A168327 -id:A168327 - cp's OEIS frontend

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

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

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

A174213 Natural numbers n such that the concatenation n//1331 is a prime number.

Original entry on oeis.org

6, 8, 9, 23, 29, 30, 32, 39, 42, 45, 53, 57, 65, 80, 92, 95, 101, 102, 108, 113, 116, 128, 141, 144, 153, 161, 182, 183, 186, 200, 206, 216, 218, 219, 225, 239, 245, 249, 260, 266, 270, 273, 279, 281, 282, 296, 311, 314, 318, 321

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 12 2010

n is no multiple of 11 as 1331 = 11^3.
Necessarily n = 3 * k or n = 3 * k + 2, but not n = 3 * k + 1, because sod(1331) = 8.
Sequence is infinite, Dirichlet's prime number theorem for naturals of the form n * 10^4 + 1331.
For prefixed 1331 and references see A173836.

			61331 = prime(6169) => a(1) = 6.
81331 = prime(7958) => a(2) = 8.

A168327, A168417, A173836

Select[Range[400],PrimeQ[#*10^4+1331]&] (* Harvey P. Dale, Jan 17 2019 *)

isok(n) = isprime(n*10^4 + 1331); \\ Michel Marcus, Aug 27 2013

A174260 Prime numbers p such that the concatenation p//1331 is a prime number.

Original entry on oeis.org

23, 29, 53, 101, 113, 239, 281, 311, 347, 353, 389, 401, 431, 617, 641, 647, 743, 797, 821, 827, 863, 911, 941, 1049, 1283, 1319, 1373, 1439, 1481, 1487, 1493, 1511, 1583, 1613, 1667, 1709, 1721, 1733, 1823, 1949

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 14 2010

Necessarily (as sod(1331) = 3 * 2 + 2): p = 6 * k - 1

See comments and references for A174213

			231331 = prime(20545) => p(1) = 23 = prime(9)
291331 = prime(25334) => p(2) = 29 = prime(10)

A168327, A168417, A173836, A174213

Select[Prime[Range[300]],PrimeQ[1331+10000#]&] (* Harvey P. Dale, Jun 22 2013 *)

A174355 Natural numbers n such that the concatenations n//1331 and 1331//n are prime numbers.

Original entry on oeis.org

53, 57, 153, 249, 279, 329, 333, 339, 347, 381, 399, 431, 471, 489, 641, 647, 711, 821, 851, 923, 959, 987, 1169, 1239, 1313, 1383, 1479, 1547, 1563, 1589, 1611, 1653, 1677, 1709, 1773, 1863, 1887, 1973, 2031, 2067

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 17 2010

See comments and references for A173836, A174213.

Intersection of A173579 and A174213. - Michel Marcus, Aug 27 2013

			531331 = prime(43928), 133153 = prime(12427), 53 is smallest term of sequence.

Marcus du Sautoy: Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004

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

Cf. A168327, A168417, A173836, A174213, A174260.

Select[Range[2100],AllTrue[{#*10^4+1331,1331*10^IntegerLength[#]+#},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 21 2015 *)

isok(n) = isprime(n*10^4 + 1331) && isprime(1331*10^(length(Str(n))) + n); \\ Michel Marcus, Aug 27 2013

A173291 Smallest prime p such that the concatenation of p and prime(n) is a prime, or 0 if no other number exists.

Original entry on oeis.org

0, 2, 0, 3, 2, 3, 3, 7, 2, 2, 3, 3, 2, 7, 3, 3, 3, 7, 3, 2, 3, 3, 2, 3, 3, 5, 7, 5, 3, 2, 7, 2, 2, 19, 11, 7, 19, 3, 3, 9, 2, 3, 3, 7, 5, 37, 7, 31, 5, 3, 5, 2, 13, 2, 3, 41, 2, 3, 31, 2, 7, 2, 3, 2, 3, 11, 3, 13, 2, 7, 11, 3, 13, 3, 19, 2, 2, 13, 17, 37, 5, 13, 5, 3, 139, 5, 3, 3, 3, 3, 2, 5, 7, 3, 3

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 15 2010

If prime(n) has k digits then a(k) is the smallest prime(m) where 10^k * prime(m) + prime(n) is a prime.

In base 10, no prime can be prefixed to 2 or 5 to make another prime.

			a(2) = 2 because prime(2) = 3, and the concatenation of 2 and 3 gives the prime 23.
a(3) = 0 because prime(3) = 5 and there is no prime to concatenate with to give another prime.
a(4) = 3 because prime(5) = 7 but the concatenation with 2 gives 27 = 3^3, so it has to be 3 in order to give 37, which is prime.

John Derbyshire, Prime obsession. Joseph Henry Press, Washington, DC 2003
Marcus du Sautoy, Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

Cf. A088606, A167764, A168327, A168417, A030469.

A174441 Primes p such that the concatenations p//1331 and 1331//p are both prime numbers (for naturals see A174355).

Original entry on oeis.org

53, 347, 431, 641, 647, 821, 1709, 1973, 2081, 2591, 2657, 2963, 4073, 4139, 4643, 4787, 5039, 5483, 5657, 6029, 6791, 6917, 6959, 7127, 7673, 8273, 8693, 8807, 8849, 9221, 9311, 9689, 10139, 10457, 11423, 12503, 12743, 13619, 13913, 14549

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 20 2010

See comments and references for A173836, A174213.

