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 1-3 of 3 results.

A273462 Rounded variance of the first n primes, for n > 1.

Original entry on oeis.org

0, 2, 5, 13, 19, 31, 41, 56, 81, 103, 136, 171, 201, 235, 280, 335, 384, 444, 505, 560, 626, 693, 772, 869, 966, 1055, 1145, 1229, 1314, 1447, 1578, 1719, 1849, 2008, 2156, 2313, 2479, 2644, 2818, 3000, 3171, 3372, 3560, 3748, 3925, 4142, 4398, 4651, 4890
Offset: 2

Views

Author

Andres Cicuttin, May 23 2016

Keywords

Links

Crossrefs

Cf. A075465, A075471, A272206.
Mean and variance of primes: A301273/A301274, A301275/A301276, A301277, A273462.

Programs

  • Mathematica
    Table[Round[Variance[Prime[Range[j]]]], {j, 2, 50}]
  • Sage
    round(variance(primes_first_n(n))) # Danny Rorabaugh, May 25 2016

Formula

a(n) = round(Sum_{i=1..n} (prime(i) - Sum_{j=1..n} prime(j)/n)^2/(n - 1)), n > 1.

A306834 Numerator of the barycenter of first n primes defined as a(n) = numerator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).

Original entry on oeis.org

1, 8, 23, 3, 53, 184, 303, 65, 331, 952, 1293, 1737, 1135, 2872, 3577, 1475, 1357, 6526, 7799, 3073, 1344, 12490, 14399, 16535, 948, 502, 24367, 9121, 7631, 33914, 37851, 42043, 1663, 51290, 56505, 20647, 33875, 73944, 80457, 87377, 47358, 34106, 1033, 119023, 31972, 137042, 146959, 157663
Offset: 1

Views

Author

Andres Cicuttin, Mar 12 2019

Keywords

Comments

It appears that lim_{n->infinity} (1/n)*(A014285(n)/A007504(n)) = k, where k is a constant around 2/3.

Links

Crossrefs

Cf. A272206, A007504, A014285.

Programs

  • Maple
    N:= 100: # for a(1)..a(N)
Primes:= map(ithprime, [$1..N]):
S1:= ListTools:-PartialSums(Primes):
S2:= ListTools:-PartialSums(zip(`*`,Primes, [$1..N])):
map(numer,zip(`/`,S2,S1)); # Robert Israel, Apr 07 2019
  • Mathematica
    a[n_]:=Sum[i*Prime[i],{i,1,n}]/Sum[Prime[i],{i,1,n}];
Table[a[n]//Numerator,{n,1,40}]
  • PARI
    a(n) = numerator(sum(i=1, n, i*prime(i))/sum(i=1, n, prime(i))); \\ Michel Marcus, Mar 15 2019

Formula

a(n) = numerator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).
a(n) = numerator(A014285(n)/A007504(n)).

A307716 Denominator of the barycenter of first n primes defined as a(n) = denominator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).

Original entry on oeis.org

1, 5, 10, 1, 14, 41, 58, 11, 50, 129, 160, 197, 119, 281, 328, 127, 110, 501, 568, 213, 89, 791, 874, 963, 53, 27, 1264, 457, 370, 1593, 1720, 1851, 71, 2127, 2276, 809, 1292, 2747, 2914, 3087, 1633, 1149, 34, 3831, 1007, 4227, 4438, 4661
Offset: 1

Views

Author

Andres Cicuttin, Apr 25 2019

Keywords

Comments

It appears that lim_{n->infinity} (1/n)*(A014285(n)/A007504(n)) = k, where k is a constant around 2/3.
a(n) = A007504(n) if and only if n is in A307414. - Robert Israel, Jul 08 2019

Links

Crossrefs

Cf. A306834 (numerators), A272206, A007504, A014285, A307414.

Programs

  • Maple
    S1:= 0:S2:= 0:
for n from 1 to 100 do
  p:= ithprime(n);
  S1:= S1 + p;
  S2:= S2 + n*p;
  A[n]:= denom(S2/S1)
od:
seq(A[i],i=1..100); # Robert Israel, Jul 08 2019
  • Mathematica
    a[n_]:=Sum[i*Prime[i], {i, 1, n}]/Sum[Prime[i], {i, 1, n}];
Table[a[n]//Denominator, {n, 1, 48}]
  • PARI
    a(n) = my(vp=primes(n)); denominator(sum(i=1, n, i*vp[i])/sum(i=1, n, vp[i])) \\ Michel Marcus, Apr 25 2019

Formula

a(n) = denominator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).
a(n) = denominator(A014285(n)/A007504(n)).
Showing 1-3 of 3 results.

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