A076815 Initial indices of five successive primes squared with integer average.

79, 258, 397, 428, 429, 502, 503, 609, 787, 788, 925, 926, 927, 1026, 1027, 1028, 1105, 1312, 1334, 1335, 1343, 1348, 1349, 1378, 1422, 1524, 1572, 1601, 1602, 1790, 1791, 1813, 2015, 2081, 2082, 2125, 2126, 2131, 2141, 2142, 2147, 2292, 2448, 2765, 2766

Offset

1,1

Comments

Or, numbers n such that sum of 5 consecutive primes squared, starting with p(n), ends with 5.

Unlike the average of two, three, four and six successive primes squared (with initial indices > 1,2,1,2, respectively), the average of five successive primes squared is rarely an integer.

Cases of sums ending with 5 are much less numerous than cases with 1, 3, 7 and 9.

E.g. for the first 20000, sums with final digits 1, 3, 5, 7 and 9 are 7238, 2380, 466, 2529 and 7386 (and 1 case with final 8, 208=A131686(1)). And for first 200000 sums the corresponding numbers are 71166, 25820, 5956, 26075, 70982.

The explanation of this "deficiency of final 5's" is simple: assuming that final digits {1,3,7,9} of primes are equally often, we get that probabilities for final digits {1,3,5,7,9} of sum of squares of five primes are {10/32,5/32,2/32,5/32,10/32}.

Links

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

Example

sum(prime(i)^2,i=79..83)/5=(401^2+409^2+419^2+421^2+431^2)/5=866645/5=173329=A076814(1),
sum(prime(i)^2,i=258..262)/5=(1627^2+1637^2+1657^2+1663^2+1667^2)/5=13617005/5=2723401=A076814(2).

Mathematica

PrimePi[Sqrt[#]]&/@Select[Partition[Prime[Range[3000]]^2, 5, 1], IntegerQ[ Mean[ #]]&][[All, 1]] (* Harvey P. Dale, Jul 15 2017 *)

Crossrefs

Cf. A076815, A076814, A131686.

Sequence in context: A347032 A142285 A265599 * A141914 A201036 A142932

Adjacent sequences: A076812 A076813 A076814 * A076816 A076817 A076818

Keyword

nonn

Author

Zak Seidov, Oct 17 2002

Extensions

Edited and merged with A131359 by Zak Seidov, May 18 2008 at the suggestion of R. J. Mathar

Status

approved