A106316 -id:A106316

A324045

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

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

FORMULA

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

PROG

(PARI)

A106315(n) = (n*numdiv(n) % sigma(n));

A106316(n) = (A106315(n) % n);

A324045(n) = (eulerphi(n) - A106316(n));

CROSSREFS

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

KEYWORD

sign

AUTHOR

Antti Karttunen, Feb 13 2019

STATUS

approved

A324046

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

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

FORMULA

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

PROG

(PARI)

A106315(n) = (n*numdiv(n) % sigma(n));

A106316(n) = (A106315(n) % n);

A324046(n) = gcd(n, A106316(n));

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 13 2019

STATUS

approved

A324047

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

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

PROG

(PARI)

A106315(n) = (n*numdiv(n) % sigma(n));

A106316(n) = (A106315(n) % n);

A324047(n) = (sigma(n) - A106316(n));

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 13 2019

STATUS

approved

A106315

Harmonic residue of n.

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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.

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(Haskell)

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

CROSSREFS

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

Cf. A000005, A000203.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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).

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10080

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

FORMULA

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

a(n) = A324058(A156552(n)).

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000005, A000203, A001599, A038040, A106315, A106316, A156552, A324045, A324046, A324047, A324058, A324122.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 15 2019

STATUS

approved

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).

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10080

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

FORMULA

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

PROG

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

CROSSREFS

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

Cf. A001599 (positions of zeros).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 15 2019

STATUS

approved

A106317

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..48.

FORMULA

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

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

PROG

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

CROSSREFS

Cf. A106315, A106316.

KEYWORD

nonn

AUTHOR

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

STATUS

approved