A167535 -id:A167535 - cp's OEIS frontend

A168327 Primes of concatenated form "1 n^3".

11, 127, 12197, 135937, 159319, 11092727, 11295029, 11860867, 12685619, 14330747, 14826809, 15000211, 15929741, 16128487, 18869743, 19393931, 124137569, 126198073, 127818127, 129503629, 138958219, 150243409, 154439939, 160698457, 175686967, 191733851, 195443993

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 23 2009

(1) It is conjectured that sequence is infinite.

(2) These are primes all with "leading" digit "1", they are concatenations of two cubic numbers: 1^3 and n^3, n is a natural.

			(1) 10^1+1^3=11 = prime(5) = a(1).
(2) 10^2+3^3=127 = prime(31) = a(2).
(3) 10^4+13^3=12197 = prime(1458) = a(3).

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A168147, A167535.

Select[FromDigits[Join[{1},IntegerDigits[#]]]&/@(Range[500]^3),PrimeQ] (* Harvey P. Dale, May 16 2012 *)

If n^3 is a d-digit number and d no multiple of 3, then p=10^d+n^3, where n is odd and no multiple of 5.

a(n) = c+10^A055642(c) where c=A167725(n). - R. J. Mathar, Nov 23 2009

Edited by Charles R Greathouse IV, Apr 24 2010

A168147 Primes of the form 10*n^3 + 1.

Original entry on oeis.org

11, 271, 641, 2161, 33751, 40961, 58321, 138241, 196831, 270001, 297911, 466561, 506531, 795071, 1326511, 1406081, 1851931, 2160001, 3890171, 4218751, 5314411, 5513681, 6585031, 7290001, 8043571, 11910161, 12597121, 12950291, 14815441

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 19 2009

(1) These primes all with end digit 1=1^3 are concatenations of two CUBIC numbers: "n^3 1".
(2) It is conjectured that the sequence is infinite.
(3) It is an open problem if 3 consecutive naturals n exist which give such a prime.
No three such integers exist, as every n = 2 (mod 3) yields 10n^3 + 1 = 0 (mod 3). - Charles R Greathouse IV, Apr 24 2010

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A030430 (primes of the form 10*n+1).
Cf. A167535 (concatenation of two square numbers which give a prime).
See A168219 for the numbers n.

[ a: n in [1..150] | IsPrime(a) where a is 10*n^3+1 ]; // Vincenzo Librandi, Jul 25 2011

Select[Table[10*n^3+1,{n,1000}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

for(n=1,2e2, isprime(n^3*10+1) && print1(n^3*10+1", "))  \\ M. F. Hasler, Jul 24 2011

a(n) = 10*A168219(n)^3 + 1. \\ M. F. Hasler, Jul 24 2011

A173836 Natural numbers n such that the concatenation 1331//n^3 is a prime number.

Original entry on oeis.org

21, 27, 29, 41, 101, 119, 141, 171, 173, 177, 191, 197, 219, 243, 267, 291, 309, 327, 333, 369, 371, 383, 411, 417, 1019, 1049, 1059, 1091, 1157, 1163, 1211, 1311, 1337, 1343, 1359, 1371, 1379, 1409, 1461, 1473, 1481, 1503, 1521, 1593, 1599, 1613, 1637

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 26 2010

Given the cube n^3 with k = A111393(n) decimal digits, we have to check whether the concatenation, 11^3 * 10^k + n^3, is a prime.
The number k of digits that 1331=11^3 is shifted is not a multiple of 3,
because the form a^3+b^3 = (a^2+a*b+b^2) * (a - b) cannot construct a prime.

			21 is in the sequence because 21^3=9261, and the concatenation is 13319261=prime(868687).
27 is in the sequence because 27^3=19683, and the concatenation is 133119683=prime(7545064).

K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985

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

Cf. A102006, A167535, A168147, A168219, A168274, A173579, A173733

Select[Range[2000],PrimeQ[FromDigits[Join[{1,3,3,1}, IntegerDigits[ #^3]]]]&] (* Harvey P. Dale, Oct 14 2011 *)

Comments sligthly rephrased - R. J. Mathar, Mar 05 2010

A066591 Primes which can be expressed as a concatenation of nonnegative squares.

