A086787 -id:A086787 - cp's OEIS frontend

A123373 a(n) = Sum_{i=1..n} (Sum_{j=1..n} prime(i)^prime(j)).

4, 48, 3598, 924780, 287358579128, 339575512147572, 836406636653653232322, 2225332017808171682043720, 21158384827910606570843063431876, 2570789828135881020104992992114519012237948

Offset: 1

Alexander Adamchuk, Oct 12 2006

nonn

p divides a(p-1) for prime p = {2, 3, 5, 23, 29, ...}.
All terms are even.
2^2 divides a(n) for n = {1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 21, 25, 26, 29, 30, 33, 37, 41, 45, 48, 49, 50, 52, ...}.
2^3 divides a(n) for n = {2, 5, 8, 12, 13, 16, 18, 21, 29, 37, 45, 48, 50, 52, 53, 58, 61, 64, 69, 76, 77, 85, 88, 93, 94, 98, ...}.
2^4 divides a(n) for n = {2, 12, 13, 18, 29, 45, 53, 58, 64, 76, 77, 93, 102, 109, 112, 125, 128, 133, 141, 149, 150, 157, 170, ...}.
2^5 divides a(n) for n = {12, 13, 18, 53, 58, 64, 77, 93, 102, 112, 141, 149, 150, 173, 178, 188, 190, 196, 205, 232, 234, 237, ...}.
2^6 divides a(n) for n = {13, 58, 64, 102, 150, 173, 178, 190, 205, 232, 234, 237, 245, 277, 285, 290, 325, 333, 382, 429, 434, ...}.
2^7 divides a(n) for n = {13, 58, 64, 150, 173, 178, 205, 234, 325, 382, 472, 573, 592, 596, 621, 628, 653, 757, 796, 893, 950, ...}.
2^8 divides a(n) for n = {13, 58, 64, 178, 205, 325, 382, 573, 653, 757, ...}.
2^9 divides a(n) for n = {13, 58, 64, 178, 325, 382, 653, 757, ...}.
2^10 divides a(n) for n = {64, 178, ...}.
2^11 divides a(n) for n = {64, 178, ...}.
2^12 divides a(n) for n = {178, ...}.

G. C. Greubel, Table of n, a(n) for n = 1..76

Cf. A086787 (Sum_{i=1..n} (Sum_{j=1..n} i^j)), A122004.

Table[Sum[Sum[Prime[i]^Prime[j],{i,1,n}],{j,1,n}],{n,1,13}]

for(n=1,10, print1(sum(j=1,n, sum(k=1,n, (prime(k))^(prime(j)))), ", ")) \\ G. C. Greubel, Oct 25 2017

from sympy import prime
def A123373(n): return sum(prime(i)**prime(j) for i in range(1,n+1) for j in range(1,n+1)) # Chai Wah Wu, Jan 08 2022

A124391 Numbers m that divide A123269(m) = Sum_{i=1..m, j=1..m, k=1..m} i^j^k.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 40, 42, 43, 44, 46, 47, 48, 49, 54, 56, 59, 60, 62, 63, 64, 66, 67, 69, 71, 72, 77, 79, 80, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 100, 103

Offset: 1

Alexander Adamchuk, Oct 30 2006

A123269(m) = Sum_{i=1..m, j=1..m, k=1..m} i^j^k = {1, 28, 7625731729896, ...}.

Primes terms are listed in A039787.

Cf. A123269, A039787, A086787.

Do[f=Sum[Mod[Sum[Mod[Sum[PowerMod[i,j^k,n], {i, 1, n}],n], {j, 1, n}],n], {k, 1, n}];If[IntegerQ[f/n],Print[n]],{n,1,103}]

A125227 A014741(n)/6 for n>2.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 49, 57, 63, 81, 147, 171, 189, 219, 243, 301, 343, 399, 441, 513, 567, 657, 729, 889, 903, 1029, 1083, 1197, 1323, 1533, 1539, 1701, 1971, 2107, 2187, 2359, 2401, 2667, 2709, 2793, 3087, 3249, 3591, 3969, 4161, 4401, 4599, 4617, 5103, 5913

Offset: 3

Alexander Adamchuk, Jan 15 2007

nonn

A014741(n) is divisible by 6 for n>2.
All powers of 3 are terms. All powers of 7 are terms. The prime divisors of terms of this sequence (for n up to 10^6) in order of their first appearance are 3, 7, 19, 73, 43, 127, 337, 163, 379, 571, 5419, 487, 2593, 439, 1459, 431, 883.
The sequence is multiplicative in the sense that if two numbers k and m are terms, then k*m is too.

			a(3) = A014741(3)/6 = 6/6 = 1.
a(4) = A014741(4)/6 = 18/6 = 3.

Amiram Eldar, Table of n, a(n) for n = 3..10000

Cf. A014741, A000295, A086787.

Select[Range[3,6000], PowerMod[2,#+1,# ]==2&]/6

a(n) = A014741(n)/6 = A014945(n-1)/3. - Max Alekseyev, Nov 14 2012

A122004 Primes p that divide A123373(p-1).

