A168147 -id:A168147 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

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

A168417 Primes q for which 1 concatenated with q^3 (A168327) is prime.

Original entry on oeis.org

3, 13, 103, 109, 139, 163, 181, 211, 379, 457, 463, 1021, 1087, 1123, 1201, 1249, 1303, 1381, 1579, 1597, 1609, 1699, 1861, 1873, 1987, 2011, 2029, 2053, 2143, 2221, 2281, 2341, 2473, 2503, 2557, 2731, 2857, 3061, 3067, 3217, 3253, 3271, 3319, 3331, 3517

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 25 2009

It is conjectured that this sequence is infinite.

			(1) "1 3^3"=10^2+3^3=127=prime(31) gives a(1)=3=prime(2)
(2) "1 103^3"=10^7+103^3=11092727=prime(732258) gives a(3)=103=prime(27)

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

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 Natural numbers n for which the concatenation p= "1 n^3" is prime

Select[Prime[Range[500]],PrimeQ[FromDigits[Join[{1},IntegerDigits[ #^3]]]]&] (* Harvey P. Dale, Jan 21 2013 *)

Edited and extended by Charles R Greathouse IV, Apr 23 2010

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

A168274 Primes p such that 10*p^3 + 1 is prime.

Original entry on oeis.org

3, 31, 37, 43, 73, 109, 163, 211, 241, 313, 337, 409, 499, 673, 739, 757, 907, 1033, 1039, 1069, 1237, 1327, 1447, 1483, 1489, 1597, 1663, 1741, 1777, 1933, 2083, 2143, 2251, 2437, 2683, 2797, 3001, 3181, 3307, 3319, 3463, 3739, 3793, 4051

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 22 2009

nonn

Subsequence of A168219.

			10*3^3+1 = 271 is prime, so prime 3 is in the sequence.
10*313^3+1 = 306642971 is prime, so prime 313 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A168219, A168147.

[ p: p in PrimesUpTo(5000) | IsPrime(10*p^3+1) ]; // Vincenzo Librandi, Jan 29 2011

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

select(p->isprime(10*p^3+1), primes(1000)) \\ Charles R Greathouse IV, Jul 16 2016

Edited by Charles R Greathouse IV, Apr 24 2010

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

A176722 Primes of the form k^3 + 13, k >= 0.

Original entry on oeis.org

13, 229, 1013, 1741, 39317, 64013, 74101, 157477, 438989, 551381, 830597, 1906637, 2000389, 4096013, 7077901, 9261013, 10941061, 15625013, 16003021, 21024589, 24897101, 27000013, 69934541, 74088013, 79507013, 93576677, 122023949

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 25 2010

nonn

Necessarily, k = 6 * j or k = 6 * j + 4.

Values of k corresponding to terms of the sequence: 0, 6, 10, 12, 34, 40, 42, 54, 76, 82, 94, 124, 126, 160, 192, 210, 222, 250, 252, 276, 292, 300, 412, 420, 430, 454, 496, 502, 570, 586, 612, 622, 640, 670, 684, 712, 720, 724, 726, 756, 784, 822, 826, 874, 882, 894, 934, 952, 964, 1006, 1056.

			0^3 + 13 = 13 = prime(6) = a(1);
6^3 + 13 = 229 = prime(50) = a(2);
300^3 + 13 = 27000013 = prime(1683067) = a(22).

H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag Berlin, 1973.

Robert Israel, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown and B. Z. Moroz, Primes Represented by Binary Cubic Forms, Proceedings of the London Math. Soc. vol. 84(2), pp. 257-288, 2002; see also.

Cf. A000040, A000578, A037396, A138375, A144953, A154648, A157772, A168147, A168219, A168327.

[a: n in [0..500]|IsPrime(a) where a is n^3+13] // Vincenzo Librandi, Dec 22 2010

select(isprime,[seq(seq((6*j+m)^3+13,m=[0,4]),j=0..1000)]); # Robert Israel, Jun 28 2018

Select[Range[0,1000]^3+13,PrimeQ]  (* Harvey P. Dale, Mar 12 2011 *)

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

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

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

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

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A168417 Primes q for which 1 concatenated with q^3 (A168327) is prime.

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

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A168274 Primes p such that 10*p^3 + 1 is prime.

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

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