Original entry on oeis.org

11, 19, 41, 101, 109, 149, 181, 191, 199, 251, 401, 409, 419, 449, 491, 499, 641, 811, 911, 919, 941, 991, 1009, 1019, 1049, 1091, 1109, 1181, 1259, 1289, 1361, 1409, 1481, 1499, 1601, 1609, 1619, 1699, 1811, 1901, 1949, 1999, 2251, 2549, 2591, 3691

Offset: 1

Amarnath Murthy, Dec 21 2001

All terms are == {1,9} mod 10. - Zak Seidov, Jul 16 2015

The surprising prime 162536496481 is the concatenation of the 6 double-digit squares in increasing order (see Prime Curios! link). - Bernard Schott, Nov 19 2020

			96181 is a term as it is a concatenation of 961 and 81 both of which are squares. 100169 is a term as it is a concatenation of 100 and 169 in one way and also that of 1, 0, 0, 16 and 9 in another way.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2103 from Robert Israel)
Chris K. Caldwell and G. L. Honaker, Jr., 1625364981, Prime Curios!

A061246 and A167535 are subsequences. - Zak Seidov, Jul 16 2015

Cf. A000290, A068493.

N:= 10^4: # to get all terms <= N
catn:= proc(x,y) if y=0 then 10*x else x*10^(ilog10(y)+1)+y fi end proc:
Sq:= {seq(i^2,i=0..floor(sqrt(N)))}: Agenda:= Sq: S:= Sq:
while Agenda <> {} do
Agenda:= select(`<=`,{seq(seq(catn(f,g),f=Agenda),g=Sq)},N) minus S;
S:= S union Agenda;
od:
sort(convert(select(isprime,S),list)); # Robert Israel, Jul 16 2015

from sympy import sieve
from itertools import count, islice
def iscat(w, A):
    return False if len(w) < 2 else any(w[:i] in A and (w[i:] in A or iscat(w[i:], A)) for i in range(1, len(w)))
def agen():
    S = {"0"}
    for d in count(2):
        S |= {str(i*i) for i in range(10**(d-2), 10**(d-1))}
        for p in sieve.primerange(10**(d-1), 10**d):
            if iscat(str(p), S):
                yield p
print(list(islice(agen(), 50))) # Michael S. Branicky, Feb 20 2024

Corrected and extended by Christopher Lund (clund(AT)san.rr.com), Apr 11 2002

A168219 Naturals n for which 1 + 10*n^3 (A168147) is prime.

Original entry on oeis.org

1, 3, 4, 6, 15, 16, 18, 24, 27, 30, 31, 36, 37, 43, 51, 52, 57, 60, 73, 75, 81, 82, 87, 90, 93, 106, 108, 109, 114, 145, 154, 159, 160, 163, 165, 171, 174, 175, 178, 196, 201, 204, 207, 208, 211, 220, 222, 225, 228, 234

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 20 2009

nonn

It is conjectured that sequence is infinite.

No three consecutive integers n are in the list. [Proof: An integer of the form n=3*k+2 generates 1+10*n^3 = 9*(9+30*k^3+60*k^2+40*k) which is divisible through 9, hence not a prime, so these n are not in the list. Since every third integer is of this form == 2 (mod 3), no more than two consecutive integers can be in the sequence.] [Zak Seidov, Nov 24 2009]

			(1) 1+10*1^3=11 gives a(1)=1
(2) 1+10*3^3=271=3^4 gives a(2)=3
(3) 1+10*37^3=506531 gives a(13)=37

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980.
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005.
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.

G. C. Greubel, Table of n, a(n) for n = 1..1139

Cf. A000040, A168147, A167535.

