A353804 -id:A353804 - cp's OEIS frontend

A353802 Multiplicative with a(p^e) = sigma(sigma(p^e)).

1, 4, 7, 8, 12, 28, 15, 24, 14, 48, 28, 56, 24, 60, 84, 32, 39, 56, 42, 96, 105, 112, 60, 168, 32, 96, 90, 120, 72, 336, 63, 104, 196, 156, 180, 112, 60, 168, 168, 288, 96, 420, 84, 224, 168, 240, 124, 224, 80, 128, 273, 192, 120, 360, 336, 360, 294, 288, 168, 672, 96, 252, 210, 128, 288, 784, 126, 312, 420, 720

Offset: 1

Antti Karttunen, May 08 2022

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

Cf. A000203, A051027, A353803, A353804, A353805, A353806.

Cf. also A336547, A336548, A353752.

A051027(n) = sigma(sigma(n));
A353802(n) = { my(f = factor(n)); prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])); };

a(n) = Product_{p^e||n} sigma(sigma(p^e)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.
a(n) = A051027(n) + A353803(n).
a(n) = A353804(n) * A353806(n).

A353806 a(n) = A353802(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 5, 16, 1, 1, 1, 1, 1, 1, 1, 112, 1, 1, 49, 13, 45, 1, 1, 1, 7, 16, 1, 5, 1, 1, 1, 16, 1, 1, 1, 1, 7, 64, 1, 1, 112, 1, 49, 16, 1, 7, 1, 1, 1, 1, 9, 784, 1, 1, 5, 720, 1, 1, 1, 1, 1, 1, 5, 7, 1, 1, 1, 16, 1, 5, 117, 1, 7, 16, 1, 16, 45, 1, 147, 16, 7

Offset: 1

Antti Karttunen, May 08 2022

Numerator of fraction A353802(n) / A051027(n).

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

Cf. A000203, A051027, A353802, A353803, A353804, A353805 (denominators).
Cf. A336547 (positions of 1's), A336548 (positions of terms > 1), see also A353807.
Cf. also A353755, A353756.

A051027(n) = sigma(sigma(n));
A353806(n) = { my(f = factor(n), u=prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); (u / gcd(A051027(n), u)); };

a(n) = A353802(n) / A353804(n) = A353802(n) / gcd(A051027(n), A353802(n)).

A353803 a(n) = Product_{p^e||n} sigma(sigma(p^e)) - sigma(sigma(n)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 42, 21, 0, 0, 0, 0, 0, 0, 0, 141, 0, 0, 72, 36, 56, 0, 0, 0, 48, 54, 0, 168, 0, 0, 0, 45, 0, 0, 0, 0, 78, 21, 0, 0, 141, 0, 108, 54, 0, 192, 0, 0, 0, 0, 64, 381, 0, 0, 168, 317, 0, 0, 0, 0, 0, 0, 168, 192, 0, 0, 0, 72, 0, 336, 188, 0, 144, 126, 0, 126, 112

Offset: 1

Antti Karttunen, May 08 2022

nonn

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

Cf. A000203, A051027, A353802, A353804, A353805, A353806.
Cf. A336547 (positions of 0's), A336548 (positions of terms > 0).
Cf. also A353753.

A051027(n) = sigma(sigma(n));
A353803(n) = { my(f = factor(n)); (prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])) - A051027(n)); };

a(n) = A353802(n) - A051027(n).

A353805 a(n) = A051027(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 31, 10, 31, 1, 1, 1, 5, 13, 1, 3, 1, 1, 1, 13, 1, 1, 1, 1, 5, 57, 1, 1, 65, 1, 31, 13, 1, 5, 1, 1, 1, 1, 7, 403, 1, 1, 3, 403, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 13, 1, 3, 70, 1, 5, 13, 1, 13, 31, 1, 85, 13, 5, 1, 1, 13

Offset: 1

Antti Karttunen, May 08 2022

Denominator of fraction A353802(n) / A051027(n).

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

Cf. A000203, A051027, A353802, A353803, A353804, A353806 (numerators).
Positions of 1's is given by the union of A336547 and A353807.
Cf. also A353755, A353756.

A051027(n) = sigma(sigma(n));
A353805(n) = { my(f = factor(n)); (A051027(n) / gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])))); };

a(n) = A051027(n) / A353804(n).

A353754 Greatest common divisor of phi(sigma(n)) and Product_{p^e||n} phi(sigma(p^e)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 4, 8, 12, 2, 4, 12, 6, 8, 4, 30, 6, 24, 8, 12, 8, 4, 8, 16, 30, 12, 16, 24, 8, 8, 16, 36, 8, 6, 8, 72, 18, 16, 12, 8, 12, 16, 20, 24, 24, 8, 16, 60, 36, 60, 12, 6, 18, 32, 8, 32, 16, 8, 16, 24, 30, 32, 48, 126, 12, 16, 32, 36, 16, 16, 24, 96, 36, 36, 60, 48, 16, 24, 32, 60, 110, 12, 24, 48, 12, 40

Offset: 1

Antti Karttunen, May 08 2022

nonn

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

Cf. A000010, A000203, A062401, A353752, A353753, A353755, A353756.

Cf. also A353804.

A062401(n) = eulerphi(sigma(n));
A353754(n) = { my(f = factor(n)); gcd(A062401(n), prod(k=1, #f~, A062401(f[k, 1]^f[k, 2]))); };

a(n) = gcd(A062401(n), A353752(n)) = gcd(A062401(n), A353753(n)) = gcd(A353752(n), A353753(n)).

a(n) = A062401(n) / A353755(n) = A353752(n) / A353756(n).

cp's OEIS Frontend

A353802 Multiplicative with a(p^e) = sigma(sigma(p^e)).

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A353806 a(n) = A353802(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

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A353803 a(n) = Product_{p^e||n} sigma(sigma(p^e)) - sigma(sigma(n)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

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A353805 a(n) = A051027(n) / gcd(A051027(n), A353802(n)), where A051027(n) = sigma(sigma(n)), and A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

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A353754 Greatest common divisor of phi(sigma(n)) and Product_{p^e||n} phi(sigma(p^e)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

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