Original entry on oeis.org

2, 3, 5, 23, 29

Offset: 1

Alexander Adamchuk, Oct 14 2006

			A123373 begins {4, 48, 3598, 924780, 287358579128, ...}.
a(1) = 2 because 2 divides A123373(1) = 4.
a(2) = 3 because 3 divides A123373(2) = 48.
a(3) = 5 because 5 divides A123373(4) = 924780.

Cf. A086787, A123373.

is(p) = isprime(p) && sum(j=1, p-1, sum(k=1, p-1, Mod(prime(k), p)^prime(j))) == 0; \\ Jinyuan Wang, Jan 16 2021

from sympy import nextprime, prime
p, A122004_list = 2, []
while p < 10**6:
    if 0 == sum(pow(prime(i),prime(j),p) for i in range(1,p) for j in range(1,p)) % p:
        A122004_list.append(p)
    p = nextprime(p) # Chai Wah Wu, Feb 19 2021

A124403 a(n) = -1 + Sum_{i=1..n} Sum_{j=1..n} i^j.

Original entry on oeis.org

0, 7, 55, 493, 5698, 82199, 1419759, 28501115, 651233660, 16676686695, 472883843991, 14705395791305, 497538872883726, 18193397941038735, 714950006521386975, 30046260016074301943, 1344648068888240941016

Offset: 1

Alexander Adamchuk, Dec 14 2006

nonn

p divides a(p-2) for prime p>2. p^k divides a(p^k-2) for prime p>2.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A086787.

List([1..30], n-> n-1 + Sum([2..n], j-> j*(j^n-1)/(j-1)) ); # G. C. Greubel, Dec 25 2019

[0] cat [n-1 + (&+[j*(j^n-1)/(j-1): j in [2..n]]): n in [2..30]]; // G. C. Greubel, Dec 25 2019

seq( n-1+add(j*(j^n-1)/(j-1), j=2..n), n=1..30); # G. C. Greubel, Dec 25 2019

Table[Sum[i^j,{i,1,n},{j,1,n}]-1,{n,1,25}]

vector(30, n, n-1 + sum(j=2,n, j*(j^n-1)/(j-1)) ) \\ G. C. Greubel, Dec 25 2019

[n-1 + sum(j*(j^n-1)/(j-1) for j in (2..n)) for n in (1..30)] # G. C. Greubel, Dec 25 2019

a(n) = -1 + Sum_{i=1..n} Sum_{j=1..n} i^j.
a(n) = n - 1 + Sum_{j=2..n} j*(j^n - 1)/(j-1).
a(n) = A086787(n) - 1.

A133298 a(n) = 1 + Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} i^(j+k).

Original entry on oeis.org

2, 41, 1727, 130917, 17245160, 3546873073, 1046002784253, 417182980579609, 215861313302976046, 140463714074395109081, 112191246261394235358555, 107867952671976721983260413, 122856922623618324408724634164

Offset: 1

Alexander Adamchuk, Oct 17 2007

nonn

p divides a(p) for prime p>3. p^2 divides a(p) for prime p=7. Nonprime n dividing a(n) are {1,15}.

G. C. Greubel, Table of n, a(n) for n = 1..210

Cf. A124405.

Cf. A062970, A086787.

List([1..20], n-> 1 + n^2 + Sum([2..n], j-> (j*(j^n-1)/(j-1))^2) ); # G. C. Greubel, Aug 02 2019

[2] cat [1+n^2 + (&+[(j*(j^n-1)/(j-1))^2: j in [2..n]]): n in [1..20]]; // G. C. Greubel, Aug 02 2019

Table[Sum[(i(i^n-1)/(i-1))^2, {i,2,n}] +n^2 +1,{n,20}]

vector(20, n, 1+n^2 + sum(j=2,n, (j*(j^n-1)/(j-1))^2)) \\ G. C. Greubel, Aug 02 2019

[1+n^2 + sum((j*(j^n-1)/(j-1))^2 for j in (2..n)) for n in (1..20)] # G. C. Greubel, Aug 02 2019

a(n) = 1 + Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} i^(j+k).

a(n) = 1 + n^2 + Sum_{j=2..n} (j*(j^n - 1)/(j-1))^2.

A124045 Numbers n such that n^2 divide A123269(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

Original entry on oeis.org

1, 2, 3, 6, 42

Offset: 1

Alexander Adamchuk, Nov 02 2006

A123269(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ] = {1, 28, 7625731729896, ...}. Numbers n that divide A123269(n) are listed in A124391(n) = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 40, 42, 43, ...}.

Cf. A123269, A124391, A039787, A086787.

Do[f=Sum[Mod[Sum[Mod[Sum[PowerMod[i, j^k, n^2], {i, 1, n}], n^2], {j, 1, n}], n^2], {k, 1, n}]; If[IntegerQ[f/n^2], Print[n]], {n, 1, 103}]

cp's OEIS Frontend

A123373 a(n) = Sum_{i=1..n} (Sum_{j=1..n} prime(i)^prime(j)).

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A124391 Numbers m that divide A123269(m) = Sum_{i=1..m, j=1..m, k=1..m} i^j^k.

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A125227 A014741(n)/6 for n>2.

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A122004 Primes p that divide A123373(p-1).

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A124403 a(n) = -1 + Sum_{i=1..n} Sum_{j=1..n} i^j.

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A133298 a(n) = 1 + Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} i^(j+k).

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A124045 Numbers n such that n^2 divide A123269(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

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