A353805 -id:A353805 - 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)).

A353755 a(n) = A062401(n) / gcd(A062401(n), A353752(n)), where A062401(n) = phi(sigma(n)), and A353752(n) = Product_{p^e||n} phi(sigma(p^e)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 7, 1, 1, 3, 1, 2, 3, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4

Offset: 1

Antti Karttunen, May 08 2022

Numerator of fraction A062401(n) / A353752(n).

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

Cf. A000010, A000203, A062401, A353752, A353753, A353754, A353756 (denominators).
Cf. A336547 (positions of 1's), A336548 (positions of terms > 1).
Cf. also A353805.

A062401(n) = eulerphi(sigma(n));
A353755(n) = { my(f = factor(n)); (A062401(n) / gcd(A062401(n), prod(k=1, #f~, A062401(f[k, 1]^f[k, 2])))); };

a(n) = A062401(n) / A353754(n) = A062401(n) / gcd(A062401(n), A353752(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).

A353804 Greatest common divisor of sigma(sigma(n)) and 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.

Original entry on oeis.org

1, 4, 7, 8, 12, 28, 15, 24, 14, 3, 28, 56, 24, 60, 12, 32, 39, 56, 42, 96, 21, 7, 60, 168, 32, 96, 90, 120, 72, 3, 63, 104, 4, 12, 4, 112, 60, 168, 24, 18, 96, 84, 84, 224, 168, 15, 124, 224, 80, 128, 39, 3, 120, 360, 3, 360, 6, 18, 168, 96, 96, 252, 210, 128, 32, 1, 126, 312, 84, 1, 195, 336, 114, 240, 224, 336

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, A353802, A353803, A353805, A353806.

Cf. also A353754.

A051027(n) = sigma(sigma(n));
A353804(n) = { my(f = factor(n)); gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); };

a(n) = gcd(A051027(n), A353802(n)) = gcd(A051027(n), A353803(n)) = gcd(A353802(n), A353803(n)).

A353807 Numbers k such that A353802(k) / sigma(sigma(k)) is an integer > 1, where A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

Original entry on oeis.org

1819, 5088, 7215, 7276, 9487, 9523, 11895, 13303, 14235, 16371, 20179, 21079, 21255, 24531, 24751, 24931, 25824, 29104, 30615, 32224, 33855, 36199, 37635, 37948, 38092, 38664, 40443, 40515, 41847, 43831, 44655, 45475, 45695, 45883, 46995, 48043, 48399, 53835, 54015, 54568, 55747, 56899, 56928, 59599, 60495, 61035

Offset: 1

Antti Karttunen, May 08 2022

nonn

Numbers k such that A353805(k) = 1, but A353806(k) > 1.

			A353802(1819) = 10920 = 2*A051027(1819) = 2*5460, therefore 1819 is included as a term.

Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A353802, A353805, A353806.

Cf. also A336561.

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])))); };
A353806(n) = { my(f = factor(n), u=prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); (u / gcd(A051027(n), u)); };
isA353807(n) = ((1==A353805(n)) && (1!=A353806(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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A353755 a(n) = A062401(n) / gcd(A062401(n), A353752(n)), where A062401(n) = phi(sigma(n)), and A353752(n) = Product_{p^e||n} phi(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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A353804 Greatest common divisor of sigma(sigma(n)) and 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.

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A353807 Numbers k such that A353802(k) / sigma(sigma(k)) is an integer > 1, where A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

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