A010848 -id:A010848 - cp's OEIS frontend

A300718 Möbius transform of A010848, number of numbers k <= n such that at least one prime factor of n is not a prime factor of k.

0, 1, 2, 1, 4, 2, 6, 2, 4, 4, 10, 4, 12, 6, 8, 4, 16, 6, 18, 8, 12, 10, 22, 8, 16, 12, 12, 12, 28, 8, 30, 8, 20, 16, 24, 10, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 36, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 16, 48, 20, 66, 32, 44, 24, 70, 20, 72, 36, 40, 36, 60, 24, 78, 32, 36, 40, 82, 24, 64, 42, 56, 40, 88, 24, 72

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A008683, A003557, A010848, A300717, A300720.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A010848(n) = (n - A003557(n));
A300718(n) = sumdiv(n,d,moebius(n/d)*A010848(d));

a(n) = Sum_{d|n} A008683(n/d)*A010848(d).
a(n) = A000010(n) - A300717(n).
a(n) = A010848(n) - A300720(n).

A300720 Difference between A010848 and its Möbius transform.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 2, 2, 5, 0, 6, 0, 7, 6, 4, 0, 9, 0, 10, 8, 11, 0, 12, 4, 13, 6, 14, 0, 21, 0, 8, 12, 17, 10, 20, 0, 19, 14, 20, 0, 29, 0, 22, 18, 23, 0, 24, 6, 25, 18, 26, 0, 27, 14, 28, 20, 29, 0, 42, 0, 31, 24, 16, 16, 45, 0, 34, 24, 45, 0, 40, 0, 37, 30, 38, 16, 53, 0, 40, 18, 41, 0, 58, 20, 43, 30, 44, 0, 63, 18

Offset: 1

Antti Karttunen, Mar 12 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A003557, A010848, A300718.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); }; \\ From A003557
A010848(n) = (n - A003557(n));
A300720(n) = -sumdiv(n,d,(dA010848(d));

a(n) = -Sum_{d|n, dA008683(n/d)*A010848(d).

a(n) = A010848(n) - A300718(n).

A326188 a(n) = A001065(n) - A003557(n), where A001065(n) = the sum of proper divisors of n, and A003557(n) = n divided by its largest squarefree divisor.

Original entry on oeis.org

-1, 0, 0, 1, 0, 5, 0, 3, 1, 7, 0, 14, 0, 9, 8, 7, 0, 18, 0, 20, 10, 13, 0, 32, 1, 15, 4, 26, 0, 41, 0, 15, 14, 19, 12, 49, 0, 21, 16, 46, 0, 53, 0, 38, 30, 25, 0, 68, 1, 38, 20, 44, 0, 57, 16, 60, 22, 31, 0, 106, 0, 33, 38, 31, 18, 77, 0, 56, 26, 73, 0, 111, 0, 39, 44, 62, 18, 89, 0, 98, 13, 43, 0, 138, 22, 45, 32, 88, 0, 141, 20

Offset: 1

Antti Karttunen, Jul 11 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A003557, A010848, A326187, A326189.

Cf. also A326185.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A326188(n) = ((sigma(n)-A003557(n))-n);

a(n) = A326187(n) - n = A000203(n) - A003557(n) - n.

a(n) = A001065(n) - A003557(n).

A326186 a(n) = n - A057521(n), where A057521 gives the powerful part of n.

Original entry on oeis.org

0, 1, 2, 0, 4, 5, 6, 0, 0, 9, 10, 8, 12, 13, 14, 0, 16, 9, 18, 16, 20, 21, 22, 16, 0, 25, 0, 24, 28, 29, 30, 0, 32, 33, 34, 0, 36, 37, 38, 32, 40, 41, 42, 40, 36, 45, 46, 32, 0, 25, 50, 48, 52, 27, 54, 48, 56, 57, 58, 56, 60, 61, 54, 0, 64, 65, 66, 64, 68, 69, 70, 0, 72, 73, 50, 72, 76, 77, 78, 64, 0, 81, 82, 80, 84

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A057521, A326185.

Cf. also A010848.

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521.
A326186(n) = (n-A057521(n));

from math import prod
from sympy import factorint
def A326186(n): return n-n//prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

a(n) = n - A057521(n).

A340806 a(n) = Sum_{k=1..n-1} (k^n mod n).

Original entry on oeis.org

0, 1, 3, 2, 10, 13, 21, 4, 27, 45, 55, 38, 78, 77, 105, 8, 136, 93, 171, 146, 210, 209, 253, 172, 250, 325, 243, 294, 406, 365, 465, 16, 528, 561, 595, 402, 666, 665, 741, 372, 820, 673, 903, 726, 945, 897, 1081, 536, 1029, 1125, 1275, 1170, 1378, 765, 1485

Offset: 1

Sebastian Karlsson, Jan 22 2021

nonn

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A010848, A048153, A048155, A094264, A121706, A195637, A195812, A202034.

a:= n-> add(k&^n mod n, k=1..n-1):
seq(a(n), n=1..55);  # Alois P. Heinz, Feb 13 2021

a(n) = sum(k=1, n-1, lift(Mod(k, n)^n)); \\ Michel Marcus, Jan 22 2021

def a(n):
    return sum([pow(k,n,n) for k in range(1, n)])
for n in range(1, 56):
    print(a(n), end=', ')

a(n) = n*A010848(n)/2, if n is odd.
a(n) = n*(n-1)/2, if n is both odd and squarefree.
a(p^e) = (1/2)*(p-1)*p^(2*e-1), if p is an odd prime.
a(2^e) = 2^(e-1).

cp's OEIS Frontend

A300718 Möbius transform of A010848, number of numbers k <= n such that at least one prime factor of n is not a prime factor of k.

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A300720 Difference between A010848 and its Möbius transform.

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A326188 a(n) = A001065(n) - A003557(n), where A001065(n) = the sum of proper divisors of n, and A003557(n) = n divided by its largest squarefree divisor.

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A326186 a(n) = n - A057521(n), where A057521 gives the powerful part of n.

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A340806 a(n) = Sum_{k=1..n-1} (k^n mod n).

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