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A351244 a(n) = n^4 * Sum_{p|n, p prime} 1/p^4. 11
0, 1, 1, 16, 1, 97, 1, 256, 81, 641, 1, 1552, 1, 2417, 706, 4096, 1, 7857, 1, 10256, 2482, 14657, 1, 24832, 625, 28577, 6561, 38672, 1, 61921, 1, 65536, 14722, 83537, 3026, 125712, 1, 130337, 28642, 164096, 1, 234193, 1, 234512, 57186, 279857, 1, 397312, 2401, 400625, 83602 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
FORMULA
a(A000040(n)) = 1.
G.f.: Sum_{k>=1} x^prime(k) * (1 + 11*x^prime(k) + 11*x^(2*prime(k)) + x^(3*prime(k))) / (1 - x^prime(k))^5. - Ilya Gutkovskiy, Feb 05 2022
Dirichlet g.f.: zeta(s-4)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^4/n^s) Sum_{p|n} 1/p^4. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^4*(p*j)^(s-4)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-4) = zeta(s-4)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 02 2023
Sum_{k=1..n} a(k) ~ A085965 * n^5/5. - Vaclav Kotesovec, Mar 03 2023
EXAMPLE
a(6) = 97; a(6) = 6^4 * Sum_{p|6, p prime} 1/p^4 = 1296 * (1/2^4 + 1/3^4) = 97.
CROSSREFS
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), this sequence (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).
Sequence in context: A036179 A309132 A099923 * A105671 A145828 A369913
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Feb 05 2022
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)