A033286 -id:A033286 - cp's OEIS frontend

A340781 a(n) = (n - 1)*prime(n + 1) mod prime(n).

0, 2, 4, 5, 8, 7, 12, 9, 2, 18, 29, 7, 24, 9, 37, 37, 32, 41, 5, 38, 47, 5, 49, 6, 96, 50, 1, 54, 3, 67, 120, 55, 64, 52, 68, 59, 59, 148, 61, 61, 80, 48, 84, 172, 88, 142, 130, 188, 96, 196, 67, 102, 38, 67, 67, 67, 112, 71, 232, 118, 34, 268, 248, 126, 256, 276

Offset: 1

Stefano Spezia, Jan 21 2021

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000040, A033286, A306192, A340128, A340649.

Table[Mod[(n-1)Prime[n+1],Prime[n]],{n,66}]

a(n) = ((n-1)*prime(n+1)) % prime(n); \\ Michel Marcus, Jan 21 2021

a(n) = A306192(n) mod A000040(n).

A351369 a(n) = Sum_{p|n, p prime} p * prime(p).

Original entry on oeis.org

0, 6, 15, 6, 55, 21, 119, 6, 15, 61, 341, 21, 533, 125, 70, 6, 1003, 21, 1273, 61, 134, 347, 1909, 21, 55, 539, 15, 125, 3161, 76, 3937, 6, 356, 1009, 174, 21, 5809, 1279, 548, 61, 7339, 140, 8213, 347, 70, 1915, 9917, 21, 119, 61, 1018, 539, 12773, 21, 396, 125, 1288, 3167

Offset: 1

Wesley Ivan Hurt, Feb 08 2022

nonn

Inverse Möbius transform of n * prime(n) * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

			a(6) = 21; a(6) = Sum_{p|6} p * prime(p) = 2*3 + 3*5 = 21.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A010051, A033286, A061397, A351368.

Join[{0},Table[Total[# Prime[#]&/@FactorInteger[n][[;;,1]]],{n,2,80}]] (* Harvey P. Dale, Jan 28 2024 *)

a(n) = Sum_{d|n} d * prime(d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025

a(p^k) = p * prime(p) for p prime and k>=1. - Wesley Ivan Hurt, Jul 16 2025

A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

Original entry on oeis.org

1, 2, 10, 74, 676, 7592, 97024, 1416200, 23015248, 412777952, 8090869984, 171435904928, 3908548404160, 95264270043776, 2470715015425024, 67913132377486208, 1971038886452490496, 60212661838223997440, 1930529043247940342272, 64801071784954698480128

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A007446, A033286, A353162, A353166.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j)*j*binomial(n, j)*j!, j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 28 2022

a[0] = 1; a[n_] := a[n] = (n-1)! Sum[k Prime[k] a[n-k]/(n-k)!, {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 28 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, prime(k)*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*prime(k)*a(n-k)/(n-k)!));

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A033286(k) * a(n-k)/(n-k)!.

A026102 a(n) = T(2n-1,n), where T is the array in A026098.

Original entry on oeis.org

1, 6, 15, 28, 55, 78, 119, 152, 230, 290, 372, 444, 574, 645, 752, 901, 1062, 1159, 1273, 1491, 1606, 1817, 1992, 2314, 2522, 2828, 2987, 3210, 3270, 3616, 4191, 4323, 4658, 5004, 5364, 5738, 6123, 6357, 6847, 7266, 7697, 7964, 8595, 8685, 9259, 9552, 10339, 10927, 11577, 11908

Offset: 1

Clark Kimberling

nonn

Cf. A001844, A026098.

