A211868 -id:A211868 - cp's OEIS frontend

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

Alois P. Heinz, Aug 07 2009

			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).

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

Cf. A072701.

Cf. A000196, A000040, A164283, A211868.

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

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);

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 *)

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

A164283 Number of ways to write n as the root-mean-square (RMS) of a set of distinct positive integers.

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 79, 225, 693, 1901, 5597, 17641, 57503, 195431, 647139, 2182987, 7344451, 25057681, 85742999, 295284367, 1028155825, 3596134963, 12659796475, 44696280143, 158226554179, 562623263251, 2006471222195, 7182910999719, 25795458946677, 92875047372825, 335362896810137

Offset: 1

Alois P. Heinz, Aug 12 2009

nonn

			a(6) = 9, because 6 is the RMS of 9 sets of distinct positive integers: 6 = RMS(6) = RMS(1,3,5,8,9) = RMS(3,4,5,7,9) = RMS(1,2,4,5,7,11) = RMS(1,3,5,6,8,9) = RMS(3,4,5,6,7,9) = RMS(1,2,3,5,7,8,10) = RMS(1,2,4,5,6,7,11) = RMS(1,2,3,5,6,7,8,10).

Eric Weisstein's World of Math, Root-Mean-Square

Cf. A163974, A066572, A066571, A072701.

Cf. A000196, A211868.

a164283 n = f [1..] 1 nn 0 where
   f (k:ks) l nl xx
     | yy > nl  = 0
     | yy < nl  = f ks (l + 1) (nl + nn) yy + f ks 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 + k * k
   nn = n ^ 2
-- Reinhard Zumkeller, Feb 13 2013

sns:= proc(i) option remember; `if`(i=1, 1, sns(i-1) +i^2) end: b:= proc(n, i, t) if n<0 or i

sns[i_] := sns[i] = If[i == 1, 1, sns[i-1] + i^2] ; b[n_, i_, t_] := Which[n < 0 || i < t, 0, n == 0, If[t == 0, 1, 0], i == 1, If[n == 1 && t == 1, 1, 0], True, b[n, i, t] = b[n, i-1, t] + b[n - i^2, i-1, t-1]]; a[n_] := a[n] = Module[{s = 1, k}, For[k = 2, sns[k] <= k*n^2, k++, s = s + b[k*n^2, Floor[Sqrt[k*n^2 - sns[k-1]]], k]]; s]; Table[Print[an = a[n]]; an, {n, 1, 29}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

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

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