  x
end proc:
P:= select(isprime,[seq(i,i=3..30000,2)]):
J:=  select(i -> isprime(rcat(P[i..i+2])) and isprime(fcat(P[i..i+2])), [$1..nops(P)-2]):
P[J];

A239974 Primes which are a concatenation of prime(k+4), prime(k+2) and prime(k) for some k.

Original entry on oeis.org

1373, 433729, 615343, 797161, 837367, 897971, 149137127, 193181173, 227211197, 337317311, 367353347, 401389379, 443433421, 557541521, 577569557, 587571563, 757743733, 811797773, 823811797, 10191009991, 10211013997, 116311511123, 120111871171, 130713011291
Offset: 1

Views

Author

K. D. Bajpai, Mar 30 2014

Keywords

Comments

All the terms in the sequence are primes which are a reverse concatenation of prime(k), prime(k+2) and prime(k+4) for some k.

Examples

			1373 is a prime and appears in the sequence because it is the concatenation of prime(2+4), prime(2+2) and prime(2).
433729 is a prime and appears in the sequence because it is the concatenation of prime(10+4), prime(10+2) and prime(10).

Links

Crossrefs

Cf. A000040, A030461, A030469, A030473, A086040, A086041, A239789.

Programs

  • Maple
    with(StringTools): KD := proc() local a, b, d, e; a:=ithprime(n+4); b:=ithprime(n+2); d:=ithprime(n);  e:= parse(cat(a, b, d)); if isprime(e) then RETURN (e); fi; end: seq(KD(), n=1..500);
  • Mathematica
    Select[Table[FromDigits[Flatten[{IntegerDigits[Prime[n+4]],IntegerDigits[Prime[n+2]], IntegerDigits[Prime[n]]}]], {n,1,500}], PrimeQ]

A244064 Primes which are concatenation of prime(n), prime(10n) and prime(100n).

Original entry on oeis.org

4173310657, 97158322307, 137221330559, 223341346447, 251390752919, 271423157191, 367552173973, 433647386371, 487729796581, 491741197813, 5097643101281, 6018831116447, 6179109119983, 6439439124577, 70910343136379, 71910459137477, 82311933155327, 82912109157739
Offset: 1

Views

Author

K. D. Bajpai, Jun 19 2014

Keywords

Examples

			4173310657 is in the sequence because concatenation of [prime(13), prime(130), prime(1300)] = 4173310657, which is a prime.
97158322307 is in the sequence because concatenation of [prime(25), prime(250), prime(2500)] = 97158322307, which is a prime.

Links

Crossrefs

Cf. A000040, A030469.
Cf. A031343 (prime(10n)), A031921 (prime(100n)).

Programs

  • Maple
    with(numtheory): with(StringTools): A244064:= proc() local a,b,c,m; a:=ithprime(n); b:=ithprime(10*n); c:=ithprime(100*n);m:=parse(cat(a,b,c)); if isprime(m) then RETURN (m); fi; end: seq(A244064 (), n=1..300);
  • Mathematica
    A244064 = {}; Do[t = FromDigits[Flatten[IntegerDigits /@ {Prime[n], Prime[10 n], Prime[100 n]}]]; If[PrimeQ[t], AppendTo[A244064, t]], {n, 300}]; A244064

A258214 Primes formed by concatenating p^2 with q, where p, q are consecutive primes.

Original entry on oeis.org

43, 257, 12113, 84131, 96137, 168143, 372167, 32041181, 120409349, 139129379, 292681547, 410881643, 516961727, 528529733, 863041937, 966289991, 10629611033, 10670891039, 11902811093, 16307291279, 21112091459, 25058891597, 29618411723, 31933691789, 35006411873
Offset: 1

Views

Author

K. D. Bajpai, May 23 2015

Keywords

Comments

All the terms in this sequence, except a(1), are congruent to 2 (mod 3).

Examples

			a(2) = 257 is prime formed by concatenation of (5^2) = 25 with 7.
a(3) = 12113 is prime formed by concatenation of (11^2) = 121 with 13.

Links

Crossrefs

Cf. A000040, A030461, A030469.

Programs

  • Magma
    [m: n in [1..300] | IsPrime(m) where m is Seqint(Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)^2))]; // Vincenzo Librandi, May 24 2015
  • Mathematica
    Select[Table[p = Prime[n]; FromDigits[Join[Flatten[ IntegerDigits[{p^2, NextPrime[p]}]]]], {n, 500}], PrimeQ]
Select[#[[1]]^2*10^IntegerLength[#[[2]]]+#[[2]]&/@Partition[Prime[ Range[ 300]],2,1],PrimeQ] (* Harvey P. Dale, Dec 05 2016 *)
  • PARI
    forprime(p = 1,5000, k=eval(concat( Str(p^2), Str(nextprime(p+1)) )); if(isprime(k), print1(k,", ")))

A174287 Smallest natural square base q = q(k) that concatenation prime(k)//prime(k+1)//q^2 (k = 1, 2, ...) is a prime number.

Original entry on oeis.org

3, 3, 1, 11, 1, 1, 1, 1, 1, 1, 3, 7, 31, 13, 9, 1, 1, 141, 53, 37, 9, 11, 1, 7, 61, 7, 17, 13, 17, 1, 17, 11, 7, 23, 7, 27, 27, 7, 1, 9, 19, 29, 7, 29, 19, 3, 3, 1, 43, 67, 1, 7, 7, 9, 9, 1, 13, 21, 7, 7, 7, 1, 1, 43, 1, 1, 57, 1, 67, 7, 17
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 15 2010

Keywords

Comments

Note two consecutive primes prime(k)//prime(k+1)
Necessarily q is odd and has end digit 1, 3, 7 or 9

Examples

			3^2=9, 239 = prime(52) => q(1) = 3
359 = prime(72) => q(2) = 3
k=18, prime(18) = 61, 141^2 = 19881, 616719881 = prime(32151650) => q(18) = 141

References

  • J.-P. Allouche, J. Shallit: Automatic Sequences, Theory, Applications, Generalizations, Cambridge University Press, 2003

Crossrefs

A000290, A030461, A030459, A030469, A171154, A174031, A174034
Previous Showing 11-17 of 17 results.

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

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