A072821 Largest prime that can appear in any representation of n as an arithmetic mean of distinct primes.

1, 1, 7, 7, 13, 13, 23, 19, 29, 29, 43, 37, 53, 47, 71, 61, 79, 73, 103, 89, 113, 109, 139, 127, 157, 139, 179, 163, 199, 181, 223, 199, 241, 227, 271, 241, 293, 271, 317, 293, 349, 317, 379, 349, 409, 379, 439, 409, 463, 439, 503, 463, 523, 499, 571, 523, 601

Offset: 2

Author

Reinhard Zumkeller, Jul 15 2002

Keywords

• nonn

Comments

Thanks to John W. Layman for inspiration.

Examples

			a(6) = 13, as 13 is the largest prime in 6 = (5+7)/2 = (2+3+13)/3 = (2+5+11)/3 = (2+3+5+7+13)/5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..100
• Reinhard Zumkeller, Representing integers as arithmetic means of primes

Crossrefs

Cf. A072701.

Programs

Maple

sp:= proc(i) option remember; `if` (i=1, 2, sp(i-1) +ithprime(i)) end:
b:= proc(n, i, t) option remember; local h; if n<0 then 0 elif n=0 then `if` (t=0, 1, 0) elif i=2 then `if` (n=2 and t=1, 2, 0) else `if` (b(n-i, prevprime(i), t-1)>0, i, b(n, prevprime(i), t)) fi end:
a:= proc(n) local s, k; s:= 1; for k from 2 while sp(k)/k<=n do s:= max (s, b(k*n, nextprime (k*n -sp(k-1)-1), k)) od: s end:
seq(a(n), n=2..40); # Alois P. Heinz, Aug 03 2009

Mathematica

sp[i_] := sp[i] = If[i == 1, 2, sp[i - 1] + Prime[i]];
b[n_, i_, t_] := b[n, i, t] = Which[n < 0, 0, n == 0, If[t == 0, 1, 0], i == 2,  If[n == 2 && t == 1, 1, 0], True, If[b[n - i, NextPrime[i, -1], t - 1] > 0, i, b[n, NextPrime[i, -1], t]]];
a[n_] := Module[{s, k}, s = 1; For[k = 2, sp[k]/k <= n, k++, s = Max[s, b[k*n, NextPrime[k*n - sp[k - 1] - 1], k]]]; s];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Feb 13 2018, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Aug 03 2009