A306834 -id:A306834 - cp's OEIS frontend

A309036 a(n) = gcd(A007504(n), A014285(n)).

2, 1, 1, 17, 2, 1, 1, 7, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 3, 8, 1, 1, 1, 20, 43, 1, 3, 4, 1, 1, 1, 28, 1, 1, 3, 2, 1, 1, 1, 2, 3, 107, 1, 4, 1, 1, 1, 2, 7, 1, 1, 10, 3, 1, 1, 30, 1, 1, 1, 2, 5, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 142, 1, 1, 3, 4, 1, 1, 11, 2, 1, 1, 1, 10

Offset: 1

Robert Israel, Jul 08 2019

nonn

a(n) is even if n == 1 (mod 4).

			a(4) = gcd(2+3+5+7, 1*2+2*3+3*5+4*7) = gcd(17,51) = 17.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007504, A014285, A306834, A307414 (a(n)=1), A307716, A309055, A309056.

p:=PrimesUpTo(1000);[Gcd(&+[p[j]:j in [1..m]],&+[j*p[j]:j in [1..m]]): m in [1..90]]; // Marius A. Burtea, Jul 09 2019

S1:= 0: S2:= 0:
for n from 1 to 100 do
  p:= ithprime(n);
  S1:= S1 + p;
  S2:= S2 + n*p;
  A[n]:= igcd(S1,S2);
od:
seq(A[i],i=1..100);

GCD @@ # & /@ Rest@ Nest[Append[#1, {#1[[-1, 1]] + #3, #1[[-1, -1]] + #2 #3}] & @@ {#1, #2, Prime@ #2} & @@ {#, Length@ #} &, {{0, 0}}, 89] (* Michael De Vlieger, Jul 08 2019 *)

a(n) = gcd(sum(k=1, n, prime(k)), sum(k=1, n, k*prime(k))); \\ Michel Marcus, Jul 09 2019

a(n) = A007504(n)/A307716(n) = A014285(n)/A306834(n).

A307414 Numbers k such that A014285(k) and A007504(k) are coprime.

Original entry on oeis.org

2, 3, 6, 7, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, 27, 30, 31, 32, 34, 35, 38, 39, 40, 44, 46, 47, 48, 51, 52, 55, 56, 58, 59, 60, 63, 64, 66, 67, 70, 71, 72, 74, 75, 76, 78, 79, 82, 83, 86, 87, 88, 91, 92, 94, 95, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 115, 116, 118, 119, 120, 122

Offset: 1

Robert Israel, Apr 07 2019

nonn

Numbers k such that A306834(k) = A014285(k).
No terms == 1 (mod 4).
Numbers k such that A309036(k)=1. - Robert Israel, Jul 09 2019

			a(3) = 6 is a term because A007504(6) = 41 and A014285(6) = 184 are coprime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007504, A014285, A306834, A309036.

N:= 1000: # for terms <= N
Primes:= map(ithprime, [$1..N]):
S1:= ListTools:-PartialSums(Primes):
S2:= ListTools:-PartialSums(zip(`*`,Primes, [$1..N])):
select(t -> igcd(S1[t],S2[t])=1, [$1..N]);

okQ[n_] := With[{pp = Prime[Range[n]]}, CoprimeQ[Total[pp], Total[pp.Range[n]]]];
Select[Range[200], okQ] (* Jean-François Alcover, Dec 05 2023 *)

isok(k) = my(vp=primes(k)); gcd(sum(i=1, k, vp[i]), sum(i=1, k, i*vp[i])) == 1; \\ Michel Marcus, Apr 07 2019

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

Andres Cicuttin, Apr 25 2019

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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

a[n_]:=Sum[i*Prime[i], {i, 1, n}]/Sum[Prime[i], {i, 1, n}];
Table[a[n]//Denominator, {n, 1, 48}]

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

a(n) = denominator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).

a(n) = denominator(A014285(n)/A007504(n)).

cp's OEIS Frontend

A309036 a(n) = gcd(A007504(n), A014285(n)).

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A307414 Numbers k such that A014285(k) and A007504(k) are coprime.

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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)).

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