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

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

Author

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

Keywords

• base
• nonn

Comments

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

Examples

			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
		

Crossrefs

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

Programs

Mathematica

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

PARI

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