A030459 -id:A030459 - cp's OEIS frontend

A036340 Concatenation of prime p and nextprime(p) is prime -> cycles of 3 steps possible.

467, 941, 959941, 3396199, 4858943, 5696101, 6475643, 7566133, 7584253, 7592261, 9305281, 9463877, 11430491, 13442243, 14374837, 15941473, 17414497, 17691997, 19584223, 21421849, 22310159, 22808459, 27601163, 29198881

Offset: 0

Patrick De Geest, Dec 15 1998

Terms from 3396199 up to 17691997 found by Jo Yeong Uk (hyukjo(AT)sigma.chungnam.ac.kr).

Carlos Rivera, Puzzle 29. Pi = P i-1&nxtprm(P i-1), Pi = prime for i => 1, The Prime Puzzles & Problems Connection.

Cf. A030459, A034593, A036339, A036341.

A034592 Cycle of 2 steps possible for 'concatenate a(n) and nextprime(a(n)) is a prime'.

Original entry on oeis.org

0, 36, 150, 284, 464, 467, 525, 584, 644, 777, 855, 941, 1058, 1120, 1179, 1362, 1788, 1855, 2368, 2520, 2547, 2550, 2576, 2743, 2988, 3063, 3273, 3339, 3410, 3930, 4054, 4370, 4739, 4843, 4910, 5253, 5445, 5550, 5671, 5967, 6512, 6721, 6987, 7131, 7216

Offset: 0

Patrick De Geest, Oct 15 1998

			a(n)=777 -> nextprime(a(n)) is 787 so '777787' is prime (=step 1); a(n2)=777787 -> nextprime(a(n2)) is 777817 so '777787777817' is prime (=step 2).

A174034 The smallest prime p such that the double-concatenation prime(n) // prime(n+1) // p is a prime number.

Original entry on oeis.org

3, 3, 7, 19, 17, 7, 17, 7, 3, 23, 11, 11, 11, 17, 3, 3, 7, 3, 11, 17, 29, 19, 13, 7, 37, 7, 23, 37, 7, 23, 7, 7, 7, 11, 7, 53, 29, 31, 31, 13, 11, 17, 7, 11, 11, 29, 23, 47, 7, 7, 7, 13, 11, 19, 67, 19, 13, 101, 59, 13, 13, 31, 17, 23, 7, 13, 29, 73, 29, 7

Offset: 1

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

It is conjectured that a(n) = 3 for infinitely many n.

			n=1: 2 // 3 // 3 = 233, which is prime, so a(1) = 3.
n=2: 3 // 5 // 2 = 352, which is not prime, but 3 // 5 // 3 = 353 is, so a(2) = 3.

Cf. A030461, A030459, A030469, A171154, A174031

A174034(n)={ n=eval(Str(prime(n),prime(n+1))); for( d=1,99, n*=10; forprime( p=10^(d-1),10^d, isprime(n+p) & return(p)))} \\ M. F. Hasler, Dec 01 2010

concat = lambda xx: Integer(''.join(map(str,xx)))
A174034 = lambda x: next((p for p in Primes() if is_prime(concat([nth_prime(x), nth_prime(x+1), p])))) # D. S. McNeil, Dec 02 2010

Edited and terms checked by D. S. McNeil, Dec 01 2010

A225135 Squares that are a concatenation of primes.

Original entry on oeis.org

25, 225, 289, 361, 529, 729, 2025, 2401, 2601, 2809, 3025, 4761, 5041, 5329, 5929, 7225, 7569, 11025, 11449, 11881, 13225, 15129, 19881, 20449, 21609, 22801, 23409, 24649, 25281, 26569, 27225, 29241, 29929, 31329, 32041, 32761, 34969, 36481, 39601, 47089

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 29 2013

Lim inf a(n)/n^2 >= 2. Is it finite? - Charles R Greathouse IV, Apr 30 2013

			25 = 5^2 and can be separated into two prime numbers: 2|5.
231361 = 481^2 and can be separated into prime numbers in six ways: 2|3|1361, 2|3|13|61, 2|31|3|61, 2|313|61, 23|1361, and 23|13|61.
Leading zeros are allowed: 2025 = 2|02|5.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), sqrt(a(n)), all possible separations of a(n) into primes for n = 1...10000

Cf. A019549, A030459, A106582.

