A332469
a(n) = Sum_{k=1..n} floor(n/k)^n.
Original entry on oeis.org
1, 5, 29, 274, 3160, 47452, 825862, 16843268, 387702833, 10009826727, 285360679985, 8918294547447, 302888236005847, 11112685321898449, 437898668488710801, 18447025705612363530, 827242514466399305122, 39346558271561286347116, 1978421007121668206129316
Offset: 1
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[&+[Floor(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 13 2020
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Table[Sum[Floor[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[(k^n - (k - 1)^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]
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a(n)={sum(k=1, n, floor(n/k)^n)} \\ Andrew Howroyd, Feb 13 2020
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from math import isqrt
def A332469(n): return -(s:=isqrt(n))**(n+1)+sum((q:=n//k)*(k**n-(k-1)**n+q**(n-1)) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023
A344527
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of ordered k-tuples (x_1, x_2, ..., x_k) with gcd(x_1, x_2, ..., x_k) = 1 (1 <= {x_1, x_2, ..., x_k} <= n).
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 11, 1, 1, 31, 79, 55, 19, 1, 1, 63, 241, 239, 115, 23, 1, 1, 127, 727, 991, 607, 181, 35, 1, 1, 255, 2185, 4031, 3091, 1199, 307, 43, 1, 1, 511, 6559, 16255, 15559, 7501, 2303, 439, 55, 1, 1, 1023, 19681, 65279, 77995, 45863, 16531, 3823, 637, 63, 1
Offset: 1
G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} mu(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 3, 7, 15, 31, 63, ...
1, 7, 25, 79, 241, 727, ...
1, 11, 55, 239, 991, 4031, ...
1, 19, 115, 607, 3091, 15559, ...
1, 23, 181, 1199, 7501, 45863, ...
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T[n_, k_] := Sum[MoebiusMu[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)
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T(n, k) = sum(j=1, n, moebius(j)*(n\j)^k);
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T(n, k) = n^k-sum(j=2, n, T(n\j, k));
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from functools import lru_cache
from itertools import count, islice
@lru_cache(maxsize=None)
def A344527_T(n,k):
if n == 0:
return 0
c, j, k1 = 1, 2, n//2
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j)*A344527_T(k1,k)
j, k1 = j2, n//j2
return n*(n**(k-1)-1)-c+j
def A344527_gen(): # generator of terms
return (A344527_T(k+1, n-k) for n in count(1) for k in range(n))
A344527_list = list(islice(A344527_gen(),30)) # Chai Wah Wu, Nov 02 2023
A344429
a(n) = Sum_{k=1..n} mu(k) * k^n.
Original entry on oeis.org
1, -3, -34, -96, -3399, 30239, -624046, -4482626, -32249230, 9768165230, -186975207617, -2150337557747, -327482869358214, 6894274639051756, 539094536846680025, 8044964790023844733, -707278869236116107432, -12275330572755863672628, -2190860499375418948848067
Offset: 1
Cf.
A002321,
A008683,
A031971,
A068340,
A321222,
A332468,
A336276,
A336277,
A336278,
A336279,
A344430.
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a[n_] := Sum[MoebiusMu[k] * k^n, {k,1,n}]; Array[a, 20] (* Amiram Eldar, May 19 2021 *)
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a(n) = sum(k=1, n, moebius(k)*k^n);
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from functools import lru_cache
from math import comb
from sympy import bernoulli
@lru_cache(maxsize=None)
def faulhaber(n,p):
""" Faulhaber's formula for calculating Sum_{k=1..n} k^p
requires sympy version 1.12+ where bernoulli(1) = 1/2
"""
return sum(comb(p+1,k)*bernoulli(k)*n**(p-k+1) for k in range(p+1))//(p+1)
@lru_cache(maxsize=None)
def A344429(n,m=None):
if n <= 1:
return 1
if m is None:
m=n
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (faulhaber(j-1,m)-faulhaber(j2-1,m))*A344429(k1,m)
j, k1 = j2, n//j2
return c+faulhaber(j-1,m)-faulhaber(n,m) # Chai Wah Wu, Nov 02 2023
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