A014148 -id:A014148 - cp's OEIS frontend

A023662 Convolution of odd numbers and primes.

2, 9, 24, 51, 96, 165, 264, 399, 576, 805, 1094, 1451, 1886, 2405, 3014, 3723, 4544, 5485, 6554, 7761, 9112, 10615, 12280, 14117, 16140, 18361, 20786, 23421, 26272, 29345, 32658, 36229, 40068, 44183, 48586, 53289, 58300, 63631, 69292

Offset: 1

Clark Kimberling

nonn

Cf. A000040, A005408, A061802 (first differences).

Cf. A167214, A062020, A316322, A014148, A007504.

A023662 := proc(n)
    add( ithprime(n-i)*(2*i+1),i=0..n-1) ;
end proc: # R. J. Mathar, Nov 29 2015

Table[Sum[Prime[n - k + 1] (2 k - 1), {k, n}], {n, 39}] (* Michael De Vlieger, Nov 29 2015 *)

a(n) = sum(i=1, n, prime(n-i+1)*(2*i-1)); \\ Michel Marcus, Sep 30 2013

a(n) = Sum_{i=0..n-1} A000040(n-i)*A005408(i). - R. J. Mathar, Nov 29 2015
a(n) = Sum_{i=0..n-1} A061802(i). - Marco Zárate, Jun 09 2024
From Ridouane Oudra, Feb 19 2025: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} min(prime(i), prime(j)).
a(n) = A167214(n) - A062020(n).
a(n) = 2*A167214(n) - A316322(n).
a(n) = A014148(n) + A014148(n-1).
a(n) = A007504(n) + 2*A014148(n-1). (End)

A023538 Convolution of natural numbers with (1, p(1), p(2), ... ), where p(k) is the k-th prime.

Original entry on oeis.org

1, 4, 10, 21, 39, 68, 110, 169, 247, 348, 478, 639, 837, 1076, 1358, 1687, 2069, 2510, 3012, 3581, 4221, 4934, 5726, 6601, 7565, 8626, 9788, 11053, 12425, 13906, 15500, 17221, 19073, 21062, 23190, 25467, 27895, 30480, 33228, 36143, 39231, 42498, 45946, 49585

Offset: 1

Clark Kimberling

nonn

F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.

Cf. A007504, A014148, A014150, A014284, A023538, A178138.

Nest[Accumulate,Join[{1},Select[Range@200,PrimeQ]],2] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)

a(n) = Sum_{k<=n} [(A158611(k+1)) * (A000027(n-k+1))] = Sum_{k<=n} [(A008578(k)) * (A000027(n-k+1))]. [Jaroslav Krizek, Aug 05 2009; Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010]

A157493 Apply partial sum operator thrice to sequence of squares of the first n primes.

Original entry on oeis.org

4, 21, 76, 218, 568, 1295, 2688, 5108, 9084, 15457, 25188, 39646, 60512, 89635, 129224, 182088, 251708, 341805, 456868, 601938, 782344, 1004327, 1274776, 1601612, 1994244, 2462873, 3018108, 3671398, 4434624, 5320555, 6345320

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Cf. A001223, A014148, A014150, A061789, A069482, A129701, A157492.

s0=s1=s2=0; Table[s0+=Prime[n]^2; s1+=s0; s2+=s1, {n,100}]
Nest[Accumulate,Prime[Range[40]]^2,3]  (* Harvey P. Dale, Jan 27 2011 *)

A330087 Permanent of a square matrix M(n) whose general element M_{i,j} is defined by i*prime(j).

Original entry on oeis.org

1, 2, 24, 1080, 120960, 33264000, 15567552000, 12967770816000, 15768809312256000, 29377291748732928000, 85194146071325491200000, 319563241913541917491200000, 1702632952915351336393113600000, 11797543730750469409867884134400000, 99429698562764956186366527484723200000

Offset: 0

Stefano Spezia, Dec 01 2019

nonn

det(M(0)) = 1, det(M(1)) = 2 and det(M(n)) = 0 for n > 1.
The trace of the matrix M(n) is A014285(n).
The antitrace of the matrix M(n) is A014148(n).
The antidiagonal of the matrix M(n) is the n-th row of the triangle A309131.

			For n = 1 the matrix M(1) is
  2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
  2, 3
  4, 6
with permanent a(2) = 24.
For n = 3 the matrix M(3) is
  2,  3,  5
  4,  6, 10
  6,  9, 15
with permanent a(3) = 1080.

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A000040, A014148, A014285, A033286, A309131.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i*ithprime(j)))):
seq(a(n), n=0..14);  # Alois P. Heinz, Dec 04 2019

M[i_, j_, n_] := i*Prime[j]; a[n_] := If[n==0,1,Permanent[Table[M[i, j, n], {i, n}, {j, n}]]]; Array[a, 14, 0]

a(n) = matpermanent(matrix(n, n, i, j, i*prime(j))); \\ Michel Marcus, Dec 04 2019

a(0) = 1 prepended by Michel Marcus, Dec 04 2019

A356178 Numbers k such that both Sum_{i=1..k} iprime(i) and Sum_{i=1..k} (k+1-i)prime(i) are prime.

