A341689
a(n) is the sum of the 4th power of the first A125907(n) primes.
Original entry on oeis.org
16, 282090779141153551270, 2210712955689035458600206881540015387708550, 48675866046797839528447895106845001955284425583991669795082795118772, 340116502128393540096171523813533871084766138971398067752157768889198596930173282496
Offset: 1
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sum = 0
for n in range(1,10000000000001):
sum += pow(prime[n],4)
if sum % n == 0:
print(n, prime[n], sum, (sum // n))
A097880
Decimal expansion of the sum for 1 to infinity of fraction sequence with numerator triangular numbers and denominator sum of 4th power of primes.
Original entry on oeis.org
1, 0, 6, 8, 3, 2, 7, 9, 0, 3, 3, 4, 6, 4, 0
Offset: 1
A118219
Smallest number k>1 such that Sum_{i=1..k} Prime[i]^n divides Product_{i=1..k} Prime[i]^n.
Original entry on oeis.org
3, 30, 17, 248, 515, 49682
Offset: 1
a(1) = 3 because 2 + 3 + 5 = 10 divides 2*3*5 = 30 but 2 + 3 = 5 does not divide 2*3 = 6.
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f[n_] := Block[{k = 2, p = 2, s = 2^n}, While[p = p*Prime@ k; s = s + Prime@ k^n; PowerMod[p, n, s] != 0, k++ ]; k]; Do[ Print@ f@n, {n, 10}] (* Robert G. Wilson v *)
A122124
Numbers n such that 25 divides Sum[ Prime[k]^n, {k,1,n}].
Original entry on oeis.org
3, 5, 7, 11, 15, 19, 23, 25, 27, 31, 35, 39, 43, 45, 47, 51, 55, 59, 63, 65, 67, 71, 75, 79, 83, 85, 87, 91, 95, 99, 103, 105, 107, 111, 115, 119, 123, 125, 127, 131, 135, 139, 143, 145, 147, 151, 155, 159, 163, 165, 167, 171, 175, 179, 183, 185, 187, 191, 195, 199
Offset: 1
There are 25 primes p < 100, p(n) = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
a(1) = because 25 divides Sum[p(n)^3,{n,1,25}] = 2^3 + 3^3 + ... + 89^3 + 97^3 = A098999[25] and does not divide Sum[p(n)^1,{n,1,25}] = A007504[25] and Sum[p(n)^2,{n,1,25}] = A024450[25].
The next a(2) = 5 because 25 divides Sum[p(n)^5,{n,1,25}] = A122103[25] and does not divide Sum[p(n)^4,{n,1,25}] = A122102[25].
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Select[Range[300],IntegerQ[Sum[ Prime[k]^#1, {k,1,25}]/25]&]
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for(n=1,100,if(sum(k=1,25,prime(k)^n)%25==0,print1(n,",")));
print;print("Alternative method not using primes:");
for(n=1,100,m=(n-1)%6;print1((n-m)*3+(n-m+if(m>1,(m-1)*12-1,m*6-1))/3,",")) \\ K. Spage, Oct 23 2009
A341690
Integer averages of first n primes to the 4th power for some n (A341689(n)/A125907(n)).
Original entry on oeis.org
16, 95591589000729770, 57770815231373815452404527382911050, 15942241394469365582203327807497328235663420076612273764, 89536555153849358635668155008982165719026544119306300984594045157568
Offset: 1
-
sum = 0
for n in range(1, 10000000000001):
sum += pow(prime[n], 4)
if sum % n == 0:
print(n, prime[n], sum, (sum // n))
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