r[d_] := Catch@ Block[{z = Length@d}, z<1 || Do[ If[ PrimeQ@ FromDigits@ Take[d, i] && r@ Take[d, i-z], Throw@ True], {i, z}]]; Select[ Range[1000]^2, r@ IntegerDigits@ # &] (* Giovanni Resta, Apr 30 2013 *)

has(n)=if(isprime(n),return(1)); if(n<202,return(isprime(n%10) && isprime(n\10))); my(k=n%10,v);if(k==5||k==2,return(if(n<6,1,n\=10;has(n/10^valuation(n,10)))));if(k%2==0,return(0));v=digits(n);for(i=1,#v,if(isprime(n%10^i) && has(n\10^i), return(1))); 0
forstep(n=5,1e3,2,if(has(n^2),print1(n^2", ")))
\\ Charles R Greathouse IV, Apr 30 2013

library(gmp); isprime2=function(x) isprime(x)>0
splithasproperty<-function(n,FUN,curdig=1,res=list(),curspl=c()) {
no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
    s=as.character(n)
    if(curdig>nchar(s)) return(res)
    if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s,curdig,nchar(s)))))) res[[length(res)+1]]=curspl
    for(i in curdig:nchar(s))
        if(FUN(as.bigz(no0(substr(s,curdig,i)))))
            res=splithasproperty(n,FUN,i+1,res,c(curspl,i))
    res
}
which(sapply(1:100,function(x) length(splithasproperty(x^2,isprime2)))>0)^2

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);

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

A225120 Square numbers whose decimal representation can be divided into two or more semiprimes.

Original entry on oeis.org

49, 64, 144, 256, 576, 625, 1156, 1296, 1444, 1521, 2209, 2916, 3364, 3844, 3969, 4096, 4356, 4489, 4624, 6889, 7744, 8649, 9025, 9216, 9409, 9604, 10201, 10404, 10609, 10816, 12321, 12996, 13456, 14161, 15129, 15376, 15625, 15876, 17956, 18496, 18769, 20164

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 29 2013

For 300 < n < 10000, 12.77*n^1.86 provides an estimate of a(n) to within 10%.
The density of squares included in the sequence asymptotically approaches 1.
There are infinitely many squares that are not in the sequence. For example, no square ending in 0 can be in the sequence. Another such infinite class is given by (50k+5)^2, for k>0. Indeed, these squares all end in "025" and since the only semiprime ending in 25 is 25 itself, then the other semiprime must end in 0, but this is impossible since the only semiprime ending in 0 is 10. - Giovanni Resta, May 03 2013

			a(50) = 25921, which is 161^2, and can be separated into semiprimes three ways: 25|921, 25|9|21, and 259|21.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), sqrt(a(n)), all possible separations of a(n) into semiprimes for n = 1...10000.

Cf. A001358, A030459, A030461, A000290.

issemipr<-function(n) ifelse(n<4,F,length(factorize(n))==2)
splithasproperty<-function(n,FUN,curdig=1,res=list(),curspl=c()) {
    no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
    s=as.character(n)
    if(curdig>nchar(s)) return(res)
    if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s,curdig,nchar(s)))))) res[[length(res)+1]]=curspl
    for(i in curdig:nchar(s))
        if(FUN(as.bigz(no0(substr(s,curdig,i)))))
            res=splithasproperty(n,FUN,i+1,res,c(curspl,i))
    res
}
which(sapply(1:100,function(x) length(splithasproperty(x^2,issemipr))>0))^2

A225136 Numbers that are concatenations of triprimes.

Original entry on oeis.org

88, 128, 188, 208, 278, 288, 308, 428, 448, 458, 508, 528, 638, 668, 688, 708, 758, 768, 788, 808, 812, 818, 820, 827, 828, 830, 842, 844, 845, 850, 852, 863, 866, 868, 870, 875, 876, 878, 888, 892, 898, 899, 928, 988, 998, 1028, 1058, 1108, 1148, 1168, 1178

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 29 2013

			88 = 8|8, both of which are triprime because 8=2*2*2.
458 = 45 | 8 = 3*3*5 | 2*2*2.
12428 can be split into triprimes in three ways: 12|428, 12|42|8, and 124|28.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), all possible separations of a(n) into triprimes for n=1..10000.

Cf. A106582, A019549, A030459.

library(gmp); istriprime=function(x) ifelse(x<8,F,length(factorize(x))==3)
splithasproperty<-function(n,FUN,curdig=1,res=list(),curspl=c()) {
no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
    s=as.character(n)
    if(curdig>nchar(s)) return(res)
    if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s,curdig,nchar(s)))))) res[[length(res)+1]]=curspl
    for(i in curdig:nchar(s))
        if(FUN(as.bigz(no0(substr(s,curdig,i)))))
            res=splithasproperty(n,FUN,i+1,res,c(curspl,i))
    res
}
which(sapply(1:500,function(x) length(splithasproperty(x,istriprime)))>0)

A225151 Squares which are a decimal concatenation of triprimes.

