A014285 -id:A014285 - 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

A114256 Prime numbers in A014285.

Original entry on oeis.org

2, 23, 42043, 378761, 462109, 667127, 1116851, 1625461, 1908787, 2637043, 2711399, 3019763, 4394603, 5405447, 7418599, 8682757, 10832561, 12652489, 13528079, 14214661, 15167443, 16413641, 23086711, 27209249, 29062339

Offset: 1

Zak Seidov, Nov 18 2005

nonn

Cf. A014285 (Sum i*prime(i); i=1..n), A114257 (numbers n such that A014285(n) is prime).

lista(nn) = for(n=1, nn, if (isprime(p=sum(k=1, n, k*prime(k))), print1(p, ", "))); \\ Michel Marcus, May 25 2018

A114257 Numbers n such that A014285(n) is prime.

Original entry on oeis.org

1, 3, 32, 63, 67, 75, 88, 99, 104, 115, 116, 120, 135, 144, 159, 167, 179, 188, 192, 195, 199, 204, 227, 239, 244, 260, 280, 283, 304, 323, 335, 348, 351, 379, 380, 384, 396, 412, 424, 436, 443, 503, 508, 523, 579, 632, 648, 651, 695, 723, 724, 736, 743, 752

Offset: 1

Zak Seidov, Nov 18 2005

nonn

Cf. A014285 (Sum i*prime(i); i=1..n), A114256 (Prime numbers in A014285).

isok(n) = isprime(sum(i=1, n, i*prime(i))); \\ Michel Marcus, May 25 2018

A062020 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i) - prime(j)).

Original entry on oeis.org

0, 1, 6, 17, 44, 81, 142, 217, 324, 485, 666, 913, 1208, 1529, 1906, 2373, 2936, 3533, 4238, 5019, 5840, 6787, 7822, 8995, 10360, 11825, 13342, 14967, 16648, 18445, 20662, 23003, 25536, 28135, 31074, 34083, 37308, 40755, 44354, 48187, 52260

Offset: 1

Amarnath Murthy, Jun 02 2001

nonn

			a(3) = (5-2) + (5-3) + (3-2) = 6.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A007504, A014285, A062021, A062022.

a[n_]:= a[n]= If[n<3, (n-1), 2*a[n-1] -a[n-2] +(n-1)*(Prime[n] -Prime[n-1])];
Table[a[n], {n, 50}] (* G. C. Greubel, May 04 2022 *)

@CachedFunction
def a(n): # A062020
    if (n<3): return (n-1)
    else: return 2*a(n-1) - a(n-2) + (n-1)*(nth_prime(n) - nth_prime(n-1))
[a(n) for n in (1..50)] # G. C. Greubel, May 04 2022

a(n) = a(n-1) + n*prime(n) - Sum_{i = 1..n} prime(i), with a(0) = 0.
a(n) = 2*a(n-1) - a(n-2) + (n-1)*(prime(n) - prime(n-1)), with a(1) = 0, a(2) = 1.
a(n) = Sum_{j=1..n} (2*j - (n+1))*prime(j) = 2*A014285(n) - (n+1)*A007504(n). - G. C. Greubel, May 04 2022

More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001

Name edited by G. C. Greubel, May 04 2022

A143121 Triangle read by rows, T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n.

Original entry on oeis.org

2, 5, 3, 10, 8, 5, 17, 15, 12, 7, 28, 26, 23, 18, 11, 41, 39, 36, 31, 24, 13, 58, 56, 53, 48, 41, 30, 17, 77, 75, 72, 67, 60, 49, 36, 19, 100, 98, 95, 90, 83, 72, 59, 42, 23, 129, 127, 124, 119, 112, 101, 88, 71, 52, 29, 160, 158, 155, 150, 143, 132, 119, 102, 83, 60, 31

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jul 26 2008

Left border = A007504, sum of first n primes: (2, 5, 10, 27, 28, 41, ...).
Right border = primes = A000040.
Row sums = A014285: (2, 8, 23, 51, 106, 184, ...).

			First few rows of the triangle are:
   2;
   5,  3;
  10,  8,  5;
  17, 15, 12,  7;
  28, 26, 23, 18, 11;
  41, 39, 36, 31, 24, 13;
  58, 56, 53, 48, 41, 30, 17;
  ...
T(5,3) = 23 = prime(3) + prime(4) + prime(5) = (5 + 7 + 11).

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A000040, A007504, A014285, A000012.

Cf. A194939 (rows reversed).

[[(&+[NthPrime(j): j in [k..n]]): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Oct 15 2018

a:=proc(n,k) add(ithprime(j),j=k..n) end: seq(seq(a(n,k),k=1..n),n=1..11); # Muniru A Asiru, Oct 15 2018

a[n_, k_] := a[n, k] = Plus@@Prime[Range[n - k + 1, n]]; Column[Table[a[n, k], {n, 15}, {k, n, 1, -1}], Center] (* Alonso del Arte, Jul 25 2011 *)
Table[Sum[Prime[j], {j, k, n}], {n, 1, 15}, {k, 1, n}]//Flatten (* G. C. Greubel, Oct 15 2018 *)

a(n,k)=my(s);forprime(p=prime(k),prime(n),s+=p);s \\ Charles R Greathouse IV, Jul 25 2011

T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n, primes = A000040.

Equals A000012 * (A000040 * 0^(n-k)) * A000012.

Corrected by Hanke Bremer, Nov 28 2008

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

Andres Cicuttin, Mar 12 2019

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

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

Cf. A272206, A007504, A014285.

