This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(19) = A001414(A072697(19)) = A001414(42) = A001414(2*3*7) = 2+3+7 = 12.
a(19) = A001222(A072697(19)) = A001222(42) = A001222(2*3*7) = 3.
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.
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
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
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 *)
a(19) = A072698(19)/A072699(19) = 12/3 = 4.
with(combinat): P:=proc(q) local a,b,c,d,j,k,x; x:=[]; for j from 1 to q do x:=[op(x),ithprime(j)]; od; for j from 1 to q do a:=choose(x,j); b:=10^40; d:=0; for k from 1 to nops(a) do c:=convert(a[k],`+`)/j; if frac(c)=0 and c
f[n_] := If[n == 0, {1}, Block[{P = Product[Prime@ i, {i, n}], lim, k = 1, c, w = ConstantArray[1, n]},lim = Prime[n + 1] P;{w}~Join~Reap[Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ w]; If[c < lim, Sow[w]; k = 1, If[k == n, Break[], k++]], {i, Infinity}]][[-1, 1]]]]; Array[Min@ Map[Times @@ # &, Select[Map[Prime@ Accumulate@ # &, f@ #], IntegerQ@ Mean@ # &]] &, 14] (* Michael De Vlieger, Mar 19 2018 *)
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