A106315 -id:A106315 - cp's OEIS frontend

A324046 a(n) = gcd(n, A106316(n)).

1, 1, 1, 1, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 3, 1, 2, 1, 10, 1, 6, 1, 4, 9, 2, 1, 12, 1, 1, 3, 2, 1, 18, 1, 8, 1, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 2, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 8, 3, 2, 1, 28, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 48, 1, 1, 9, 4, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Feb 13 2019

nonn

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

Cf. A000005, A000203, A106315, A106316, A324045, A324047.

A106315(n) = (n*numdiv(n) % sigma(n));
A106316(n) = (A106315(n) % n);
A324046(n) = gcd(n,A106316(n));

a(n) = gcd(n, A106316(n)).

A324047 a(n) = A000203(n) - A106316(n).

Original entry on oeis.org

1, 2, 2, 6, 2, 12, 2, 13, 12, 14, 2, 24, 2, 16, 12, 29, 2, 27, 2, 26, 12, 20, 2, 48, 18, 22, 39, 56, 2, 48, 2, 60, 45, 26, 39, 76, 2, 28, 51, 80, 2, 90, 2, 72, 42, 32, 2, 112, 24, 72, 63, 80, 2, 102, 68, 88, 69, 38, 2, 120, 2, 40, 101, 124, 76, 114, 2, 96, 81, 86, 2, 183, 2, 46, 121, 104, 76, 126, 2, 130, 79, 50, 2, 196, 92, 52

Offset: 1

Antti Karttunen, Feb 13 2019

nonn

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

Cf. A000005, A000203, A106315, A106316, A324045, A324046.

A106315(n) = (n*numdiv(n) % sigma(n));
A106316(n) = (A106315(n) % n);
A324047(n) = (sigma(n) - A106316(n));

a(n) = A000203(n) - A106316(n).

A324122 a(n) = sigma(n) - gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

0, 2, 2, 6, 4, 0, 6, 14, 12, 16, 10, 24, 12, 16, 12, 30, 16, 36, 18, 36, 28, 32, 22, 48, 30, 40, 36, 0, 28, 48, 30, 60, 36, 52, 44, 90, 36, 56, 52, 80, 40, 48, 42, 72, 72, 64, 46, 120, 54, 90, 60, 96, 52, 96, 68, 112, 76, 88, 58, 144, 60, 88, 102, 126, 80, 96, 66, 120, 84, 128, 70, 192, 72, 112, 122, 136, 92, 144, 78, 184, 120, 124, 82

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..117800

Cf. A000005, A000203, A038040, A106315, A106316, A324045, A324046, A324047, A324121.

Cf. A001599 (positions of zeros).

A324122(n) = (sigma(n) - gcd(sigma(n),n*numdiv(n)));

a(n) = A000203(n) - A324121(n) = A000203(n) - gcd(A000203(n), A038040(n)).

A106317 Numbers k such that the remainder of the harmonic residue of k when divided by k is k-1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Offset: 1

George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Apr 29 2005

nonn

Cf. A106315, A106316.

is(n) = {my(f = factor(n)); n*numdiv(f) % sigma(f) == n - 1;} \\ Amiram Eldar, Jan 09 2024

It appears that k is in the sequence iff k is prime or k is in {1, 21, 822857} (Verified to 3.1*10^6). It is true that if k is the product of two distinct primes, then k=21. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Apr 30 2005, R. J. Mathar, Jan 25 2017

There are no other nonprime terms below 10^11. - Amiram Eldar, Jan 09 2024

cp's OEIS Frontend

A324046 a(n) = gcd(n, A106316(n)).

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A324047 a(n) = A000203(n) - A106316(n).

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PARI

Formula

A324122 a(n) = sigma(n) - gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

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PARI

Formula

A106317 Numbers k such that the remainder of the harmonic residue of k when divided by k is k-1.

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Author

Keywords

Crossrefs

Programs

PARI

Formula