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.

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A217263 Composite numbers such that every concatenation of their prime factors is prime.

Original entry on oeis.org

21, 33, 51, 63, 93, 111, 133, 177, 201, 219, 247, 253, 327, 411, 427, 549, 573, 589, 657, 679, 687, 763, 793, 813, 833, 889, 993, 1077, 1081, 1119, 1127, 1243, 1339, 1347, 1401, 1411, 1497, 1501, 1603, 1623, 1651, 1671, 1821, 1839, 1843, 1851, 1981, 2009, 2019
Offset: 1

Views

Author

Dmitri Kamenetsky, Sep 29 2012

Keywords

Comments

The smallest term with 3 or more prime factors is 63 = 3*3*7 (see A217264).
The smallest term with 4 or more prime factors is 21249 = 3*3*3*787 (see A217265).
The smallest term with 5 or more prime factors is 146461 = 7*7*7*7*61.
There is no term under 10^8 with 6 or more prime factors.
The smallest term with 3 or more distinct prime factors is 3311 = 7*11*43 (see A180679).
There is no term under 10^8 with 4 or more distinct prime factors.

Examples

			21 is 3*7. Both 37 and 73 are prime, so 21 is in the sequence.
63 is 3*3*7. 337, 373 and 733 are all prime, so 63 is in the sequence.

Crossrefs

Cf. A217264, A217265, A180679, A181559. Related sequences: A105184, A019549 and A106582.

A238499 Primes which are the concatenation of two primes in exactly three ways.

Original entry on oeis.org

3137, 3797, 13997, 19937, 19997, 23911, 23929, 29173, 29311, 31193, 37337, 37397, 43397, 59929, 73331, 78737, 79337, 103997, 109397, 127997, 139967, 173347, 173359, 193337, 193373, 193877, 199337, 199373, 199967, 229373, 233113, 233329, 233353, 233617
Offset: 1

Views

Author

Colin Barker, Feb 27 2014

Keywords

Examples

			13997 is in the sequence because (13, 997), (139, 97), (1399, 7) are all primes, so there are three ways.

Links

Crossrefs

Cf. A105184, A238056, A238057, A238500.

Programs

  • Mathematica
    spl[n_] := Block[{d = IntegerDigits@n, c = 0, z}, z = Length@d; Do[If[PrimeQ@ FromDigits@ Take[d, k] && d[[k + 1]] > 0 && PrimeQ@ FromDigits@ Take[d, k - z], c++], {k, z - 1}]; c]; Select[ Prime@ Range@ 20000, spl[#] == 3 &] (* Giovanni Resta, Mar 03 2014 *)

Extensions

Example clarified by Harvey P. Dale, Jun 09 2025

A121610 Composite numbers that can be written as concatenation of two composite numbers in decimal representation.

Original entry on oeis.org

44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 104, 106, 108, 124, 126, 128, 129, 144, 146, 148, 154, 156, 158, 159, 164, 166, 168, 169, 184, 186, 188, 189, 204, 206, 208, 209, 214, 216, 218, 219, 224, 226, 228, 244, 246, 248, 249, 254, 256, 258
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 10 2006

Keywords

Examples

			A002808(328) = 408 = 40*10+8 = A002808(27)*10+A002808(3),
therefore 408 is a term: a(99) = 408;
A002808(331) = 412 = 4*100+12 = A002808(1)*100+A002808(6),
therefore 412 is a term: a(100) = 412.

Crossrefs

Cf. A121609, A105184, A121608.

A166503 Numbers k with property that k^2 is the concatenation of two or more prime numbers.

Original entry on oeis.org

5, 15, 17, 19, 23, 27, 49, 51, 53, 69, 73, 77, 85, 87, 107, 109, 115, 123, 141, 147, 151, 153, 157, 159, 163, 165, 171, 173, 177, 181, 187, 191, 199, 219, 229, 231, 233, 235, 239, 241, 243, 267, 269, 277, 279, 281, 289, 299, 319, 327, 331, 335, 337, 343, 357
Offset: 1

Views

Author

Zak Seidov, Oct 15 2009

Keywords

Comments

Only odd numbers are eligible.

Links

Crossrefs

Cf. A019549, A038692, A105184, A129800.

Programs

Formula

a(n) = sqrt(A038692(n)).

Extensions

Terms updated according to stricter definition of A152242 by M. F. Hasler, Oct 15 2009

A238500 Primes which are the concatenation of two primes in exactly four ways.