Cf. A033286. [R. J. Mathar, Oct 22 2008]

T[1, 1] = 1; T[2, 1] = 3; T[2, 2] = 2; T[n_, 1] := Prime[n];
T[n_, k_] := T[n, k] = Module[{m, mp, jtt}, For[m = 1, True, m++, mp = m Prime[n+1-k]; jtt = Join[Table[T[i, j], {i, 1, n-1}, {j, 1, i}] // Flatten, Table[T[n, j], {j, 1, k-1}]]; If[FreeQ[jtt, mp], Return[mp]]]];
a[n_] := T[2n-1, n]; (* Jean-François Alcover, Sep 05 2019 *)

lista(nn) = {my(all = []); for (n=1, nn, my(row = getrow(n, all)); if (n % 2, print1(row[(n+1)/2], ", ")); all = Set(concat(all, row)););} \\ uses getrow from A026098; Michel Marcus, Sep 04 2019

a(n) = A026098(A001844(n)). - Sean A. Irvine, Sep 16 2019

Corrected and extended by Michel Marcus, Sep 04 2019

A106033 a(n) is the least number k such that n*prime(n)+k is a perfect square.

Original entry on oeis.org

2, 3, 1, 8, 9, 3, 2, 17, 18, 34, 20, 40, 43, 23, 24, 52, 21, 58, 23, 24, 67, 26, 27, 73, 75, 78, 28, 29, 88, 91, 32, 33, 103, 35, 114, 40, 120, 47, 48, 136, 57, 142, 68, 157, 160, 62, 83, 112, 113, 214, 217, 116, 223, 135, 26, 156, 43, 158, 41, 40, 161, 59, 259, 260, 104, 103

Offset: 1

Zak Seidov, May 05 2005

			a(10)=34 because 10*prime(10)+34 = 10*29+34 = 324 = 18^2.

Cf. A033286 (n*prime(n)), A080883 (distance of n to next square).

a[n_]:=(Floor[Sqrt[n*Prime[n]]]+1)^2-n*Prime[n]
lnk[n_]:=With[{c=n Prime[n]},(Floor[Sqrt[c]]+1)^2-c]; Array[lnk,70] (* Harvey P. Dale, Feb 17 2024 *)

a(n) = (floor(sqrt(n*prime(n)))+1)^2 - n*prime(n).

a(n) = A080883(A033286(n)). - Michel Marcus, Mar 29 2020

A161522 prime(n)*( prime(n)-n ).

Original entry on oeis.org

2, 3, 10, 21, 66, 91, 170, 209, 322, 551, 620, 925, 1148, 1247, 1504, 1961, 2478, 2623, 3216, 3621, 3796, 4503, 4980, 5785, 6984, 7575, 7828, 8453, 8720, 9379, 12192, 12969, 14248, 14595, 16986, 17365, 18840, 20375, 21376, 23009, 24702, 25159, 28268, 28757

Offset: 1

Juri-Stepan Gerasimov, Jun 12 2009

nonn

Cf. A000040, A014689.

f[n_]:=Module[{pn=Prime[n]},pn(pn-n)];Array[f,50] (* Harvey P. Dale, Sep 23 2011 *)

a(n) = A000040(n)*A014689(n) = A001248(n)-A033286(n).

Entries checked by R. J. Mathar, Sep 23 2009

A268467 Smallest prime that is the (sum, k*prime(k),k=m,..n+m-1) for some m, or 0 if no such m exists.

Original entry on oeis.org

2, 43, 23, 0, 1109, 1187, 929, 0, 4973, 1291, 11197, 0, 26099, 15583, 4423, 0, 42139, 10729, 21283, 0, 36899, 27179, 21563, 0, 24359, 33863, 27361, 0, 223423, 51239, 293467, 42043, 67699, 56503, 118361, 0, 80449, 94693, 136739, 0, 127837, 136991, 387913, 0, 304259, 192013, 321721, 0, 339517, 357683

