A106316 -id:A106316 - cp's OEIS frontend

A324045 a(n) = A000010(n) - A106316(n).

1, 0, 0, 1, 0, 2, 0, 2, 5, 0, 0, 0, 0, -2, -4, 6, 0, -6, 0, -8, -8, -6, 0, -4, 7, -8, 17, 12, 0, -16, 0, 13, 17, -12, 15, -3, 0, -14, 19, 6, 0, 6, 0, 8, -12, -18, 0, 4, 9, -1, 23, 6, 0, 0, 36, -8, 25, -24, 0, -32, 0, -26, 33, 29, 40, -10, 0, 2, 29, -34, 0, 12, 0, -32, 37, 0, 40, -18, 0, -24, 12, -36, 0, -4, 48, -38, 35, -36, 0, -30, 44, -4, 37, -42, 52, -16, 0

Offset: 1

Antti Karttunen, Feb 13 2019

sign

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

Cf. A000005, A000010, A000203, A106316, A324046, A324047.

A106315(n) = (n*numdiv(n) % sigma(n));
A106316(n) = (A106315(n) % n);
A324045(n) = (eulerphi(n) - A106316(n));

a(n) = A000010(n) - A106316(n).

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

Original entry on oeis.org

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).

A106315 Harmonic residue of n.

Original entry on oeis.org

0, 1, 2, 5, 4, 0, 6, 2, 1, 4, 10, 16, 12, 8, 12, 18, 16, 30, 18, 36, 20, 16, 22, 12, 13, 20, 28, 0, 28, 24, 30, 3, 36, 28, 44, 51, 36, 32, 44, 50, 40, 48, 42, 12, 36, 40, 46, 108, 33, 21, 60, 18, 52, 72, 4, 88, 68, 52, 58, 48, 60, 56, 66, 67, 8, 96, 66, 30, 84, 128, 70, 84, 72, 68, 78

Offset: 1

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

nonn

The harmonic residue is the remainder when n*d(n) is divided by sigma(n), where d(n) is the number of divisors of n and sigma(n) is the sum of the divisors of n. If n is perfect, the harmonic residue of n is 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A106316, A106317, A001599 (positions of zeros).

Cf. A000005, A000203.

a106315 n = n * a000005 n `mod` a000203 n -- Reinhard Zumkeller, Apr 06 2014

A106315 := proc(n)
    modp(n*numtheory[tau](n),numtheory[sigma](n)) ;
end proc:
seq(A106315(n),n=1..100) ; # R. J. Mathar, Jan 25 2017

HarmonicResidue[n_]=Mod[n*DivisorSigma[0, n], DivisorSigma[1, n]]; HarmonicResidue[ Range[ 80]]

a(n) = A038040(n) - A000203(n) * A240471(n) . - Reinhard Zumkeller, Apr 06 2014

Mathematica program completed by Harvey P. Dale, Feb 29 2024

A324121 a(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

1, 1, 2, 1, 2, 12, 2, 1, 1, 2, 2, 4, 2, 8, 12, 1, 2, 3, 2, 6, 4, 4, 2, 12, 1, 2, 4, 56, 2, 24, 2, 3, 12, 2, 4, 1, 2, 4, 4, 10, 2, 48, 2, 12, 6, 8, 2, 4, 3, 3, 12, 2, 2, 24, 4, 8, 4, 2, 2, 24, 2, 8, 2, 1, 4, 48, 2, 6, 12, 16, 2, 3, 2, 2, 2, 4, 4, 24, 2, 2, 1, 2, 2, 112, 4, 4, 12, 4, 2, 18, 28, 24, 4, 8, 20, 36, 2, 3, 6, 1, 2, 24, 2, 2, 24

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Records 1, 2, 12, 56, 112, 120, 336, 720, 992, 2016, 4368, 8640, 14880, 16256, 26208, 59520, 78624, 120960, 131040, 191520, 227584, 297600, ... occur at positions: 1, 3, 6, 28, 84, 120, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 27846, 30240, 32760, 55860, 105664, 117800, ... . Note that A001599 is not a subsequence of the latter, as at least 18620 (present in A001599) is missing.

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, A001599, A038040, A106315, A106316, A156552, A324045, A324046, A324047, A324058, A324122.

Table[GCD[n DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Feb 17 2023 *)

PARI
```
A324121(n) = gcd(sigma(n),n*numdiv(n));
```

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

a(n) = A324058(A156552(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

A324045 a(n) = A000010(n) - A106316(n).

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

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

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A106315 Harmonic residue of n.

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A324121 a(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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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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A106317 Numbers k such that the remainder of the harmonic residue of k when divided by k is k-1.

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