A045533 -id:A045533 - cp's OEIS frontend

A030459 Prime p concatenated with next prime is also prime.

2, 31, 83, 151, 157, 167, 199, 233, 251, 257, 263, 271, 331, 353, 373, 433, 467, 509, 523, 541, 601, 653, 661, 677, 727, 941, 971, 1013, 1033, 1181, 1187, 1201, 1223, 1259, 1367, 1453, 1459, 1657, 1669, 1709, 1741, 1861, 1973, 2069, 2161

Offset: 1

Patrick De Geest

Terms 157, 257, 263, 541, 1187, 1459, 2179 also belong to A030460. - Carmine Suriano, Jan 27 2011

All terms, except for the first one, must be either in A185934 or in A185935, i.e., have the same residue (mod 6) as the subsequent prime. - M. F. Hasler, Feb 06 2011

M. F. Hasler and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4110 terms from Hasler)

See A030461 for the concatenated primes.

Cf. A000720, A030460, A045533, A151799.

Select[Prime[Range[500]],PrimeQ[FromDigits[Join[IntegerDigits[#], IntegerDigits[ NextPrime[#]]]]]&] (* Harvey P. Dale, Jun 20 2011 *)

o=2;forprime(p=3,1e4, isprime(eval(Str(o,o=p))) & print1(precprime(p-1)","))  \\ M. F. Hasler, Feb 06 2011

a(n) = A151799(A030460(n)).

A030461(n) = concat(a(n), A030460(n)) = A045533(A000720(a(n))).

A030461 Primes that are concatenations of two consecutive primes.

Original entry on oeis.org

23, 3137, 8389, 151157, 157163, 167173, 199211, 233239, 251257, 257263, 263269, 271277, 331337, 353359, 373379, 433439, 467479, 509521, 523541, 541547, 601607, 653659, 661673, 677683, 727733, 941947, 971977, 10131019

Offset: 1

Patrick De Geest

Any term in the sequence (apart from the first) must be a concatenation of consecutive primes differing by a multiple of 6. - Francis J. McDonnell, Jun 26 2005

			a(2) is 3137 because 31 and 37 are consecutive primes and after concatenation 3137 is also prime. - _Enoch Haga_, Sep 30 2007

Georg Fischer, Table of n, a(n) for n = 1..5720 [First 1000 terms from Zak Seidov]

Cf. A030459.
Cf. A185934, A185935.
Subsequence of A045533.

a030461 n = a030461_list !! (n-1)
a030461_list = filter ((== 1) . a010051') a045533_list
-- Reinhard Zumkeller, Apr 20 2012

[Seqint( Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)) ): n in [1..200 ]| IsPrime(Seqint( Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)) )) ]; // Marius A. Burtea, Mar 21 2019

conc:=proc(a,b) local bb: bb:=convert(b,base,10): 10^nops(bb)*a+b end: p:=proc(n) local w: w:=conc(ithprime(n),ithprime(n+1)): if isprime(w)=true then w else fi end: seq(p(n),n=1..250); # Emeric Deutsch

Select[Table[p=Prime[n]; FromDigits[Join[Flatten[IntegerDigits[{p,NextPrime[p]}]]]],{n,170}],PrimeQ] (* Jayanta Basu, May 16 2013 *)

{digits(n) = if(n==0,[0],u=[];while(n>0,d=divrem(n,10);n=d[1];u=concat(d[2],u));u)} {m=1185;p=2;while(pKlaus Brockhaus

o=2;forprime(p=3,1e4, isprime(eval(Str(o,o=p))) & print1(precprime(p-1),p",")) \\ M. F. Hasler, Feb 06 2011

A030461(n) = concat(A030459(n),A030460(n)) = A045533( A000720( A030459(n))). - M. F. Hasler, Feb 06 2011

Edited by N. J. A. Sloane, Apr 19 2009 at the suggestion of Zak Seidov

A095958 Twin prime pairs concatenated in decimal representation.

