A163974 Number of ways to write n as the root-mean-square (RMS) of a set of distinct primes.

0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 7, 0, 3, 3, 11, 1, 11, 2, 11, 3, 37, 0, 44, 18, 52, 24, 103, 50, 147, 52, 214, 170, 475, 229, 711, 375, 1116, 587, 2101, 542, 3009, 1940, 4870, 1680, 8961, 5923, 16712, 4190, 24098, 11552, 42715, 11347, 69608, 32495, 103914, 50493, 189499, 103581, 304367, 152520, 453946, 203153, 783817, 246991, 1345661

Offset: 1

Author

Alois P. Heinz, Aug 07 2009

Keywords

• nice
• nonn

Examples

			a(13) = 7 because 13 is the RMS of 7 sets of distinct primes: 13 = RMS(13) = RMS(7,17) = RMS(5,11,19) = RMS(7,13,17) = RMS(5,11,13,19) = RMS(5,7,11,17,19) = RMS(5,7,11,13,17,19).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Root-Mean-Square

Crossrefs

Cf. A072701.

Cf. A000196, A000040, A164283, A211868.

Programs

Haskell

a163974 n = f a000040_list 1 nn 0 where
   f (p:ps) l nl xx
     | yy > nl   = 0
     | yy < nl   = f ps (l + 1) (nl + nn) yy + f ps l nl xx
     | otherwise = if w == n then 1 else 0
     where w = if r == 0 then a000196 m else 0
           (m, r) = divMod yy l
           yy = xx + p * p
   nn = n ^ 2
-- Reinhard Zumkeller, Feb 13 2013

Maple

sps:= proc(i) option remember; `if`(i=1, 4, sps(i-1) +ithprime(i)^2) 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=4 and t=1, 1, 0) else b(n, i, t):= b(n, prevprime(i), t) +b(n-i^2, prevprime(i), t-1) fi end: a:= proc(n) option remember; local s, k; s:= `if`(isprime(n), 1, 0); for k from 2 while sps(k)<=k*n^2 do s:= s +b(k*n^2, nextprime(floor(sqrt(k*n^2 -sps(k-1)))-1), k) od; s end: seq(a(n), n=1..30);

Mathematica

sps[i_] := sps[i] = If[i == 1, 4, sps[i - 1] + Prime[i]^2]; b[n_, i_, t_] := b[n, i, t] = If[ n < 0 , 0 , If[ n == 0 , If[t == 0, 1, 0], If[ i == 2 , If[n == 4 && t == 1, 1, 0], b[n, NextPrime[i, -1], t] + b[n - i^2, NextPrime[i, -1], t - 1]]]]; a[n_] := a[n] = (s = Boole[PrimeQ[n]]; For[k = 2, sps[k] <= k*n^2, k++, s = s + b[k*n^2, NextPrime[ Floor[ Sqrt[k*n^2 - sps[k - 1]]] - 1], k]]; s); Table[ Print[a[n]]; a[n], {n, 1, 58}] (* Jean-François Alcover, Jul 11 2012, translated from Maple *)

Extensions

Terms a(59)-a(67) by Reinhard Zumkeller, Feb 13 2013