Original entry on oeis.org

1, 3, 199, 351, 1583, 1955, 2579, 2627, 3251, 3407, 3503, 5311, 6359, 6819, 7295, 7547, 8791, 9663, 10143, 10591, 11499, 11579, 12199, 12443, 14527, 15563, 15583, 16051, 16543, 16655, 18047, 18319, 20691, 20847, 23979, 24079, 24575, 25667, 26491, 28235, 30395, 30627, 32235, 32259, 33091, 33287, 33527

Offset: 1

J. M. Bergot and Robert Israel, Jul 28 2022

nonn

Numbers k such that A014148(k) and A014285(k) are both prime.

a(n) == 3 (mod 4) for n > 1.

			a(2) = 3 is a term because Sum_{i=1..3} i*prime(i) = 1*2 + 2*3 + 3*5 = 23 and Sum_{i=1..3} (4-i)*prime(i) = 3*2 + 2*3 + 1*5 = 17 are prime.

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

Cf. A014148, A014285.

S1:= 2: S2:= 2: S3:= 2*S2-S1: R:= 1: count:= 1: p:= 2:
for n from 2 to 40000 do
  p:= nextprime(p);
  S1:= S1 + n*p;
  S2:= S2 + p;
  if n mod 4 = 3 and isprime(S1) then
    S3:= (n+1)*S2 - S1;
    if isprime(S3) then
       count:= count+1; R:= R, n;
    fi
  fi;
od:
R;

r = Range[35000]; p = Prime[r]; Intersection[Position[Accumulate[r*p], ?PrimeQ], Position[Accumulate[Accumulate[p]], ?PrimeQ]] // Flatten (* Amiram Eldar, Jul 28 2022 *)

isok(k) = my(vp=primes(k)); isprime(sum(i=1, k, i*vp[i])) && isprime(sum(i=1, k, (k+1-i)*vp[i])); \\ Michel Marcus, Jul 29 2022

A157494 Primes in A014150.

Original entry on oeis.org

2, 1429, 32869, 3189059, 5157791, 62701339, 139181423, 296686879, 522304883, 5070516751, 6276844867, 7098350179, 8983996079, 9331926623, 21211375343, 31177858939, 34861039007, 38865340309, 39918757589, 62858815181

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Cf. A001223, A014148, A014150, A061789, A069482, A122382, A129701, A157492, A157493

s0=s1=s2=0;lst={};Do[p=Prime[n];s0+=p;s1+=s0;s2+=s1;If[PrimeQ[s2],AppendTo[lst,s2]],{n,7!}];lst
Select[Nest[Accumulate[#]&,Prime[Range[700]],3],PrimeQ] (* Harvey P. Dale, Jul 11 2025 *)

A293210 a(n) = [x^n] (1/(1 - x)^n)Sum_{k>=1} prime(k)x^k.

Original entry on oeis.org

0, 2, 7, 26, 97, 366, 1388, 5288, 20225, 77618, 298766, 1153018, 4460072, 17287558, 67129566, 261095420, 1016994627, 3966529870, 15488964428, 60549061804, 236932924494, 927984726826, 3637661249946, 14270586372348, 56024073085546, 220089137078792, 865154426179408, 3402841810234762, 13391422390407194

Offset: 0

Ilya Gutkovskiy, Oct 02 2017

nonn

Diagonal of A254858.
Cf. A000040, A007504, A014148, A014150, A178138, A254784.
CF. A007442, A082594.

Table[SeriesCoefficient[1/(1 - x)^n Sum[Prime[k] x^k, {k, 1, n}], {x, 0, n}], {n, 0, 28}]

a(n) = A254858(n,n).

cp's OEIS Frontend

A023662 Convolution of odd numbers and primes.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A023538 Convolution of natural numbers with (1, p(1), p(2), ... ), where p(k) is the k-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A157493 Apply partial sum operator thrice to sequence of squares of the first n primes.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A330087 Permanent of a square matrix M(n) whose general element M_{i,j} is defined by i*prime(j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A356178 Numbers k such that both Sum_{i=1..k} i*prime(i) and Sum_{i=1..k} (k+1-i)*prime(i) are prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A157494 Primes in A014150.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A293210 a(n) = [x^n] (1/(1 - x)^n)*Sum_{k>=1} prime(k)*x^k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A356178 Numbers k such that both Sum_{i=1..k} iprime(i) and Sum_{i=1..k} (k+1-i)prime(i) are prime.

A293210 a(n) = [x^n] (1/(1 - x)^n)Sum_{k>=1} prime(k)x^k.