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A072701
Number of ways to write n as the arithmetic mean of a set of distinct primes.
10
0, 1, 1, 2, 3, 4, 5, 10, 9, 18, 19, 40, 37, 80, 79, 188, 163, 385, 355, 855, 738, 1815, 1555, 3796, 3237, 8281, 6682, 17207, 13967, 35370, 28575, 74385, 58831, 153816, 119948, 312288, 244499, 643535, 495011, 1309267, 997381, 2629257, 2004295, 5334522
OFFSET
1,4
COMMENTS
a(n) = #{ m | A072700(m)=n }.
a(n) < A066571(n).
EXAMPLE
a(6) = 4, as 6 = (5+7)/2 = (2+3+13)/3 = (2+5+11)/3 = (2+3+5+7+13)/5;
a(7) = 5, as 7 = 7/1 = (3+11)/2 = (3+5+13)/3 = (3+7+11)/3 = (3+5+7+13)/4.
MAPLE
sp:= proc(i) option remember; `if`(i=1, 2, sp(i-1) +ithprime(i)) end: b:= proc(n, i, t) 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 b(n, i, t):= b(n, prevprime(i), t) +b(n-i, prevprime(i), t-1) fi end: a:= proc(n) local s, k; s:= `if`(isprime(n), 1, 0); for k from 2 while sp(k)/k<=n do s:= s +b(k*n, nextprime(k*n -sp(k-1)-1), k) od; s end: seq(a(n), n=1..28); # Alois P. Heinz, Jul 20 2009
MATHEMATICA
Needs["DiscreteMath`Combinatorica`"]; a = Drop[ Sort[ Subsets[ Table[ Prime[i], {i, 1, 20}]]], 1]; b = {}; Do[c = Apply[Plus, a[[n]]]/Length[a[[n]]]; If[ IntegerQ[c], b = Append[b, c]], {n, 1, 2^20 - 1}]; b = Sort[b]; Table[ Count[b, n], {n, 1, 20}]
t = Table[0, {200}]; k = 2; lst = Prime@Range@25; While[k < 2^25+1, slst = Flatten@Subsets[lst, All, {k}]; If[Mod[Plus @@ slst, Length@slst] == 0, t[[(Plus @@ slst)/(Length@slst)]]++ ]; k++ ]; t (* Robert G. Wilson v *)
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, b[n, NextPrime[i, -1], t] + b[n - i, NextPrime[i, -1], t - 1]];
a[n_] := Module[{s, k}, s = If[PrimeQ[n], 1, 0]; For[k = 2, sp[k]/k <= n, k++, s = s + b[k*n, NextPrime[k*n - sp[k - 1] - 1], k]]; s];
Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Feb 13 2018, after Alois P. Heinz *)
PROG
(Haskell)
a072701 n = f a000040_list 1 n 0 where
f (p:ps) l nl x
| y > nl = 0
| y < nl = f ps (l + 1) (nl + n) y + f ps l nl x
| otherwise = if y `mod` l == 0 then 1 else 0
where y = x + p
-- Reinhard Zumkeller, Feb 13 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 04 2002 and Jul 15 2002
EXTENSIONS
Corrected by John W. Layman, Jul 11 2002
More terms from Alois P. Heinz, Jul 20 2009
STATUS
approved