A326066 -id:A326066 - cp's OEIS frontend

A326065 Sum of divisors of the largest proper divisor of n: a(n) = sigma(A032742(n)).

1, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 12, 1, 8, 6, 15, 1, 13, 1, 18, 8, 12, 1, 28, 6, 14, 13, 24, 1, 24, 1, 31, 12, 18, 8, 39, 1, 20, 14, 42, 1, 32, 1, 36, 24, 24, 1, 60, 8, 31, 18, 42, 1, 40, 12, 56, 20, 30, 1, 72, 1, 32, 32, 63, 14, 48, 1, 54, 24, 48, 1, 91, 1, 38, 31, 60, 12, 56, 1, 90, 40, 42, 1, 96, 18, 44, 30, 84, 1, 78, 14

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n).

Cf. A000203, A013661, A020639, A032742, A067029, A143112, A326066, A326067, A326068, A326069, A326135.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326065(n) = sigma(A032742(n));

a(n) = A000203(A032742(n)) = A000203(n) - A326066(n).
a(n) = A326135(n) * A000203(A020639(n)^(A067029(n)-1)).
Sum_{k=1..n} a(k) ~ (zeta(2)/2) * c * n^2, where c = Sum_{p prime} ((p/((p-1)^2*(p+1))) * Product_{primes q <= p} ((q-1)^2*(q+1)/q^3)) = 0.3076135997... . - Amiram Eldar, Dec 21 2024

A326067 a(n) = sigma(n) - sigma(A032742(n)) - n, where A032742 gives the largest proper divisor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

-1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 3, 0, 0, 8, 0, 4, 3, 2, 0, 8, 0, 2, 0, 4, 0, 18, 0, 0, 3, 2, 5, 16, 0, 2, 3, 8, 0, 22, 0, 4, 9, 2, 0, 16, 0, 12, 3, 4, 0, 26, 5, 8, 3, 2, 0, 36, 0, 2, 9, 0, 5, 30, 0, 4, 3, 26, 0, 32, 0, 2, 18, 4, 7, 34, 0, 16, 0, 2, 0, 44, 5, 2, 3, 8, 0, 66, 7, 4, 3, 2, 5, 32, 0, 16, 9, 24, 0, 42, 0, 8, 39

Offset: 1

Antti Karttunen, Jun 06 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
Index entries for sequences related to sigma(n).

Cf. A000203, A013661, A032742, A033879, A246655 (positions of zeros), A326065, A326066, A326068, A326069.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326066(n) = (sigma(n) - sigma(A032742(n)));
A326067(n) = (A326066(n) - n);

a(n) = A326066(n) - n = A000203(n) - A000203(A032742(n)) - n.
a(n) = A326068(n) - A033879(n).
a(p^k) = 0 for all primes and all exponents k >= 1.
Sum_{k=1..n} a(k) ~ ((zeta(2) * (1 - c) - 1)/2) * n^2, where c is defined in the corresponding formula in A326065. . - Amiram Eldar, Dec 21 2024

A326068 a(n) = n - sigma(A032742(n)), where sigma is the sum of divisors of n and A032742 gives the largest proper divisor of n.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 1, 5, 4, 10, 0, 12, 6, 9, 1, 16, 5, 18, 2, 13, 10, 22, -4, 19, 12, 14, 4, 28, 6, 30, 1, 21, 16, 27, -3, 36, 18, 25, -2, 40, 10, 42, 8, 21, 22, 46, -12, 41, 19, 33, 10, 52, 14, 43, 0, 37, 28, 58, -12, 60, 30, 31, 1, 51, 18, 66, 14, 45, 22, 70, -19, 72, 36, 44, 16, 65, 22, 78, -10, 41, 40, 82, -12, 67, 42, 57, 4, 88, 12

Offset: 1

Antti Karttunen, Jun 06 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000203, A013661, A032742, A033879, A326065, A326066, A326067, A326069.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326065(n) = sigma(A032742(n));
A326068(n) = (n - A326065(n));

a(n) = n - A326065(n) = n - A000203(A032742(n)).
a(n) = A326067(n) + A033879(n).
Sum_{k=1..n} a(k) ~ ((1 - zeta(2) * c)/2) * n^2, where c = Sum_{p prime} ((p/((p-1)^2*(p+1))) * Product_{prime q <= p} ((q-1)^2*(q+1)/q^3)) = 0.307613599749... . - Amiram Eldar, Dec 21 2024

A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 4, 12, 2, 3, 1, 16, 1, 18, 2, 1, 2, 22, 4, 19, 2, 14, 4, 28, 6, 30, 1, 3, 2, 1, 1, 36, 2, 1, 2, 40, 2, 42, 4, 3, 2, 46, 4, 41, 1, 3, 2, 52, 2, 1, 8, 1, 2, 58, 12, 60, 2, 1, 1, 1, 6, 66, 2, 3, 2, 70, 1, 72, 2, 2, 4, 1, 2, 78, 2, 41, 2, 82, 4, 1, 2, 3, 4, 88, 6, 7, 4, 1, 2, 5, 4, 96, 1, 3, 1, 100, 6, 102, 2, 3

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A032742, A326065, A326066, A326067, A326068.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326066(n) = (sigma(n) - sigma(A032742(n)));
A326067(n) = (A326066(n) - n);
A326065(n) = sigma(A032742(n));
A326068(n) = (n - A326065(n));
A326069(n) = gcd(A326067(n), A326068(n));

a(n) = gcd(A326067(n), A326068(n)) = gcd(A326066(n) - n, n - A326065(n)).

A326136 a(n) = sigma(n) - sigma(A028234(n)), where sigma is the sum of divisors of n, and A028234 gives n without any occurrence of its smallest prime factor.

Original entry on oeis.org

0, 2, 3, 6, 5, 8, 7, 14, 12, 12, 11, 24, 13, 16, 18, 30, 17, 26, 19, 36, 24, 24, 23, 56, 30, 28, 39, 48, 29, 48, 31, 62, 36, 36, 40, 78, 37, 40, 42, 84, 41, 64, 43, 72, 72, 48, 47, 120, 56, 62, 54, 84, 53, 80, 60, 112, 60, 60, 59, 144, 61, 64, 96, 126, 70, 96, 67, 108, 72, 96, 71, 182, 73, 76, 93, 120, 84, 112, 79, 180, 120, 84, 83

Offset: 1

Antti Karttunen, Jun 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000203, A013661, A028234, A326066, A326135.

A028234(n) = { my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); }; \\ From A028234
A326136(n) = (sigma(n) - sigma(A028234(n)));

a(n) = A000203(n) - A000203(A028234(n)).
From Amiram Eldar, Dec 21 2024: (Start)
a(n) = A000203(n) - A326135(n).
Sum_{k=1..n} a(k) ~ (zeta(2)/2) * (1 - c) * n^2, where c is defined in the corresponding formula in A326135. (End)

cp's OEIS Frontend

A326065 Sum of divisors of the largest proper divisor of n: a(n) = sigma(A032742(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A326067 a(n) = sigma(n) - sigma(A032742(n)) - n, where A032742 gives the largest proper divisor of n, and sigma is the sum of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A326068 a(n) = n - sigma(A032742(n)), where sigma is the sum of divisors of n and A032742 gives the largest proper divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A326136 a(n) = sigma(n) - sigma(A028234(n)), where sigma is the sum of divisors of n, and A028234 gives n without any occurrence of its smallest prime factor.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula