A167535 -id:A167535 - cp's OEIS frontend

A345314 Primes that can be constructed by concatenating two squares >= 4.

449, 499, 1009, 1699, 2549, 4289, 4441, 4729, 6449, 6481, 8419, 9619, 12149, 14449, 16361, 16529, 16729, 16981, 19681, 21169, 22549, 24019, 25121, 25169, 25841, 28099, 28949, 30259, 34819, 36529, 38449, 41521, 41681, 41849, 42209, 43481, 43721, 43969, 45329, 46889

Offset: 1

Tanya Khovanova, Jun 13 2021

If we allow 1, we get sequence A167535.

			449 is a prime that is a concatenation of two squares: 4 and 49.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A167535.

zcat:= proc(a,b) 10^(1+ilog10(b))*a+b end proc:
select(t -> t <= 10^5 and isprime(t), {seq(seq(zcat(a^2,b^2),a=2..100),b=3..1000,2)}); # Robert Israel, Jun 17 2021

Take[Select[Union[Flatten[Table[FromDigits[Join[IntegerDigits[n^2],IntegerDigits[k^2]]], {n, 2, 300}, {k, 2, 300}]]], PrimeQ[#] &], 60]

from sympy import isprime
def aupto(lim):
  s = list(i**2 for i in range(2, int(lim**(1/2))+2))
  t = set(int(str(a)+str(b)) for a in s for b in s)
  return sorted(filter(isprime, filter(lambda x: x<=lim, t)))
print(aupto(49000)) # Michael S. Branicky, Jun 13 2021

A364359 Primes that are the concatenation of a square and a prime that is the concatenation of two squares.

Original entry on oeis.org

419, 911, 919, 941, 1181, 1499, 1619, 1811, 4919, 8111, 9181, 9491, 9811, 11699, 12119, 12251, 14411, 14419, 16481, 16811, 19001, 22511, 22541, 32411, 32441, 36251, 44111, 44119, 44729, 49499, 49811, 52919, 57641, 64499, 64811, 67619, 72911, 81181, 90011, 90019, 91009, 92251, 94441, 97841, 98419

Offset: 1

Robert Israel, Oct 20 2023

Primes that are the concatenation of a square and a member of A167535.

			a(5) = 1181 is a term because it is the concatenation of 1^2 = 1, 1^2 =1 and 9^2 = 81, and 181 and 1181 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A167535.

for d from 1 to 3 do
  m1:= ceil(10^((d-1)/2));
  m2:= floor(sqrt(10^d - 1));
  S[d]:= {seq(i^2, i=m1..m2)};
  if m1::even then m1:= m1+1 fi;
  So[d]:= {seq(i^2, i=m1..m2,2)};
od:
for d from 2 to 4 do P2[d]:= select(isprime, {seq(seq(seq(10^i*s+t, t=So[i]),s=S[d-i]),i=1..d-1)}) od:
for d from 3 to 5 do P3[d]:= select(isprime, {seq(seq(seq(10^i*s+t, t=P2[i]),s=S[d-i]),i=2..d-1)}) od:
sort([seq](op(P3[d]),d=3..5));

A168487 Primes of the form 100n^3 + 27.

Original entry on oeis.org

127, 827, 6427, 12527, 34327, 219727, 491327, 1562527, 2438927, 3276827, 8518427, 16637527, 22698127, 43897627, 45653327, 51200027, 77868827, 119101627, 129502927, 140492827, 156089627, 177156127, 190662427, 251545627, 257135327

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 27 2009

nonn

(1) These primes all with the end digits 2 and 7 are concatenations of two CUBIC numbers: "n^3 3^3".

(2) It is conjectured that sequence is infinite.

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Friedhelm Padberg, Elementare Zahlentheorie, Spektrum Akademischer Verlag, 2. Auflage 1991

A167535 Concatenation of two square numbers which give a prime
A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
A168327 Primes of concatenated form p = "1 n^3"

Select[100Range[140]^3+27,PrimeQ] (* Harvey P. Dale, Aug 22 2011 *)

Edited by Charles R Greathouse IV, Apr 24 2010

A168568 Primes p such that the concatenation p//29 is a squared prime.

Original entry on oeis.org

5, 53, 3329, 4583, 7691, 12611, 14957, 17609, 20249, 35081, 56501, 71663, 134909, 191231, 237851, 305477, 339539, 351293, 394007, 418997, 432569, 466079, 574109, 611993, 619841, 628373, 659831, 701741, 709469, 744251, 752903, 1386977, 1398779

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Nov 30 2009

Subsequence of A168545: p such that p//29 = m^2.
(1) Conjecture: the sequence is infinite.
(2) 29 = prime(10) is the smallest prime which can appear as the least significant digits of perfect squares.
(3) The set of possible least significant digit pairs of m is {23, 27, 73 or 77}.
(4) Only four 2-digit primes are least significant digits of perfect squares: 29, 41, 61 and 89.
(5) There are no squares of the form p//41 = m^2 because only even numbers (no primes) concatenated with 41 are squares.

			a(1) = 5 = prime(3) because 529=23^2 and 23=prime(9).
7 = prime(4) is not in the sequence because 729=27^2 and 27=3^3 is not a prime.
a(2) = 53 = prime(16) because 5329=73^2 and 73=prime(21).
a(3) = 3329 = prime(469) because 332929=577^2 and 577=prime(106).