			531331 = prime(43928), 133153 = prime(12427) => p(1) = 53 = prime(16).
3471331 = prime(248286), 1331347 = prime(102237) => p(2) = 347 = prime(69).
139131331 = prime(7865788), 133113913 = prime(7544750) => p(39) = 13913 = prime(1645).

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

Cf. A168327, A168417, A173836, A174213, A174260, A174355.

Select[Prime[Range[2000]],AllTrue[{#*10^4+1331,1331*10^IntegerLength[ #]+#}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 08 2016 *)

isok(n) = isprime(n) && isprime(n*10^4 + 1331) && isprime(1331*10^(length(Str(n))) + n); \\ Michel Marcus, Aug 27 2013

A176600 Numbers n such that concatenations n//13 and n//31 are consecutive primes.

Original entry on oeis.org

19, 190, 250, 346, 378, 400, 402, 456, 516, 553, 567, 586, 664, 759, 762, 853, 931, 972, 1140, 1156, 1161, 1242, 1266, 1284, 1314, 1317, 1338, 1398, 1440, 1645, 1744, 1785, 1840, 1875, 1930, 1944, 2227, 2248, 2271, 2287, 2316, 2397, 2401, 2467, 2568, 2602

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 21 2010

p = n//13 = n * 10^2 + 13 = prime(i) , q = n//31 = n * 10^2 + 31 = prime(i+1)
p and q are formed by the same digits (counted with multiplicity)
n = m//k (k = 0, 1, ...,9)
List of m < 10^3
0//13: 19, 25, 40, 114, 144, 184, 193, 280, 411, 415, 567, 604, 634, 777, 852, 862, 870, 943 (18)
1//13: 93, 116, 227, 240, 392, 462, 543, 570, 611, 675, 689, 734, 759, 821, 822, 878, 969, 986 (18)
2//13: 40, 76, 97, 124, 260, 338, 365, 415, 505, 545, 599, 625, 788, 809 (14)
3//13: 55, 85, 312, 349, 421, 424, 451, 454, 619, 622, 724, 928 (12)
4//13: 66, 128, 131, 174, 194, 293, 345, 414, 657, 687, 702, 741, 752, 867, 870, 939 (16)
5//13: 164, 178, 187, 277, 379, 416, 481, 536, 754, 824, 935, 974, 995 (13)
6//13: 34, 45, 51, 58, 115, 126, 231, 336, 402, 432, 439, 489, 502, 541, 705, 780, 838, 850, 909, 985 (20)
7//13: 56, 131, 222, 228, 239, 246, 309, 480, 530, 716, 732, 747, 761, 792, 831, 936, 981 (17)
8//13: 37, 133, 139, 224, 256, 286, 301, 304, 497, 518, 550, 559, 562, 728, 856, 907 (16)
9//13: 1, 75, 526, 558, 681, 720, 765, 916, 943 (9)
The sequence could be defined as "Numbers n such that 100n+13 and 100n+31 are consecutive primes". In that sense it could be considered to be independent of the decimal numeral system. - M. F. Hasler, Dec 04 2010

			19//13 = 1913 = prime(293), 19//31 = 1931 = prime(294), 19 is 1st term
190//13 = 19013 = prime(2161), 190//31 = 19031 = prime(2162), 190 is 2nd term

Cf. A168327, A168417, A173836, A174213, A174260, A174355, A174441.

Select[Range[3000],PrimeQ[# 100+13]&&NextPrime[# 100+13]==# 100+31&] (* Harvey P. Dale, Jun 23 2022 *)

A176600(n,print_all=0)={ for(k=1,1e9,isprime(100*k+13) || next;nextprime(100*k+17)==100*k+31||next;print_all & print1(k",");n-- || return(k))} \\ M. F. Hasler, Dec 04 2010

A176601 Primes p that p//13 and p//31 are consecutive primes.

Original entry on oeis.org

19, 853, 2287, 2467, 4243, 4513, 4621, 5431, 5701, 7243, 7477, 7591, 7927, 8221, 8317, 9283, 9439, 9817, 10039, 12781, 13933, 14461, 14923, 15727, 16693, 17443, 18199, 18217, 19207, 20749, 21139, 22147, 23761, 25471, 26701, 26953, 27481, 28111, 28447, 28579

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 21 2010

See A176600.

			19//13 = 1913 = prime(293), 19//31 = 1931 = prime(294), 19 = prime(8) is 1st term.
853//13 = 85313 = prime(8306), 853//31 = 85331 = prime(8307), 853 = prime(147) is 2nd term.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A168327, A168417, A173836, A174213, A174260, A174355, A174441, A176600.

okQ[n_]:=Module[{idn=IntegerDigits[n],p13,p31},p13=FromDigits[ Join[ idn,{1,3}]];p31=FromDigits[Join[idn,{3,1}]];PrimeQ[p13]&&NextPrime[p13] == p31]; Select[Prime[Range[16000]],okQ] (* Harvey P. Dale, Jan 21 2012 *)

More terms from Harvey P. Dale, Jan 21 2012

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 *)

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A168417 Primes q for which 1 concatenated with q^3 (A168327) 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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A174213 Natural numbers n such that the concatenation n//1331 is a prime number.

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A174260 Prime numbers p such that the concatenation p//1331 is a prime number.

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A174355 Natural numbers n such that the concatenations n//1331 and 1331//n are prime numbers.

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A173291 Smallest prime p such that the concatenation of p and prime(n) is a prime, or 0 if no other number exists.

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A174441 Primes p such that the concatenations p//1331 and 1331//p are both prime numbers (for naturals see A174355).

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A176600 Numbers n such that concatenations n//13 and n//31 are consecutive primes.

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