Original entry on oeis.org

35, 57, 1113, 1719, 2931, 4143, 5961, 7173, 101103, 107109, 137139, 149151, 179181, 191193, 197199, 227229, 239241, 269271, 281283, 311313, 347349, 419421, 431433, 461463, 521523, 569571, 599601, 617619, 641643, 659661

Offset: 1

Reinhard Zumkeller, Jul 14 2004

a(n) mod 3 = 0 for n>1, proof: A007953(a(n)) = 2*A007953(A001359(n)+1) and A007953(A001359(n)) mod 3 = 2 for n>1, therefore A007953(a(n)) mod 3 = 0.

			29 = A001359(5), 29 + 2 = 31 = A006512(5): a(5) = 2931.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A045533, A001359, A006512.

Cf. A077800, subsequence of A045533.

a095958 n = a095958_list !! (n-1)
a095958_list = f $ map show a077800_list :: [Integer] where
   f (t:t':ts) = read (t ++ t') : f ts
-- Reinhard Zumkeller, Apr 20 2012

[Seqint( Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)) ): n in [1..150 ]| NthPrime(n+1)-NthPrime(n) eq 2 ]; // Marius A. Burtea, Mar 21 2019

concat[{a_,b_}]:=FromDigits[Flatten[IntegerDigits/@{a,b}]]; concat/@ Select[Partition[ Prime[ Range[150]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Apr 20 2012 *)

A132903 Numbers formed by concatenating 3 consecutive prime numbers.

Original entry on oeis.org

235, 357, 5711, 71113, 111317, 131719, 171923, 192329, 232931, 293137, 313741, 374143, 414347, 434753, 475359, 535961, 596167, 616771, 677173, 717379, 737983, 798389, 838997, 8997101, 97101103, 101103107, 103107109, 107109113, 109113127

Offset: 1

Omar E. Pol, Sep 04 2007

			Prime numbers.
2
3
5 -----------> a(1) = 235
7 -----------> a(2) = 357
11 ----------> a(3) = 5711

Harvey P. Dale, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Prime numbers: A000040. Cf. A007795, A045533, A091762.

FromDigits[Flatten[IntegerDigits[#]]]&/@Partition[Prime[Range[40]],3,1] (* Harvey P. Dale, Jan 03 2021 *)

A030460 Previous prime concatenated with this prime p is a prime.

Original entry on oeis.org

3, 37, 89, 157, 163, 173, 211, 239, 257, 263, 269, 277, 337, 359, 379, 439, 479, 521, 541, 547, 607, 659, 673, 683, 733, 947, 977, 1019, 1039, 1187, 1193, 1213, 1229, 1277, 1373, 1459, 1471, 1663, 1693, 1721, 1747, 1867, 1979, 2081, 2179

Offset: 1

Patrick De Geest

Terms 157, 257, 263, 541, 1187, 1459, 2179 also belong to A030459. [Carmine Suriano, Jan 27 2011]

Harvey P. Dale, Table of n, a(n) for n = 1..10000 [extending M.F. Hasler's prior b-File beyond n=4110]

Transpose[Select[Partition[Prime[Range[400]],2,1], PrimeQ[FromDigits[ Flatten[IntegerDigits/@#]]]&]][[2]] (* Harvey P. Dale, Dec 23 2011 *)

c=0; o=2; s=1; forprime(p=3,default(primelimit), p>s & s*=10; isprime(o*s+o=p) & write("b030460.txt",c++," ",p))  \\ M. F. Hasler, Feb 06 2011

A030459(n) = A151799(a(n)). - M. F. Hasler, Feb 06 2011

A030461(n) = concat(A030459(n),a(n)) = A045533( A000720( A030459(n))). - M. F. Hasler, Feb 06 2011

Offset changed from 0 to 1 by M. F. Hasler, Feb 06 2011

A229814 Primes which are a concatenation of prime(i) and prime(prime(i)) for some i.

Original entry on oeis.org

23, 1759, 2383, 41179, 61283, 71353, 83431, 127709, 2271433, 3372269, 3532381, 4993559, 5033593, 5714153, 7275503, 8876899, 9117109, 9377351, 9717649, 10618513, 142711909, 157913297, 166314107, 169314437, 170914591, 187116073, 187716127, 190716451, 194916901

Offset: 1

K. D. Bajpai, Sep 30 2013

			a(2)=1759: prime(7)= 17 and prime(17)= 59. Concatenating 17 and 59 gives 1759 which is prime.
a(4)=41179: prime(13)= 41 and prime(41)= 179. Concatenating 41 and 179 gives 41179 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A030459, A045533.

