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%I #15 Aug 04 2024 17:59:09
%S 19,190,250,346,378,400,402,456,516,553,567,586,664,759,762,853,931,
%T 972,1140,1156,1161,1242,1266,1284,1314,1317,1338,1398,1440,1645,1744,
%U 1785,1840,1875,1930,1944,2227,2248,2271,2287,2316,2397,2401,2467,2568,2602
%N Numbers n such that concatenations n//13 and n//31 are consecutive primes.
%C p = n//13 = n * 10^2 + 13 = prime(i) , q = n//31 = n * 10^2 + 31 = prime(i+1)
%C p and q are formed by the same digits (counted with multiplicity)
%C n = m//k (k = 0, 1, ...,9)
%C List of m < 10^3
%C 0//13: 19, 25, 40, 114, 144, 184, 193, 280, 411, 415, 567, 604, 634, 777, 852, 862, 870, 943 (18)
%C 1//13: 93, 116, 227, 240, 392, 462, 543, 570, 611, 675, 689, 734, 759, 821, 822, 878, 969, 986 (18)
%C 2//13: 40, 76, 97, 124, 260, 338, 365, 415, 505, 545, 599, 625, 788, 809 (14)
%C 3//13: 55, 85, 312, 349, 421, 424, 451, 454, 619, 622, 724, 928 (12)
%C 4//13: 66, 128, 131, 174, 194, 293, 345, 414, 657, 687, 702, 741, 752, 867, 870, 939 (16)
%C 5//13: 164, 178, 187, 277, 379, 416, 481, 536, 754, 824, 935, 974, 995 (13)
%C 6//13: 34, 45, 51, 58, 115, 126, 231, 336, 402, 432, 439, 489, 502, 541, 705, 780, 838, 850, 909, 985 (20)
%C 7//13: 56, 131, 222, 228, 239, 246, 309, 480, 530, 716, 732, 747, 761, 792, 831, 936, 981 (17)
%C 8//13: 37, 133, 139, 224, 256, 286, 301, 304, 497, 518, 550, 559, 562, 728, 856, 907 (16)
%C 9//13: 1, 75, 526, 558, 681, 720, 765, 916, 943 (9)
%C 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
%e 19//13 = 1913 = prime(293), 19//31 = 1931 = prime(294), 19 is 1st term
%e 190//13 = 19013 = prime(2161), 190//31 = 19031 = prime(2162), 190 is 2nd term
%t Select[Range[3000],PrimeQ[# 100+13]&&NextPrime[# 100+13]==# 100+31&] (* _Harvey P. Dale_, Jun 23 2022 *)
%o (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
%Y Cf. A168327, A168417, A173836, A174213, A174260, A174355, A174441.
%K base,nonn
%O 1,1
%A Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 21 2010
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