A030461 -id:A030461 - cp's OEIS frontend

A099727 Concatenations of six consecutive primes forming a prime.

113127131137139149, 569571577587593599, 727733739743751757, 733739743751757761, 739743751757761769, 102110311033103910491051, 105110611063106910871091, 110911171123112911511153, 118111871193120112131217, 138113991409142314271429

Offset: 1

Ray G. Opao, Nov 07 2004

			The prime 113127131137139149 is a concatenation of the consecutive primes 113, 127, 131, 137, 139 and 149.

K. D. Bajpai, Table of n, a(n) for n = 1..1470

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

select(isprime, [seq(parse(cat([seq(ithprime(i), i=n+0..n+5)][])), n=1..500)])[]; # K. D. Bajpai, Mar 24 2014

Select[FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[Prime[Range[ 300]],6,1],PrimeQ] (* Harvey P. Dale, Apr 30 2020 *)

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

A239789 Primes which are a concatenation of prime(k), prime(k+2) and prime(k+4) for some k.

Original entry on oeis.org

172331, 233141, 717989, 137149157, 191197211, 197211227, 223229239, 229239251, 257269277, 331347353, 353367379, 359373383, 467487499, 521541557, 617631643, 619641647, 647659673, 677691709, 733743757, 787809821, 797811823, 103310491061, 106110691091, 109711091123

Offset: 1

K. D. Bajpai, Mar 26 2014

			172331 is a prime and appears in the sequence because it is the concatenation of prime(7), prime(7+2) and prime(7+4).
233141 is a prime and appears in the sequence because it is the concatenation of prime(9), prime(9+2) and prime(9+4).

K. D. Bajpai, Table of n, a(n) for n = 1..3738

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

with(StringTools): KD := proc() local a,b,d,e; a:=ithprime(n); b:=ithprime(n+2); d:=ithprime(n+4);
e:= parse(cat(a,b,d)); if isprime(e) then RETURN (e); fi; end: seq(KD(), n=1..500);

Select[Table[FromDigits[Flatten[{IntegerDigits[Prime[n]], IntegerDigits[Prime[n+2]], IntegerDigits[Prime[n+4]]}]], {n,1,500}],PrimeQ]

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

Derek Orr, Oct 03 2014

			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

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

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

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,", "))))

Terms and program corrected by Derek Orr to match definition, thanks to Jens Kruse Andersen

A248208 Primes p such that p^3 is the concatenation of two k-digit primes where k is half the number of decimal digits in p^3.

Original entry on oeis.org

3, 11, 47, 83, 1063, 1637, 1699, 7529, 7673, 23059, 28097, 29573, 34157, 34961, 36587, 40897, 43609, 44711, 101839, 102763, 103423, 104087, 104393, 106363, 117437, 117499, 124471, 125407, 126011, 129419, 134753, 135007, 137393, 139487, 143879, 143971, 145037

Offset: 1

Derek Orr, Oct 03 2014

			47 is prime and 47^3 = 103823 is the concatenation of two primes (103 and 823) that are of the same length (here, their length is 3). So, 47 is a member of this sequence.
73 is not in the sequence since 73^3 = 389017, where 389 is a 3-digit prime but 017 is a 2-digit prime. - _Jens Kruse Andersen_, Oct 06 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A030078, A105184, A030461, A153048, A080906, A248046.

forprime(p=1,10^6,d=digits(p^3);if((#d)%2==0,if(isprime((p^3)\(10^(#d/2)))&&isprime((p^3)%(10^(#d/2)))&&#Str((p^3)%(10^(#d/2)))==#d/2,print1(p,", "))))

Terms and PARI program corrected by Jens Kruse Andersen, Oct 06 2014

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

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

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

K. D. Bajpai, Mar 30 2014

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.

			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).

K. D. Bajpai, Table of n, a(n) for n = 1..3742

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

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);

Select[Table[FromDigits[Flatten[{IntegerDigits[Prime[n+4]],IntegerDigits[Prime[n+2]], IntegerDigits[Prime[n]]}]], {n,1,500}], PrimeQ]

A253245 Primes that are the concatenation of prime(n) and prime(n+2).

Original entry on oeis.org

37, 1117, 1319, 1723, 4759, 89101, 97103, 101107, 113131, 151163, 181193, 223229, 227233, 239251, 251263, 293311, 313331, 337349, 389401, 421433, 461467, 491503, 587599, 631643, 647659, 683701, 691709, 701719, 739751, 761773, 809821

Offset: 1

Altug Alkan, Aug 22 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A030461.

Module[{nn=300,pr},pr={#[[1]],#[[3]]}&/@Partition[Prime[Range[nn]],3,1];Select[Table[FromDigits[Flatten[IntegerDigits/@pr[[n]]]],{n, Length[ pr]}],PrimeQ]] (* Harvey P. Dale, Nov 29 2015 *)

for(n=1, 1e3, if(isprime(k=eval(Str(prime(n), prime(n+2)))), print1(k", ")))

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

K. D. Bajpai, May 23 2015

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

			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.

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

Cf. A000040, A030461, A030469.

[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

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 *)

forprime(p = 1,5000, k=eval(concat( Str(p^2), Str(nextprime(p+1)) )); if(isprime(k), print1(k,", ")))

cp's OEIS Frontend

A099727 Concatenations of six consecutive primes forming 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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A239789 Primes which are a concatenation of prime(k), prime(k+2) and prime(k+4) for some k.

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

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A248208 Primes p such that p^3 is the concatenation of two k-digit primes where k is half the number of decimal digits in p^3.

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PARI

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

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R

A225151 Squares which are a decimal concatenation of triprimes.

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

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