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A168556 Smallest primes p = p(k) with (p(k)+p(k+1)+p(k+2))/15 an integer. 0

%I

%S 3,4691,6397,6911,8893,10181,11833,12113,13063,13267,14251,16661,

%T 17851,20563,21341,21881,22229,22877,23339,23633,24859,26561,26591,

%U 27337,27361,28151,29531,30029,30211,30881,30983,31271,33961,34439,35227

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

%C (1) Denominator 15 = p(2)+ p(3)+ p(4) is sum of the first 3 consecutive odd primes.

%C (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.

%C (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.

%C (4) Note the "gap" for a(1) and a(2).

%D Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005

%D Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

%e (1) 3 + 5 + 7 = 15 * 1 => a(1)=3 for k=2.

%e (2) 4691 + 4703 + 4721 = 14115 = 15 * 941 => a(2)=4691 for k=634.

%e (3) 6397 + 6421 + 6427 = 19245 = 15 * 1283 => a(3)=6397 for k=834.

%e (4) 8893 + 8923 + 8929 = 26745 = 15 * 1783 => a(5)=8893 for k=1108.

%e (5) 23339 + 23357 + 23369 = 70065 = 15 * 4671 => a(19)=23339 for k=2603.

%o (PARI) 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

%Y Cf. A000040 The prime numbers

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

%K nonn

%O 1,1

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

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Last modified September 25 20:01 EDT 2021. Contains 347659 sequences. (Running on oeis4.)