Select[Range[100], PrimeQ[1 + 10*#^3] &] (* G. C. Greubel, Jul 16 2016 *)

for(n=1,2e2, isprime(n^3*10+1) && print1(n", "))  \\ M. F. Hasler, Jul 24 2011

A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

Original entry on oeis.org

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 164, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196

Offset: 1

Klaus Brockhaus, Jun 19 2011

Complement of A193096; A193095(a(n)) > 0; A038670, A039686, A167535, A192993, A193097 and A193144 are subsequences. [Reinhard Zumkeller, Jul 17 2011]

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A000290, A192993.

import Data.List (findIndices)
a191933 n = a191933_list !! (n-1)
a191933_list = findIndices (> 0) $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

CheckSplits:=function(n); v:=false; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then v:=true; end if; end for; return v; end function; [ p: p in [1..1200] | CheckSplits(p) ];

Take[Union[Flatten[Table[FromDigits[Flatten[{IntegerDigits[m^2], IntegerDigits[n^2]}]], {m, 20}, {n, 20}]]], 50] (* Alonso del Arte, Aug 11 2011 *)
squareQ[n_] := IntegerQ[Sqrt[n]]; okQ[n_] := MatchQ[IntegerDigits[n], {a__ /; squareQ[FromDigits[{a}]], b__ /; First[{b}] > 0 && squareQ[FromDigits[ {b}]]}]; Select[Range[2000], okQ] (* Jean-François Alcover, Dec 13 2016 *)

A173579 Natural numbers n which give primes when 1331 = 11^3 is prefixed.

Original entry on oeis.org

3, 17, 21, 53, 57, 69, 83, 87, 107, 119, 123, 153, 207, 227, 243, 249, 251, 261, 269, 279, 293, 299, 327, 329, 333, 339, 347, 377, 381, 383, 399, 411, 431, 437, 443, 471, 489, 497, 513, 521, 527, 549, 567, 573, 579, 587, 591, 597, 599, 611, 633, 641, 647, 657

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 22 2010

Concatenation of N = 1331 = 11^3 = palindrome(113) and natural n is a prime. No zeros "between" N and n.
13 = emirp(1) = prime(6), R(13) = 31 = emirp(3) = prime(11).
Necessarily n = 3 * k or n = 3 * k + 2, but not n = 3 * k + 1, because sod(1331) = 8. So no prime twins are terms of the sequence.

			13313 = prime(1581) => a(1) = 3.
133117 = prime(12425) => a(2) = 17.
133103, 133109 are prime, but "0" included: "03" resp. "09" are no terms of the sequence.

Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985
Theo Kempermann: Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

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

A102006, A167535, A168147, A168219, A168274

Select[Range[700],PrimeQ[1331*10^IntegerLength[#]+#]&] (* Harvey P. Dale, Jun 25 2020 *)

isok(n) = isprime(n + 1331*10^(length(Str(n)))); \\ Michel Marcus, Aug 27 2013

A168375 Natural numbers n for which the concatenation p= "1 n^3" (A168327) is prime.

Original entry on oeis.org

1, 3, 13, 33, 39, 103, 109, 123, 139, 163, 169, 171, 181, 183, 207, 211, 289, 297, 303, 309, 339, 369, 379, 393, 423, 451, 457, 463, 1021, 1027, 1047, 1053, 1057, 1081, 1087, 1111, 1123, 1161, 1189, 1201, 1249, 1273, 1293, 1303, 1329, 1339, 1351, 1381, 1387

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 24 2009

It is conjectured that sequence is infinite

			(1) "1 1^3"=10^1+1^3=11=prime(5) gives a(1)=1
(2) "1 3^3"=10^2+3^3=127=prime(31) gives a(2)=3
(3) "1 13^3"=10^4+13^3=12197=prime(1458) gives a(3)=13

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
Friedhelm Padberg, Elementare Zahlentheorie, Spektrum Akademischer Verlag, 2. Auflage 1991
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996

Cf. A000040 The prime numbers
Cf. A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
Cf. A168327 Primes of concatenated form p= "1 n^3"
Cf. A167535 Concatenation of two square numbers which give a prime

If n^3 is a d-digit natural number, odd and no multiple of 5, and d no multiple of 3, then p=10^d+n^3

Edited by Charles R Greathouse IV, Apr 23 2010

A173733 Primes p which give primes when 1331 = 11^3 is prefixed (see A173579).