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

a[n_]:=Sum[i*Prime[i],{i,1,n}]/Sum[Prime[i],{i,1,n}];
Table[a[n]//Numerator,{n,1,40}]

a(n) = numerator(sum(i=1, n, i*prime(i))/sum(i=1, n, prime(i))); \\ Michel Marcus, Mar 15 2019

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

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

A316322 Sum of piles of first n primes: a(n) = Sum(prime(i)(2i-1): 1<=i<=n).

Original entry on oeis.org

2, 11, 36, 85, 184, 327, 548, 833, 1224, 1775, 2426, 3277, 4302, 5463, 6826, 8469, 10416, 12551, 15030, 17799, 20792, 24189, 27924, 32107, 36860, 42011, 47470, 53355, 59568, 66235, 73982, 82235, 91140, 100453, 110734, 121455, 132916, 145141, 158000, 171667, 186166, 201189, 217424, 234215, 251748

Offset: 1

N. J. A. Sloane, Jul 03 2018, based on Reinhard Zumkeller's A083215

nonn

			............................................ 7
........................... 5 ............ 7 5 7
............ 3 .......... 5 3 5 ........ 7 5 3 5 7
2 ........ 3 2 3 ...... 5 3 2 3 5 .... 7 5 3 2 3 5 7
a(1)=2 ... a(2)=11 .... a(3)=36 ...... a(4)=85.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A083215, A014285, A007504, A167214, A023662, A062020.

seq(add((2*i-1)*ithprime(i),i=1..n), n=1..80); # Ridouane Oudra, Feb 19 2025

nxt[{n_, a_}] := {n + 1, a + Prime[n + 1] (2 n + 1)}; NestList[nxt,{1,2},50][[All,2]] (* Harvey P. Dale, Jul 05 2018 *)

a(n) = sum(i=1, n, prime(i)*(2*i-1)); \\ Michel Marcus, Jan 22 2022

From Ridouane Oudra, Feb 19 2025: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(prime(i), prime(j)).
a(n) = 2*A014285(n) - A007504(n).
a(n) = 2*A167214(n) - A023662(n).
a(n) = A167214(n) + A062020(n). (End)

A194939 Table T read by rows, where T(n, k) is the sum of the largest k primes up to and including prime(n), for 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 5, 8, 10, 7, 12, 15, 17, 11, 18, 23, 26, 28, 13, 24, 31, 36, 39, 41, 17, 30, 41, 48, 53, 56, 58, 19, 36, 49, 60, 67, 72, 75, 77, 23, 42, 59, 72, 83, 90, 95, 98, 100, 29, 52, 71, 88, 101, 112, 119, 124, 127, 129, 31, 60, 83, 102, 119, 132, 143, 150, 155, 158, 160

Offset: 1

Alonso del Arte, Sep 07 2011

From the left, the second column gives the sums of two consecutive primes, the third column gives the sums of three consecutive primes, etc. Thus, from the right, the rightmost column gives the running sum of all prime numbers up to that row.
This triangle is the mirror image of A143121: left border are the primes (right border in the other one) while the right border is the sum of the first n primes (A007504, left border in the other one). Row sums are given by A014285, just like the other triangle.
On odd numbered rows, the central entry is exactly the same as the corresponding position in A143121: T(n, (n + 1)/2) = A143121(n, (n + 1)/2). The rest of the row is of course the reverse.

			First few rows of triangle are:
2
3,   5
5,   8, 10
7,  12, 15, 17
11, 18, 23, 26, 28
...
T(5, 2) = 18 because the sum of the fourth and fifth primes (two consecutive primes) is 7 + 11 = 18.
T(5, 3) = 23 because the sum of the third, fourth and fifth primes (three consecutive primes) is 5 + 7 + 11 = 23.

Michel Marcus, Table of n, a(n) for n = 1..5050 (first 100 rows)

Cf. A143121 (rows reversed), A014285 (row sums).
Cf. A000040 (column k=1), A007504 (main diagonal).
Cf. A067377.

a[n_, k_] := a[n, k] = Plus@@Prime[Range[n - k + 1, n]]; Column[Table[a[n, k], {n, 15}, {k, n}], Center]

T(n, k) = Sum_{i = n-k+1..n} prime(i), where prime(i) is the i-th prime number.

More terms from Michel Marcus, Aug 31 2020

New name from David A. Corneth, Aug 31 2020

A272206 Rounded barycenter of first n primes defined as a(n) = round(sum_{i=1..n}(i*prime(i)) / sum_{i=1..n}prime(i)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45

Offset: 1

Andres Cicuttin, May 21 2016

nonn

Cf. A007504, A014285.

a[n_] := Sum[i*Prime[i], {i, 1, n}]/Sum[Prime[i], {i, 1, n}];
Table[a[n] // Round, {n, 1, 64}];

a(n) = round(sum(k=1, n, k*prime(k))/ sum(k=1, n, prime(k))); \\ Michel Marcus, May 22 2016

a(n) = round(sum_{i=1..n}(i*prime(i)) / sum_{i=1..n}prime(i)).

a(n) = round(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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A114256 Prime numbers in A014285.

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A114257 Numbers n such that A014285(n) is prime.

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A062020 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i) - prime(j)).

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A143121 Triangle read by rows, T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n.

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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)).

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A316322 Sum of piles of first n primes: a(n) = Sum(prime(i)(2i-1): 1<=i<=n).