Peter Bundschuh, Einfuehrung in die Zahlentheorie, Springer, 4. Auflage 1998.
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005.
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.

David Corneth, Table of n, a(n) for n = 1..10000 (first 150 terms from Harvey P. Dale)

Cf. A000040, A167535, A158896, A168545.

Select[Prime[Range[120000]],PrimeQ[Sqrt[100#+29]]&] (* Harvey P. Dale, Jan 15 2019 *)

Keyword:base added by R. J. Mathar, Dec 05 2009

A351925 Squares which are the concatenation of two primes.

Original entry on oeis.org

25, 289, 361, 529, 729, 2401, 2601, 2809, 4761, 5329, 5929, 7569, 11449, 11881, 15129, 19881, 21609, 22801, 23409, 24649, 25281, 26569, 29241, 29929, 31329, 34969, 36481, 39601, 47961, 52441, 53361, 54289, 57121, 58081, 59049, 71289, 77841, 83521, 89401

Offset: 1

Max Z. Scialabba, Feb 25 2022

The first term that is the concatenation of two primes in more than one way is a(11) = 5929 = 5 | 929 = 59 | 29. - Robert Israel, Oct 01 2023

			25 is the concatenation of 2 and 5, both primes.
4761 is the concatenation of 47 and 61, both primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290 (squares), A039686, A106582, inverse of A167535.

Cf. A038692, A225135.

L:= NULL: count:=0:
for x from 1 by 2 while count < 100 do
  xs:= x^2;
  for i from 1 to ilog10(xs) do
    a:= xs mod 10^i;
    if a > 10^(i-1) and isprime(a) then
      b:= (xs-a)/10^i;
      if isprime(b) then
        L:= L, xs; count:= count+1; break
      fi fi
od od:
L; # Robert Israel, Oct 01 2023

isb(n)={my(d=10); while(dAndrew Howroyd, Feb 26 2022

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        s = str(k*k)
        if any(s[i] != '0' and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s))):
            yield k*k
print(list(islice(agen(), 39))) # Michael S. Branicky, Feb 26 2022

Intersection of A106582 and A000290.

A366106 Primes that are the concatenation of three squares in base 10.

Original entry on oeis.org

101, 109, 149, 191, 199, 401, 409, 419, 449, 491, 499, 911, 919, 941, 991, 1049, 1181, 1259, 1361, 1481, 1499, 1601, 1609, 1619, 1699, 1811, 1949, 2549, 2591, 3691, 4049, 4259, 4481, 4649, 4909, 4919, 4999, 6449, 6491, 8101, 8111, 8191, 9049, 9161, 9181, 9491, 9649, 9811, 9949, 10009, 10091

Offset: 1

Robert Israel, Sep 29 2023

The three squares need not be distinct.
At least one of the squares must be divisible by 9.
The first term that is a concatenation of three squares in two different ways is 14411, the concatenation of 1 = 1^2, 441 = 21^2 and 1 = 1^2 and also 144 = 12^2, 1 = 1^2 and 1 = 1^2.
The first term that is a concatenation of three squares in three different ways is 1961441, the concatenation of 196 = 14^2, 144 = 12^2 and 1 = 1^2, of 196, 1 and 441 = 21^2, and of 1, 961 = 31^2 and 441.

			a(16) = 1049 is a term because it is the concatenation of 1 = 1^2, 0 = 0^2 and 49 = 7^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A167535.

M:= 5: # for terms < 10^M
S:= {}:
for a from 1 while a^2 < 10^(M-2) do
  x:= a^2; mx:= length(x);
  for b from 0 while b^2 < 10^(M-1-mx) do
    y:= b^2; my:= max(1,length(y));
    for c from 0 while c^2 < 10^(M-mx-my) do
      v:= parse(cat(x,y,c^2));
      if isprime(v) then S:= S union {v} fi;
od od od:
sort(convert(S,list));

a[maxSquareIndex_Integer?Positive]:=Select[Flatten[Table[ToExpression[IntegerString[a^2]<>IntegerString[b^2]<>IntegerString[c^2]],{a,1,maxSquareIndex},{b,0,maxSquareIndex},{c,0,maxSquareIndex}]],PrimeQ]//Sort;a[10][[1;;51]] (* Robert P. P. McKone, Oct 02 2023 *)

A169586 Primes p in A168540 for which q = 3^3 + 10^2*p^3 (A168487) is prime.

