| Displaying 1-3 of 3 results found.
page
1
Rounded variance of the first n primes, for n > 1.
+10
6
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
FORMULA
a(n) = round(Sum_{i=1..n} (prime(i) - Sum_{j=1..n} prime(j)/n)^2/(n - 1)), n > 1.
MATHEMATICA
Table[Round[Variance[Prime[Range[j]]]], {j, 2, 50}]
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)).
+10
4
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
COMMENTS
It appears that lim_{n->infinity} (1/n)*( A014285(n)/ A007504(n)) = k, where k is a constant around 2/3.
FORMULA
a(n) = numerator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).
MAPLE
N:= 100: # for a(1)..a(N)
Primes:= map(ithprime, [$1..N]):
S1:= ListTools:-PartialSums(Primes):
S2:= ListTools:-PartialSums(zip(`*`, Primes, [$1..N])):
MATHEMATICA
a[n_]:=Sum[i*Prime[i], {i, 1, n}]/Sum[Prime[i], {i, 1, n}];
Table[a[n]//Numerator, {n, 1, 40}]
PROG
(PARI) a(n) = numerator(sum(i=1, n, i*prime(i))/sum(i=1, n, prime(i))); \\ Michel Marcus, Mar 15 2019
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)).
+10
2
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
COMMENTS
It appears that lim_{n->infinity} (1/n)*( A014285(n)/ A007504(n)) = k, where k is a constant around 2/3.
FORMULA
a(n) = denominator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).
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:
MATHEMATICA
a[n_]:=Sum[i*Prime[i], {i, 1, n}]/Sum[Prime[i], {i, 1, n}];
Table[a[n]//Denominator, {n, 1, 48}]
PROG
(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
Search completed in 0.004 seconds
|