Offset: 1

Zak Seidov, Feb 05 2016

nonn

Smallest prime that is the sum of n consecutive terms of A033286.
Apparently a(n) exists for any odd n.
Values of m = {1, 3, 1, 0, 7, 6, 4, 0, 9, 2, 12, 0, 17, 11, 2, 0, 17, 4, 8, 0, 11, 7, 4, 0, 3, 5, 2, 0, 27, 5, 30, 1, 5, 2, 10, 0, 3, 4, 8, 0, 5, 5, 22, 0, 15, 6, 14, 0, 13, 13, ...}. - Michael De Vlieger, Feb 05 2016

			n=1: m=1 and 1*prime(1) = 1*2 = 2 = a(1),
n=2: m=3 and 3*prime(3)+4*prime(4) = 3*5+4*7 = 43 = a(2),
n=3: m=1 and 1*prime(1)+2*prime(2)+3*prime(3) = 1*2+2*3+3*15 = 23 = a(3),
n=4: no solution => a(4) = 0,
n=5: m=7 and 7*prime(7)+..11*prime(11) = 119+152+207+290+341 = 1109 = a(5).

Cf. A000040, A033286, A105455, A268465, A105454, A152117, A119487.

Table[If[# == 0, 0, Sum[k Prime@ k, {k, #, n + # - 1}]] &@(SelectFirst[Range[10^3], PrimeQ@ Sum[k Prime@ k, {k, #, n + # - 1}] &] /. x_ /; MissingQ@ x -> 0), {n, 50}] (* Michael De Vlieger, Feb 05 2016, Version 10.2 *)

A356868 a(n) = n^2 * prime(n).

Original entry on oeis.org

2, 12, 45, 112, 275, 468, 833, 1216, 1863, 2900, 3751, 5328, 6929, 8428, 10575, 13568, 17051, 19764, 24187, 28400, 32193, 38236, 43907, 51264, 60625, 68276, 75087, 83888, 91669, 101700, 122047, 134144, 149193, 160684, 182525, 195696, 214933, 235372, 254007, 276800, 300899

Offset: 1

Alex Ratushnyak, Sep 01 2022

Cf. A000040, A000290, A004232, A033286, A196421.

a[n_] := n^2 * Prime[n]; Array[a, 40] (* Amiram Eldar, Sep 02 2022 *)

from sympy import prime
def a(n): return n**2 * prime(n)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 01 2022

A369867 a(n) = n * Sum_{p|n, p prime} prime(n/p) / p.

Original entry on oeis.org

0, 2, 2, 6, 2, 21, 2, 28, 15, 61, 2, 106, 2, 125, 70, 152, 2, 285, 2, 318, 134, 347, 2, 596, 55, 539, 207, 630, 2, 1073, 2, 848, 356, 1009, 174, 1542, 2, 1279, 548, 1572, 2, 2213, 2, 1766, 912, 1915, 2, 2984, 119, 2715, 1018, 2654, 2, 3879, 396, 3148, 1288, 3167

Offset: 1

Wesley Ivan Hurt, Feb 03 2024

Cf. A000040, A351368, A369748, A369749, A369750, A369866.

Cf. A003557, A033286, A246655.

Table[n*DivisorSum[n, Prime[n/#]/# &, PrimeQ[#] &], {n, 60}]

a(n) = my(vp=primes([1, n])); n*sum(i=1, #vp, if (!(n % vp[i]), prime(n/vp[i])/vp[i])); \\ Michel Marcus, May 11 2024

From Wesley Ivan Hurt, May 10 2024: (Start)
a(p^k) = p^(k-1) * prime(p^(k-1)) for primes p and k >= 1.
a(A246655(n)) = A033286(A003557(A246655(n))). (End)

cp's OEIS Frontend

A340781 a(n) = (n - 1)*prime(n + 1) mod prime(n).

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A351369 a(n) = Sum_{p|n, p prime} p * prime(p).

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A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

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A026102 a(n) = T(2n-1,n), where T is the array in A026098.

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A106033 a(n) is the least number k such that n*prime(n)+k is a perfect square.

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A161522 prime(n)*( prime(n)-n ).

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A268467 Smallest prime that is the (sum, k*prime(k),k=m,..n+m-1) for some m, or 0 if no such m exists.

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A356868 a(n) = n^2 * prime(n).

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A369867 a(n) = n * Sum_{p|n, p prime} prime(n/p) / p.

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