A168417 -id:A168417 - cp's OEIS frontend

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

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

A174229 Natural numbers n such that the concatenation n^3//1331, i.e., a cube and 11^3, is a prime number.

Original entry on oeis.org

2, 6, 8, 14, 21, 38, 39, 51, 54, 65, 68, 78, 80, 93, 104, 107, 114, 117, 119, 125, 135, 137, 146, 147, 152, 153, 158, 159, 167, 186, 206, 225, 243, 246, 248, 257, 258, 269, 270, 272, 278, 284, 290, 291, 306, 311, 317, 321, 323, 324, 369, 372, 387, 390, 399, 404, 410, 414, 425

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 13 2010

See comments and references for A174213.

			2^3 = 8, 81331 = prime(7958) => a(1) = 2;
6^3 = 216, 2161331 = prime(160048) => a(2) = 6.

Cf. A168327, A168417, A173836, A174213.

isok(n) = isprime(eval(concat(Str(n^3), Str(1331)))); \\ Michel Marcus, Jul 20 2017

More terms from Michel Marcus, Jul 20 2017

A174409 Prime numbers p such that the concatenation p^3//1331 is a prime number.

Original entry on oeis.org

2, 107, 137, 167, 257, 269, 311, 317, 557, 593, 761, 773, 809, 911, 1103, 1151, 1283, 1289, 1481, 1487, 1559, 1709, 1787, 1931, 2111, 2141, 2243, 2339, 2357, 2657, 2687, 2777, 2909, 3137, 3209, 3251, 3359, 3371, 3389, 3449

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 19 2010

See comments at A174213.
p^3//1331 is the concatenation of the cubes of two primes.
With the exception of a(1)=2, each term is necessarily of the form 6*k-1.

			The first prime is 2; 2^3 = 8, and 81331 = prime(7958), so a(1)=2.
The smallest prime p > 2 such that p^3//1331 yields a prime is p=107: 107^3 = 1225043, and 12250431331 = prime(552342812), so a(2)=107.

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

Cf. A168327, A168417, A173836, A174213, A174260, A174229, A174355.

[p: p in PrimesUpTo(5000) | IsPrime(Seqint(Intseq(1331) cat Intseq(p^3)))]; // Vincenzo Librandi, Mar 05 2018

Select[Prime[Range[500]],PrimeQ[10000#^3+1331]&] (* Harvey P. Dale, May 30 2017 *)

cp's OEIS Frontend

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

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A174229 Natural numbers n such that the concatenation n^3//1331, i.e., a cube and 11^3, is a prime number.

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