K := proc(n) local a,b,d; a :=ithprime(n);  b:=ithprime(a); d:=parse(cat(a,b)); if isprime(d) then return (d) end if; end proc:
seq(K(n), n=1..1000);

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

K. D. Bajpai, Jun 18 2014

The semiprimes in A045533.

			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.

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

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);

A244057 = {}; Do[t = FromDigits[Flatten[IntegerDigits /@ {Prime[n], Prime[n + 1]}]]; If [PrimeOmega[t] == 2, AppendTo[A244057, t]], {n, 100}]; A244057

A132901 Numbers formed by concatenating 11 consecutive prime numbers.

Original entry on oeis.org

235711131719232931, 3571113171923293137, 57111317192329313741, 711131719232931374143, 1113171923293137414347, 1317192329313741434753, 1719232931374143475359, 1923293137414347535961, 2329313741434753596167

Offset: 1

Omar E. Pol, Sep 04 2007

Matthew House, Table of n, a(n) for n = 1..10000

Prime numbers: A000040. Cf. A007795, A045533, A059932, A091762.

A132905 Numbers formed by concatenating 5 consecutive prime numbers.

Original entry on oeis.org

235711, 3571113, 57111317, 711131719, 1113171923, 1317192329, 1719232931, 1923293137, 2329313741, 2931374143, 3137414347, 3741434753, 4143475359, 4347535961, 4753596167, 5359616771, 5961677173, 6167717379, 6771737983

Offset: 1

Omar E. Pol, Sep 04 2007

Prime numbers: A000040. Cf. A007795, A045533, A059932, A091762.

A153353 List of pairs of concatenated adjacent primes [p(n),p(n+1)] such that the concatenation is divisible by n.

Original entry on oeis.org

23, 57, 127131, 937941, 2087320879, 6557965581, 925039925051, 14073611407383, 89811798981191, 94647139464723, 1553629915536303, 122504623122504633, 200602291200602313, 495401873495401911, 133201458751133201458773

Offset: 1

Gil Broussard, Dec 24 2008

The corresponding values of n are listed in A125085.

Subset of A045533. [R. J. Mathar, Jan 03 2009]

			23 is a term because 2 and 3 are adjacent primes and 23 is divisible by 1 (the position of 2 in the sequence of primes).
127131 is a term because 127 and 131 are adjacent primes and 127131 is divisible by 31 (the position of 127 in the sequence of primes).

Cf. A007795, A045533, A095958.

Cf. A000040.

A055642 := proc(n) max(1,ilog10(n)+1) ; end: cat2 := proc(a,b) a*10^A055642(b)+b ; end: A045533 := proc(n) cat2(ithprime(n),ithprime(n+1)) ; end: for n from 1 to 800000 do if A045533(n) mod n = 0 then printf("%d,",A045533(n)) ; fi; od: # R. J. Mathar, Jan 03 2009

p = q = 2; c = 0; lst = {}; Do[c++; q = NextPrime@q; r = FromDigits@ Flatten[{IntegerDigits@ p, IntegerDigits@ q}]; If[Mod[r, c] == 0, AppendTo[lst, r]; Print[{c, r}]]; c++; q = NextPrime@ q; p = q, {n, 174254000}]; lst (* Robert G. Wilson v, Jan 23 2009 *)

3 more terms from R. J. Mathar, Jan 03 2009
a(9)-a(14) from Robert G. Wilson v, Jan 23 2009
a(15) from Donovan Johnson, Aug 08 2010

cp's OEIS Frontend

A030459 Prime p concatenated with next prime is also prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A030461 Primes that are concatenations of two consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A095958 Twin prime pairs concatenated in decimal representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

A132903 Numbers formed by concatenating 3 consecutive prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A030460 Previous prime concatenated with this prime p is a prime.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

Extensions

A229814 Primes which are a concatenation of prime(i) and prime(prime(i)) for some i.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A244057 Semiprimes which are concatenation of two consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple