A072820 Largest number of distinct primes to represent n as arithmetic mean.

1, 1, 3, 3, 5, 4, 5, 5, 7, 6, 9, 8, 9, 9, 11, 10, 11, 11, 13, 12, 13, 13, 15, 14, 17, 15, 17, 16, 17, 17, 19, 18, 19, 19, 21, 20, 23, 21, 23, 22, 23, 23, 25, 24, 25, 25, 27, 26, 27, 27, 29, 28, 29, 28, 29, 29, 31, 30, 31, 31, 33, 32, 33, 33, 35, 33, 35, 34, 37, 35, 37, 36, 37, 37

Offset: 2

Author

Reinhard Zumkeller, Jul 15 2002

Keywords

• nonn

Examples

			a(20) = 13: (2+3+5+7+11+13+17+19+23+29+31+41+59)/13 = 20. [corrected by _Jean-François Alcover_, Nov 10 2020]
		

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) 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, 1, 0) else h := b(n, prevprime(i), t); b(n, i, t):= `if`(h>0, h, b(n-i, prevprime(i), t-1)) fi end:
a:= proc(n) local i, k; if n<4 then 1 else for k from 2 while sp(k)/k<=n do od: do k:= k-1; if b(k*n, nextprime(k*n -sp(k-1)-1), k)>0 then break fi od; k fi end: seq(a(n), n=2..50); # 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] = Module[{h}, Which[n < 0, 0, n == 0, If[t == 0, 1, 0], i == 2, If[n == 2 && t == 1, 1, 0], True, h = b[n, NextPrime[i, -1], t]; If[h > 0, h, b[n - i, NextPrime[i, -1], t - 1]]]];
a[n_] := a[n] = Module[{k}, If[n < 4, 1, For[k = 2, sp[k]/k <= n, k++]; While[True, k = k - 1; If[b[k n, NextPrime[k n - sp[k - 1] - 1], k] > 0, Break[]]]; k]];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 100}] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Aug 03 2009