A326135 -id:A326135 - 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

A326066 a(n) = sigma(n) - sigma(A032742(n)), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

0, 2, 3, 4, 5, 8, 7, 8, 9, 12, 11, 16, 13, 16, 18, 16, 17, 26, 19, 24, 24, 24, 23, 32, 25, 28, 27, 32, 29, 48, 31, 32, 36, 36, 40, 52, 37, 40, 42, 48, 41, 64, 43, 48, 54, 48, 47, 64, 49, 62, 54, 56, 53, 80, 60, 64, 60, 60, 59, 96, 61, 64, 72, 64, 70, 96, 67, 72, 72, 96, 71, 104, 73, 76, 93, 80, 84, 112, 79, 96, 81, 84, 83

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

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

Cf. A000203, A013661, A020639, A032742, A246655 (positions of fixed points), A247180, A326065, A326067, A326135, A326136.

Join[{0},Table[DivisorSigma[1,n]-DivisorSigma[1,Divisors[n][[-2]]],{n,2,100}]] (* Harvey P. Dale, Jan 12 2022 *)

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

a(n) = A000203(n) - A326065(n) = A000203(n) - A000203(A032742(n)).
a(1) = 0; for n > 1, if n is of the form p^k (p prime and exponent k >= 1), then a(n) = n, otherwise a(n) > n.
For terms in A247180, i.e., when n = A020639(n) * A032742(n), with the smallest prime factor A020639(n) unitary, a(n) = A020639(n) * A326065(n).
Sum_{k=1..n} a(k) ~ (zeta(2)/2) * (1 - c) * n^2, where c is defined in the corresponding formula in A326065. . - Amiram Eldar, Dec 21 2024

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)).

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A326066 a(n) = sigma(n) - sigma(A032742(n)), where A032742 gives the largest proper divisor of n.

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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.

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