Original entry on oeis.org

12544, 15376, 19044, 20164, 27556, 28561, 42436, 45369, 45796, 75076, 81796, 86436, 87025, 89401, 98596, 114244, 116964, 123201, 124609, 125316, 126025, 127449, 128164, 131044, 139876, 141376, 150544, 174724, 175561, 184041, 188356, 190969, 191844, 207025

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 30 2013

			a(10) is 75076, which splits into 75|076. 75 = 3*5*5; 76 = 2*2*19.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), sqrt(a(n)), all possible separations of a(n) into triprimes

Cf. A014612, A030459, A030461, A000290, A019547.

library(gmp); istriprime=function(x) ifelse(x<8, F, length(factorize(x))==3)
splithasproperty<-function(n, FUN, curdig=1, res=list(), curspl=c()) {
no0<-function(s){ while(substr(s, 1, 1)=="0" & nchar(s)>1) s=substr(s, 2, nchar(s)); s}
    s=as.character(n)
    if(curdig>nchar(s)) return(res)
    if(length(curspl)>0) if(FUN(as.bigz(no0(substr(s, curdig, nchar(s)))))) res[[length(res)+1]]=curspl
    for(i in curdig:nchar(s))
        if(FUN(as.bigz(no0(substr(s, curdig, i)))))
            res=splithasproperty(n, FUN, i+1, res, c(curspl, i))
    res
}
which(sapply(1:500, function(x) length(splithasproperty(x^2, istriprime)))>0)^2

A225575 Primes p such that if q is the next prime after p then the concatenation of p with q and the concatenation of q with p are both primes.

Original entry on oeis.org

199, 233, 257, 353, 523, 653, 971, 1973, 2333, 3259, 3637, 3761, 4283, 4993, 5927, 6353, 6529, 6563, 7907, 8831, 9293, 9851, 10711, 10861, 11731, 13037, 13177, 13681, 15241, 16381, 16693, 16931, 18341, 18899, 19577, 21787, 23857, 24071, 28621, 31657, 32911

Offset: 1

W. Edwin Clark, May 10 2013

This sequence was suggested by Puzzle 687 at Prime Puzzles (see Links).
If q = p + 2 (that is, p is the lesser of a twin prime, A001359), then p is not in the sequence. - Alonso del Arte, May 10 2013
In general, q-p == 0 (mod 6). - Zak Seidov, May 11 2013

			The prime following 199 is 211 and both 199211 and 211199 are prime.

Marius A. Burtea, Table of n, a(n) for n = 1..10000 (terms 1..250 from Paolo P. Lava)
Carlos Rivera, Puzzle 687: 17769643, The Prime Puzzles & Problems Connection.

Cf. A030459.

conc:=func; [p:p in PrimesUpTo(17000)| IsPrime(conc(p,(NextPrime(p)))) and IsPrime(conc(NextPrime(p),p))]; // Marius A. Burtea, Jan 25 2020

a:=NULL;
for i from 1 to 2000 do
  p:=ithprime(i);
  q:=nextprime(p);
  s:=convert(p,string);
  t:=convert(q,string);
  if isprime(parse(cat(s,t))) and isprime(parse(cat(t,s))) then a:=a,p; fi;
od:
a;

concatQ[{a_,b_}]:=Module[{idna=IntegerDigits[a],idnb=IntegerDigits[b]},AllTrue[ FromDigits/@ {Join[idna,idnb], Join[idnb,idna]},PrimeQ]]; Transpose[Select[ Partition[ Prime[Range[2000]],2,1],concatQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 16 2015 *)

cp's OEIS Frontend

A036340 Concatenation of prime p and nextprime(p) is prime -> cycles of 3 steps possible.

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A034592 Cycle of 2 steps possible for 'concatenate a(n) and nextprime(a(n)) is a prime'.

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A174034 The smallest prime p such that the double-concatenation prime(n) // prime(n+1) // p is a prime number.

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A225135 Squares that are a concatenation of primes.

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Mathematica

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

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Maple

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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Maple

A225120 Square numbers whose decimal representation can be divided into two or more semiprimes.

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R

A225136 Numbers that are concatenations of triprimes.

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R

A225151 Squares which are a decimal concatenation of triprimes.

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