A030461 -id:A030461 - cp's OEIS frontend

A267692 Terms of A030461 that give successive gap records.

23, 3137, 199211, 523541, 16691693, 1393313963, 2428124317, 3498135023, 4028940343, 191353191413, 221327221393, 507217507289, 843911844001, 25654632565559, 55778515577959, 82237498223863, 1127656111276687, 1280935912809491, 2038858320388727, 3338451733384667, 5272111352721287, 9280102992801209

Offset: 1

Jean-Marc Rebert, Jan 19 2016

Subsequence of A030461.

			a(1) = A030461(1) = 23. gap = 3 - 2 = 1.
a(2) = 3137, because 3137 is the first term in A030461 > 23, with gap = 37 - 31 = 6 > 1.
a(3) = 199211, because 199211 is the first term in A030461 > 3137, with gap = 211 - 199 = 12 > 6.

Jean-Marc Rebert, Table of n, a(n) for n = 1..23

Cf. A030461, A267721.

Reap[For[record = 0; p = 2, p < 10^8, p = NextPrime[p], If[PrimeQ[pp = FromDigits[Join[IntegerDigits[p], IntegerDigits[np = NextPrime[p]]]]], If[np - p > record, record = np - p; Print[pp]; Sow[pp]]]]][[2, 1]] (* Jean-François Alcover, Mar 02 2016 *)

A267721 a(n) is the least term of A030461 with gap = 6*n between consecutive primes or 0 if no such term exists.

Original entry on oeis.org

3137, 199211, 523541, 16691693, 1393313963, 2428124317, 3498135023, 7318973237, 4028940343, 191353191413, 221327221393, 507217507289, 937253937331, 10402271040311, 843911844001, 25654632565559, 81661078166209, 55778515577959, 82237498223863

Offset: 1

Jean-Marc Rebert, Jan 20 2016

Subsequence of A030461.

a(n) is the concatenation of the smallest prime p and the next prime q, such that p + 6n = q and the concatenations of these 2 primes is also prime. a(n) = 0 if no such term exists.

			a(1) = A030461(2) = 3137. gap =  37 - 31 = 6 = 6 * 1.
a(2) = 199211, because 199211 is the first term in A030461, with gap = 211 - 199 = 12 = 6 * 2.

Jean-Marc Rebert, Table of n, a(n) for n = 1..32

Cf. A030459, A030460, A030461, A058193, A267692.

Primes:= select(isprime,[seq(i,i=3..10^7,2)]):
cati:= (x,y) -> 10^(1+ilog10(y))*x+y;
for i from 1 to nops(Primes)-1 do
  g:= Primes[i+1]-Primes[i];
  if g mod 6 <> 0 then next fi;
  if assigned(A[g/6]) then next fi;
  z:= cati(Primes[i],Primes[i+1]);
  if isprime(z) then A[g/6]:= z fi;
od:
seq(A[i],i=1..max(map(op,[indices(A)]))); # Robert Israel, Jan 24 2016

A030459 Prime p concatenated with next prime is also prime.

Original entry on oeis.org

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))).

A045533 Concatenate the n-th and (n+1)st prime.

Original entry on oeis.org

23, 35, 57, 711, 1113, 1317, 1719, 1923, 2329, 2931, 3137, 3741, 4143, 4347, 4753, 5359, 5961, 6167, 6771, 7173, 7379, 7983, 8389, 8997, 97101, 101103, 103107, 107109, 109113, 113127, 127131, 131137

Offset: 1

N. J. A. Sloane, Dec 11 1999

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

Cf. A077800, A095958 (subsequence), A030461 (primes).

a045533 n = a045533_list !! (n-1)
a045533_list = f $ map show a000040_list :: [Integer] where
   f (t:ts@(t':_)) = read (t ++ t') : f ts
-- Reinhard Zumkeller, Apr 20 2012

[Seqint(Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n))): n in [1..50 ] ]; // Marius A. Burtea, Mar 21 2019

#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Jun 06 2015 *)

a(n) = eval(concat(Str(prime(n)), Str(prime(n+1)))); \\ Michel Marcus, May 11 2014

a(n) = prime(n)*(10^floor(log_10(prime(n+1)))+1) + prime(n+1). - Conner L. Delahanty, May 10 2014

Offset changed to 1, in agreement with (almost?) all references to this sequence, by M. F. Hasler

A030469 Primes which are concatenations of three consecutive primes.

Original entry on oeis.org

5711, 111317, 171923, 313741, 414347, 8997101, 229233239, 239241251, 263269271, 307311313, 313317331, 317331337, 353359367, 359367373, 383389397, 389397401, 401409419, 409419421, 439443449, 449457461

Offset: 1

Patrick De Geest

a(n) = "p(k) p(k+1) p(k+2)" where p(k) is k-th prime

It is conjectured that sequence is infinite. - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009

			(1) 5=p(3), 7=p(4), 11=p(5) gives a(1).
(2) 7=p(4), 11=p(5), 13=p(6), but 71113 = 7 x 10159

Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer 2005 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
John Derbyshire: Prime obsession, Joseph Henry Press, Washington, DC 2003 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
Marcus du Sautoy: Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A030461, A167517, A132903, A068655, A030997, A030473, A086041, A099727.

