cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A072701 Number of ways to write n as the arithmetic mean of a set of distinct primes.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jul 04 2002 and Jul 15 2002

Keywords

Comments

a(n) = #{ m | A072700(m)=n }.
a(n) < A066571(n).

Examples

			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.

Links

Crossrefs

Cf. A072700, A072697, A072820, A072821.
Cf. A066571, A163974.

Programs

  • 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
  • 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 *)

Extensions

Corrected by John W. Layman, Jul 11 2002
More terms from Alois P. Heinz, Jul 20 2009

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

Views

Author

Alois P. Heinz, Aug 12 2009

Keywords

Examples

			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).

Links

Crossrefs

Cf. A163974, A066572, A066571, A072701.
Cf. A000196, A211868.

Programs

  • Haskell
    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
  • Maple
    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
  • Mathematica
    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 *)

A211868 Number of ways to write n as the root-mean-square (RMS) of a set of distinct odd integers.

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 3, 0, 9, 1, 19, 2, 59, 13, 161, 50, 413, 123, 1201, 352, 3463, 689, 10921, 1585, 35365, 5409, 110773, 20950, 359725, 82702, 1192801, 320873, 3998397, 1096384, 13584075, 3417934, 45973713, 10657777, 157515581, 33447019, 543663919, 111463220
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 13 2013

Keywords

Examples

			a(5) = 3: 5 = RMS(5) = RMS(1,7) = RMS(1,5,7);
a(7) = 3: 7 = RMS(7) = RMS(1,5,11) = RMS(1,5,7,11);
a(9) = 9: 9 = RMS(9) = RMS(5,7,13) = RMS(5,7,9,13) = RMS(3,5,11,13) = RMS(3,5,9,11,13) = RMS(1,3,7,11,15) = RMS(1,3,7,9,11,15) = RMS(1,3,5,17) = RMS(1,3,5,9,17);
a(10) = 1: 10 = RMS(1,3,5,7,9,11,15,17);
a(11) = 19: 11 = RMS(11) = RMS(3,9,13,15) = RMS(3,9,11,13,15) = RMS(5,7,17) = RMS(5,7,11,17) = RMS(1,5,13,17) = RMS(1,5,11,13,17) = RMS(1,3,9,15,17) = RMS(1,3,9,11,15,17) = RMS(3,5,7,9,13,15,17) = RMS(3,5,7,9,11,13,15,17) = RMS(1,5,7,13,19) = RMS(1,5,7,11,13,19) = RMS(1,3,7,9,15,19) = RMS(1,3,7,9,11,15,19) = RMS(3,5,7,9,21) = RMS(3,5,7,9,11,21) = RMS(1,3,5,9,13,21) = RMS(1,3,5,9,11,13,21);
a(12) = 2: 12 = RMS(1,5,7,9,11,15,17,19) = RMS(1,3,5,7,9,13,17,23).

Links

Crossrefs

Cf. A000196, A005408, A163974, A164283.

Programs

  • Haskell
    a211868 n = f a005408_list 1 nn 0 where
   f (o:os) l nl xx
     | yy > nl   = 0
     | yy < nl   = f os (l + 1) (nl + nn) yy + f os 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 + o * o
   nn = n ^ 2

Extensions

a(37)-a(40) from Alois P. Heinz, Feb 25 2013
a(41)-a(42) from Alois P. Heinz, May 03 2015

A339556 Number of subsets of the first n primes whose elements have a prime root-mean-square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 16, 19, 30, 41, 54, 69, 106, 177, 272, 397, 686, 1299, 2416, 4225, 7196, 11701, 20352, 36305, 70134, 132721, 248722, 473391, 894318, 1674923, 3054022, 5452067, 9626552, 16696543, 29086462, 51830095, 96887612, 192393735, 397875694
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 08 2020

Keywords

Examples

			a(7) = 11 subsets: {2}, {3}, {5}, {7}, {11}, {13}, {17}, {7, 17}, {5, 7, 17}, {7, 13, 17} and {5, 7, 11, 17}.

Links

Crossrefs

Cf. A000040, A071810, A127542, A163974, A309160, A339454, A339485.

Extensions

a(10)-a(40) from Alois P. Heinz, Dec 08 2020
Showing 1-4 of 4 results.

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