Original entry on oeis.org

233347, 233911, 239929, 337397, 373613, 379397, 733331, 796337, 1321997, 1933331, 2333347, 2333533, 2339929, 2392333, 2393257, 2393761, 2939971, 3136373, 3165713, 3217337, 3319733, 3499277, 3539311, 3727397, 3733967, 3739103, 3739199, 3739397, 3739433
Offset: 1

Views

Author

Colin Barker, Feb 27 2014

Keywords

Examples

			233347 is in the sequence because 2, 33347, 23, 3347, 233, 347, 2333 and 47 are all primes, so there are four ways.

Links

Crossrefs

Cf. A105184, A238056, A238057, A238499.

Programs

  • Mathematica
    spl[n_] := Block[{d = IntegerDigits@n, c = 0, z}, z = Length@d; Do[If[PrimeQ@ FromDigits@ Take[d, k] && d[[k + 1]] > 0 && PrimeQ@ FromDigits@ Take[d, k - z], c++], {k, z - 1}]; c]; Select[ Prime@ Range@ 250000, spl[#] == 4 &] (* Giovanni Resta, Mar 03 2014 *)

A245759 Primes p such that concatenation of p with its digit sum is also prime.

Original entry on oeis.org

61, 83, 137, 139, 197, 199, 223, 241, 281, 313, 337, 353, 373, 397, 421, 449, 557, 577, 647, 719, 773, 809, 881, 937, 953, 971, 991, 1033, 1039, 1091, 1093, 1097, 1129, 1187, 1217, 1277, 1291, 1297, 1303, 1321, 1361, 1381, 1523, 1543, 1567, 1657, 1693, 1723, 1907
Offset: 1

Views

Author

K. D. Bajpai, Jul 31 2014

Keywords

Examples

			61 is in the sequence because it is prime and the concatenation[ 61 with (6 + 1)] = 617 is also prime.
197 is in the sequence because it is prime and the concatenation[ 197 with (1 + 9 + 7)] = 19717 is also prime.

Links

Crossrefs

Cf. A000040, A019518, A105184, A244424, A246428.

Programs

  • Maple
    with(StringTools): A245759 := proc() local a, b, d, e; a:=ithprime(m); b:=add( i,i = convert((a), base, 10))(a); d:=parse(cat(a, b)); e:= parse(cat(b, a)); if isprime(d) then RETURN (a); fi; end: seq(A245759 (), m=1..1000);
  • PARI
    for(n=1,10^3,p=prime(n);if(isprime(eval(concat(Str(p),Str(sumdigits(p))))),print1(p,", "))) \\ Derek Orr, Jul 31 2014

A133187 Prime numbers formed by the concatenation of q and p, where q > p are also primes.

Original entry on oeis.org

53, 73, 113, 137, 173, 193, 197, 233, 293, 313, 317, 373, 433, 593, 613, 617, 673, 677, 733, 797, 977, 1013, 1033, 1093, 1097, 1277, 1373, 1493, 1637, 1733, 1913, 1933, 1973, 1993, 1997, 2113, 2237, 2273, 2293, 2297, 2311, 2333, 2393, 2417, 2633, 2693, 2713
Offset: 1

Views

Author

Tom Mueller (muel4503(AT)uni-trier.de), Dec 17 2007

Keywords

Comments

These numbers are called Caesar primes because the birth date of Julius Caesar (July 13th) provides one example of such a number, i.e. p=7 and q=13 give the prime 137.

Links

Crossrefs

Cf. A019549, A105184.

Programs

  • Mathematica
    lim=2700;plim=Max[FromDigits[Rest[IntegerDigits[lim]]],FromDigits[Drop[IntegerDigits[lim],-1]]];f2p[{p_,q_}]:=FromDigits[Join[IntegerDigits[q],IntegerDigits[p]]];p=Prime[Range[PrimePi[plim]]];p2=Subsets[p,{2}];Union[Select[f2p/@p2,PrimeQ[#]&&#<=lim&]] (* James C. McMahon, Mar 12 2025 *)
  • Python
    from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return any(s[i]!='0' and (q:=int(s[:i])) > (p:=int(s[i:])) and isprime(q) and isprime(p) for i in range(1, len(s)))
print([k for k in range(2800) if ok(k)]) # Michael S. Branicky, Apr 05 2025

Extensions

a(27)-a(47) from James C. McMahon, Mar 12 2025

A168529 Primes n that are concatenation of two 3-digit primes p

Original entry on oeis.org

101107, 101113, 101149, 101173, 101197, 101281, 101293, 101347, 101359, 101383, 101419, 101449, 101467, 101503, 101599, 101641, 101653, 101701, 101719, 101797, 101839, 101863, 101929, 101977, 103307, 103349, 103409, 103421, 103457, 103577
Offset: 1

Views

Author

Zak Seidov, Nov 28 2009

Keywords

Comments

Or, primes of the form 10^3*p+q, p,q primes, 100
There are exactly 1407 such primes from 101107 to 971977.

