A033448 Initial prime in set of 4 consecutive primes in arithmetic progression with common difference 18.

74453, 76543, 132893, 182243, 202823, 297403, 358793, 485923, 655453, 735883, 759113, 780613, 797833, 849143, 1260383, 1306033, 1442173, 1531093, 1534153, 1586953, 1691033, 1717063, 1877243, 1945763, 1973633, 2035513, 2067083, 2216803, 2266993, 2542513, 2556803, 2565203, 2805773

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Up to n = 10^4, the smallest difference a(n+1) - a(n) is 60 and occurs at n = 8571. - M. F. Hasler, Oct 26 2018

Each term is congruent to 3 mod 10 (as noted by Zak Seidov in the SeqFan email list). This means the three following consecutive primes are always congruent to 1, 9, and 7 mod 10, respectively (i.e., final digits for these primes are 3, 1, 9, 7, in that order). There cannot be a set of 5 such consecutive primes because a(n) + 4*18 == 5 (mod 10) so is a multiple of 5. - Rick L. Shepherd, Mar 27 2023

Examples

			{74453, 74471, 74489, 74507} is the first such set of 4 consecutive primes with common difference 18, so a(1) = 74453.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Analogous sequences (start of CPAP-4 with common difference in square brackets): A033451 [6], A033447 [12], A033448 [this: 18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].

Cf. A033450, A052189.

Programs

Mathematica

A033448 = Reap[For[p = 2, p < 2100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 18, p3 = NextPrime[p2]; If[p3 - p2 == 18, p4 = NextPrime[p3]; If[p4 - p3 == 18, Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *)
Transpose[Select[Partition[Prime[Range[160000]],4,1],Union[ Differences[ #]] == {18}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

PARI

A033448(n,show_all=1,g=18,p=2,o,c)={forprime(q=p+1,,if(p+g!=p=q,next, q!=o+2*g, c=3, c++>4, print1(o-g","); n--||break); o=q-g);o-g} \\ Can be used as nxt(p)=A033448(1,,,p+1), e.g.: {p=0;vector(20,i,p=nxt(p))} or {p=0;for(i=1,1e4,write("b.txt",i" "nxt(p)))}. - M. F. Hasler, Oct 26 2018

Extensions

More terms from Labos Elemer, Jan 31 2000

Definition clarified by Harvey P. Dale, Jun 17 2014

Example reflecting final digits given by Rick L. Shepherd, Mar 27 2023