Select[Table[FromDigits[Flatten[IntegerDigits/@{Prime[n],Prime[n+1],Prime[n+2]}]],{n,11000}],PrimeQ] (* Zak Seidov, Oct 16 2009 *)
concat[{a_,b_,c_}]:=FromDigits[Flatten[IntegerDigits/@{a,b,c}]]; Select[ concat/@ Partition[ Prime[ Range[200]],3,1],PrimeQ] (* Harvey P. Dale, Sep 06 2017 *)

for(i=1,999, isprime(p=eval(Str(prime(i),prime(i+1),prime(i+2)))) & print1(p," ")) \\ M. F. Hasler, Nov 10 2009

A132903 INTERSECT A000040. - R. J. Mathar, Nov 11 2009

A086041 Primes that are concatenations of 5 consecutive primes.

Original entry on oeis.org

711131719, 4753596167, 5359616771, 6771737983, 97101103107109, 101103107109113, 149151157163167, 401409419421431, 431433439443449, 479487491499503, 487491499503509, 757761769773787, 827829839853857

Offset: 1

Chuck Seggelin, Jul 07 2003

			a(1)=711131719 because 711131719 is prime and the concatenation of 7, 11, 13, 17 and 19.

Zak Seidov, Table of n, a(n) for n = 1..1358

Cf. A030461, A030469, A030473, A086040.

A030997 Smallest prime which is a concatenation of n consecutive primes.

Original entry on oeis.org

2, 23, 5711, 2357, 711131719, 113127131137139149, 29313741434753, 107109113127131137139149, 211223227229233239241251257, 691701709719727733739743751757, 2329313741434753596167

Offset: 1

Patrick De Geest and Warut Roonguthai

			a(5) = 711131719 is the smallest prime which is the concatenation of five consecutive primes 7, 11, 13, 17 and 19.

Hans Havermann, Table of n, a(n) for n = 1..100

Cf. A030996, A068655, A030473, A086041, A099727, A167517.

Cf. A030461 (primes that are concatenations of two primes), A030469 (three primes), A030473 (four primes), A086041 (five primes).

for(k=1,19, for(i=0,1e9, isprime( eval( p=concat( vector( k,j,Str( prime( i+j )))))) & break); print1(p,", ")) \\ M. F. Hasler, Nov 10 2009

A030473 Primes which are concatenations of 4 consecutive primes.

Original entry on oeis.org

2357, 67717379, 838997101, 139149151157, 149151157163, 383389397401, 503509521523, 557563569571, 577587593599, 587593599601, 601607613617, 613617619631, 727733739743, 937941947953, 1181118711931201

Offset: 1

Patrick De Geest

Zak Seidov and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1205 from Zak Seidov)

Cf. A030461, A030469, A086040, A086041.

P:= select(isprime,[2,seq(p,p=3..10^4,2)]):
select(isprime,[seq(P[i+3]+10^(1+ilog10(P[i+3]))*P[i+2] + 10^(2+ilog10(P[i+3])+ilog10(P[i+2]))*P[i+1] + 10^(3+ilog10(P[i+3])+ilog10(P[i+2])+ilog10(P[i+1]))*P[i], i=1..nops(P)-3)]); # Robert Israel, Apr 14 2016

Edited by Charles R Greathouse IV, Apr 23 2010

A174031 The smallest integer k>0 such that the double-concatenation prime(n) // prime(n+1) // k is a prime number.

Original entry on oeis.org

3, 3, 1, 19, 1, 1, 1, 1, 1, 1, 9, 11, 11, 17, 3, 1, 1, 3, 11, 17, 21, 19, 1, 7, 37, 7, 23, 37, 7, 1, 7, 7, 7, 11, 7, 33, 29, 31, 1, 13, 11, 17, 7, 11, 11, 9, 9, 1, 7, 7, 1, 13, 11, 19, 67, 1, 13, 21, 49, 13, 13, 1, 1, 23, 1, 1, 29, 1, 29, 7

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 06 2010

Leading zeros in k are not allowed.
All entries k are odd with final digit 1, 3, 7 or 9.
Dirichlet's prime number theorem for arithmetic progressions says that the sequence is infinite.
Conjecture: 1 appears infinitely often.

			n=1: 2//3//1 = 231 = 3 * 7 * 11 is not prime, so k<>1. 233 = prime(51), therefore 3 is the first entry.
n=2: 3//5//1 = 351 = 3^3 * 13 is not prime, so k <> 1, but 353 = prime(71), therefore 3 is the second entry.
n=30: 113//127//1 = 1131271 = prime(87976), so the 30th entry is 1.

Cf. A030461, A030459, A030469, A171154

read("transforms") ;
A174031 := proc(n) for e from 1 do if isprime(digcatL([ithprime(n),ithprime(n+1),e])) then return e ; end if; end do: end proc:

Entries checked; replaced variables by OEIS standard names - R. J. Mathar, Nov 17 2010

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

cp's OEIS Frontend

A267692 Terms of A030461 that give successive gap records.

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A267721 a(n) is the least term of A030461 with gap = 6*n between consecutive primes or 0 if no such term exists.

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A030459 Prime p concatenated with next prime is also prime.

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A045533 Concatenate the n-th and (n+1)st prime.

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A030469 Primes which are concatenations of three consecutive primes.

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A086041 Primes that are concatenations of 5 consecutive primes.

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A030997 Smallest prime which is a concatenation of n consecutive primes.

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A030473 Primes which are concatenations of 4 consecutive primes.

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