A168069 -id:A168069 - cp's OEIS frontend

A168556 Smallest primes p = p(k) with (p(k)+p(k+1)+p(k+2))/15 an integer.

3, 4691, 6397, 6911, 8893, 10181, 11833, 12113, 13063, 13267, 14251, 16661, 17851, 20563, 21341, 21881, 22229, 22877, 23339, 23633, 24859, 26561, 26591, 27337, 27361, 28151, 29531, 30029, 30211, 30881, 30983, 31271, 33961, 34439, 35227

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 29 2009

nonn

(1) Denominator 15 = p(2)+ p(3)+ p(4) is sum of the first 3 consecutive odd primes.
(2) First three consecutive odd primes 3, 5, 7 are an exceptional case, because of p(k+1)=p(3)=5, no other prime has end digit 5.
(3) Four out of 19 possibilities for end digit permutations (which all "happen") of three consecutive primes p(k), p(k+1) and p(k+2) are permutations of (1,1,3), (1,7,7), (3,3,9) and (7,9,9), as sum of these digits has to be necessarily a multiple of 5.
(4) Note the "gap" for a(1) and a(2).

			(1) 3 + 5 + 7 = 15 * 1 => a(1)=3 for k=2.
(2) 4691 + 4703 + 4721 = 14115 = 15 * 941 => a(2)=4691 for k=634.
(3) 6397 + 6421 + 6427 = 19245 = 15 * 1283 => a(3)=6397 for k=834.
(4) 8893 + 8923 + 8929 = 26745 = 15 * 1783 => a(5)=8893 for k=1108.
(5) 23339 + 23357 + 23369 = 70065 = 15 * 4671 => a(19)=23339 for k=2603.

Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

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

Cf. A000040 The prime numbers

Cf. A168069 a(n) is the index k of the smallest prime p(k) with (p(k) + p(k+1))/n an integer

Select[Partition[Prime[Range[4000]],3,1],IntegerQ[Total[#]/15]&][[;;,1]] (* Harvey P. Dale, Jul 20 2025 *)

lista(nn) = {vp = primes(nn); for (k = 1, nn - 2, if ((vp[k] + vp[k+1] + vp[k+2]) % 15 == 0, print1(vp[k], ", ")););} \\ Michel Marcus, Aug 27 2013

cp's OEIS Frontend

A168556 Smallest primes p = p(k) with (p(k)+p(k+1)+p(k+2))/15 an integer.

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