Links

Crossrefs

Subsequence of A105184.

Programs

  • Mathematica
    p3=Select[Range[100,999],PrimeQ];Le=Length@p3;Reap[Do[p=10^3*p3[[i]];Do[If[PrimeQ[n=p+p3[[k]]],Sow[n]],{k,i+1,Le}],{i,Le-1}]][[2,1]];
  • PARI
    concat(apply(p->select(n->isprime(n) && isprime(n%1000),[1000*p+p+1..1000*p+997]), select(isprime,[101..996]))) \\ Andrew Howroyd, Feb 22 2018

Extensions

Terms a(25) and beyond in b-file corrected by Andrew Howroyd, Feb 22 2018

A244862 List of pairs of prime numbers (p,q) starting with (2, 3) such that p || q (where || denotes concatenation) is a prime number and the sequence is always extended with the smallest prime not yet present in the sequence.

Original entry on oeis.org

2, 3, 5, 23, 7, 19, 11, 17, 13, 61, 29, 53, 31, 37, 41, 59, 43, 73, 47, 83, 67, 79, 71, 167, 89, 101, 97, 103, 107, 137, 109, 139, 113, 131, 127, 157, 149, 173, 151, 163, 179, 233, 181, 193, 191, 227, 197, 257, 199, 211, 223, 229, 239, 251, 241, 271, 263, 269, 277, 331, 281, 317, 283, 397, 293, 311, 307, 337, 313, 373, 347, 359, 349, 379, 353
Offset: 1

Views

Author

Michel Lagneau, Jul 25 2014

Keywords

Examples

			The first few pairs are (2,3),(5,23),(7,19),(11,17),(13,61),(29,53), ..., giving the primes 23, 523, 719, 1117, 1361, 2953, ...

Links

Crossrefs

Cf. A000040, A105184.
A373794 is a very similar sequence (they first differ at term 69).

Programs

  • Maple
    with(numtheory):nn:=60:lst:={2,3}: printf ( "%d %d \n",2,3):
   for a from 2 to nn do:
     p:=ithprime(a):ii:=0:
      for b from 1 to nn while(ii=0)do:
        q:=ithprime(b):s:=p*10^(length(q))+q:
         if type(s,prime)=true and lst intersect {p,q}={}
          then
          lst:=lst union {p,q}:ii:=1:printf(`%d, `,p):printf(`%d, `,q):
          else
        fi:
      od:
    od:
[I have been informed that this program may be incorrect. - N. J. A. Sloane, Jul 03 2024]
# alternative version
P:=proc(q) local a,b,k,i,j,n,ok; a:=[2,3];
for n from 1 to q do k:=3; ok:=1; for i do if ok=1 then k:=nextprime(k);
if numboccur(k,a)=0 then b:=k;
for j from k do k:=nextprime(k); if numboccur(k,a)=0 then
if isprime(b*10^length(k)+k) then a:=[op(a),b,k]; ok:=0; break; fi; fi; od; fi;
else break;fi; od; od; print(op(a)); end: P(500);	# Paolo P. Lava, Jul 03 2024

Extensions

Edited by N. J. A. Sloane, Jul 03 2024. More than the usual number of terms are shown in order to distinguish this from A373794.

A248046 Primes p such that p^2 is the concatenation of two k-digit primes where k is half the length of p^2.

Original entry on oeis.org

5, 73, 337, 409, 701, 827, 5449, 5477, 5939, 6841, 7417, 8353, 8573, 9109, 9227, 9311, 9733, 9767, 32569, 34319, 34327, 34501, 35933, 35999, 38371, 38449, 38923, 38953, 39023, 39367, 39671, 40531, 40973, 42701, 43543, 44651, 45259, 46021, 47623, 48311, 49531, 50923, 54133, 54437, 54547
Offset: 1

Views

Author

Derek Orr, Oct 03 2014

Keywords

Examples

			73 is prime, and 73^2 = 5329 is the concatenation of two 2-digit primes (53 and 29). So 73 is a member of this sequence.
929 is not in the sequence since 929^2 = 863041, where 863 is a 3-digit prime but 041 is a 2-digit prime. - _Jens Kruse Andersen_, Oct 06 2014

Links

Crossrefs

Cf. A105184, A030461, A153048, A153164, A080906, A248208.

Programs

  • PARI
    forprime(p=1,10^5,d=digits(p^2);if((#d)%2==0,if(isprime((p^2)\(10^(#d/2)))&&isprime((p^2)%(10^(#d/2)))&&#Str((p^2)%(10^(#d/2)))==#d/2,print1(p,", "))))

Extensions

Terms and program corrected by Derek Orr to match definition, thanks to Jens Kruse Andersen
Previous Showing 11-20 of 27 results. Next

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

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