Original entry on oeis.org

3, 17, 53, 83, 107, 227, 251, 269, 293, 347, 383, 431, 443, 521, 587, 599, 641, 647, 683, 719, 761, 773, 821, 857, 929, 1031, 1097, 1217, 1223, 1301, 1367, 1409, 1433, 1451, 1619, 1637, 1709, 1787, 1973, 2081, 2087, 2129, 2399, 2477, 2591, 2633, 2657, 2693

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 23 2010

N = 1331 = 11^3, p k-digit prime, to check if q = N * 10^k + p is prime

With exception of 3 necessarily p of form 3k+2, as sod(1331 = 8)

			13313 = prime(1581) => a(1) = prime(2) = 3
133117 = prime(12425) => a(2) = prime(7) = 17
133153 = prime(12427) => a(3) = prime(16) = 53
13311217 = prime(868166) => a(28) = prime(199) = 1217
13311223 = prime(868167) => a(29) = prime(200) = 1223
Note: two consecutive primes P = prime(n), Q = prime(n+1) yield consecutive prime concatenations "N P" = prime(m) and "N Q" = prime(m+1)

K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985
Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983

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

Cf. A102006, A167535, A168147, A168219, A168274, A173579

Select[Prime[Range[400]],PrimeQ[FromDigits[Join[{1,3,3,1}, IntegerDigits[ #]]]]&] (* Harvey P. Dale, Jun 09 2015 *)

Edited and extended by Charles R Greathouse IV, Apr 24 2010

A168545 Primes p such that the concatenation of p and 29 is a square number: "p 29" = N = m^2.

Original entry on oeis.org

5, 7, 53, 59, 151, 313, 1069, 1789, 1823, 2237, 2777, 3329, 3881, 3931, 4583, 5227, 6037, 7621, 7691, 9467, 12611, 13759, 14957, 17609, 20249, 28123, 35081, 36979, 49417, 56311, 56501, 63857, 69011, 71663, 79693, 85439, 94433, 114041, 117443

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Nov 29 2009

(1) It is conjectured that the sequence is infinite.
(2) 29 = prime(10) is the smallest prime with the property that its digits can be the final two digits of a square.
(3) The possible final digits of m are necessarily e = 23, 27, 73 or 77.
(4) Elementary proof of (3) with (10^2 * k + e)^2 = "n 29" for these four values of e only.
(5) Note 23 + 77 = 27 + 73 = 10^2.

			(1) 529 = 23^2, 5 = prime(3) = a(1);
(2) 729 = 27^2, 7 = prime(4) = a(2);
(3) 5329 = 73^2, 53 = prime(16) = a(3);
(4) 16129 = 127^2, but 161 = 7 * 23 is composite => 161 is not a term of the sequence;
(5) 31329 = 177^2, 313 = prime(65) gives a(6) = 313.

Andreas Bartholome, Josef Rung, Hans Kern: Zahlentheorie für Einsteiger, Vieweg & Sohn 1995
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

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

Cf. A000040 (the prime numbers).
Cf. A167535 (concatenation of two square numbers which give a prime).
Cf. A158896 (primes whose squares are a concatenation of 2 with some prime).

A:= NULL:
count:= 0:
for m from 0 while count < 100 do
  for q in [23,27,73,77] do
    r:= floor((100*m + q)^2/100);
    if isprime(r) then A:= A, r; count:= count+1; fi
od od:
A; # Robert Israel, Nov 23 2015

isok(n) = isprime(n) && issquare(100*n + 29) \\ Michel Marcus, Jul 22 2013; corrected Jun 13 2022

cp's OEIS Frontend

A168327 Primes of concatenated form "1 n^3".

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A168147 Primes of the form 10*n^3 + 1.

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A173836 Natural numbers n such that the concatenation 1331//n^3 is a prime number.

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A066591 Primes which can be expressed as a concatenation of nonnegative squares.

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A168219 Naturals n for which 1 + 10*n^3 (A168147) is prime.

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A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

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A173579 Natural numbers n which give primes when 1331 = 11^3 is prefixed.

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