Original entry on oeis.org

2, 5, 7, 13, 17, 29, 61, 109, 137, 149, 191, 223, 227, 269, 311, 331, 337, 359, 389, 397, 409, 433, 457, 467, 491, 587, 619, 653, 661, 709, 727

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 02 2009

nonn

It is conjectured that sequence is infinite

			(1) 3^3+10^2*2^3=827=prime(144) gives a(1)=2=prime(1)
(2) 3^3+10^2*5^3=12527=prime(1496) gives a(2)=5=prime(3)
(3) 3^3+10^2*13^3=219727=prime(19588) gives a(4)=13=prime(6)

Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005
Arnold Scholz, Bruno Schoeneberg: Einführung in die Zahlentheorie, Walter de Gruyter, 5. Auflage 1973

A000040 The prime numbers
A167535 Concatenation of two square numbers which give a prime
A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
A168327 Primes of concatenated form p= "1 n^3"
A168375 Naturals n for which the concatenation p= "1 n^3"is prime
A168487 Primes of form p = 3^3 + 10^2*n^3 with a natural number n
A168540 Naturals n for which the concatenation p = 3^3 + 10^2*n^3 is prime

A203815 Primes which are the decimal concatenation of two squares in more than one way.

Original entry on oeis.org

14449, 4651249, 14592449, 25281961, 64516961, 144552049, 4963984001, 10241442401, 121828921681, 1089936421201, 1589776384961, 4640334443681, 19713620187049, 25593280441009, 81105458016049, 121521961692601, 445309309225961

Offset: 1

Emmanuel Vantieghem, Jan 06 2012

Subsequence of A167535.

			a(1) = 14449 = the concatenation of 12^2 and 7^2 and of 38^2 and 3^2.
a(5) = 64516961 = the concatenation of 8^2 and 719^2 and of 254^2 and 31^2.

Cf. A167535.

K[m_Integer, n_Integer] := n+m*10^IntegerLength[n]; k=-1; V={}; g=5000000; W={}; While[k10 (2 g)^2, n=2 g, If[PrimeQ[a], V={V, a}]]]]; V=Flatten[V]; W1=Sort[V]; W2=Union[V](*= A167535*); i=0; While[i

A364369 a(n) is the least prime that is the concatenation of n squares, where the concatenations of the last k of these squares are prime for 2 <= k <= n.

Original entry on oeis.org

11, 419, 4919, 181919, 1981919, 49936919, 991981919, 9991981919, 16999369225919, 136999369225919, 99361981818199181, 1729936999369225919, 3681225936999369225919, 132481225936999369225919, 99362500576936999369225919, 8199362500576936999369225919

Offset: 2

Robert Israel, Oct 20 2023

At most 2 of the n squares are not divisible by 9.

			a(5) = 181919 because it is the concatenation of five squares 1^2 = 1, 9^2 = 81, 3^2 = 9, 1^2 = 1 and 3^2 = 9, and 19, 919, 81919 and 181919 are all prime, and this is the least number that works.

Cf. A167535, A364359.

for d from 1 to 8 do
  m1:= ceil(10^((d-1)/2));
  m2:= floor(sqrt(10^d - 1));
  S[d]:= {seq(i^2, i=m1..m2)};
  if m1::even then m1:= m1+1 fi;
  So[d]:= {seq(i^2, i=m1..m2,2)};
od:
for d from 2 to 9 do P[2,d]:= select(isprime, {seq(seq(seq(10^i*s+t, t=So[i]),s=S[d-i]),i=1..d-1)}) od:
L[2]:= 11:
mm[2]:= 2:
for m from 3 do
   found:= false;
   for d from mm[m-1]+1 to m+7 do
     P[m,d]:= select(isprime, {seq(seq(seq(10^i*s+t, t=P[m-1,i]),s=S[d-i]),i=mm[m-1]..d-1)});
     if P[m,d] <> {} and not found then mm[m]:= d; found:= true: L[m]:= min(P[m,d]) fi;
   od;
   if not found then break fi;
od:
seq(L[i],i=2..m-1);

cp's OEIS Frontend

A345314 Primes that can be constructed by concatenating two squares >= 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A364359 Primes that are the concatenation of a square and a prime that is the concatenation of two squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A168487 Primes of the form 100n^3 + 27.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Extensions

A168568 Primes p such that the concatenation p//29 is a squared prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A351925 Squares which are the concatenation of two primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Python

Formula

A366106 Primes that are the concatenation of three squares in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A169586 Primes p in A168540 for which q = 3^3 + 10^2*p^3 (A168487) is prime.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A203815 Primes which are the decimal concatenation of two squares in more than one way.

Views

Author