A174441 -id:A174441 - cp's OEIS frontend

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

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

cp's OEIS Frontend

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

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