A329372 -id:A329372 - cp's OEIS frontend

A329374 a(1) = 0; for n > 1, a(n) = A000265(A329372(n)), where A329372 is Dirichlet convolution of the identity function with A156552.

0, 1, 1, 5, 1, 3, 1, 17, 3, 11, 1, 11, 1, 5, 1, 49, 1, 61, 1, 39, 7, 19, 1, 33, 1, 71, 25, 17, 1, 19, 1, 129, 13, 137, 11, 209, 1, 133, 47, 115, 1, 1, 1, 63, 37, 131, 1, 89, 5, 159, 89, 227, 1, 15, 5, 49, 85, 1039, 1, 63, 1, 129, 31, 321, 35, 29, 1, 429, 83, 25, 1, 605, 1, 4115, 111, 409, 15, 101, 1, 307, 45, 8213, 1, 13, 65, 8203, 655, 179, 1, 335, 25

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..32768
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A061395, A156552, A297112, A297167, A329372, A329373.

A000265(n) = (n/2^valuation(n, 2));
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A329372(n) = sumdiv(n,d,sigma(n/d)*A297112(d));
A329374(n) = if(1==n, 0, A000265(A329372(n)));

a(1) = 0; and for n > 1, a(n) = A000265(A329372(n)).

A329347 Dirichlet convolution of the identity function with bigomega.

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 11, 5, 9, 1, 23, 1, 11, 10, 26, 1, 28, 1, 31, 12, 15, 1, 59, 7, 17, 18, 39, 1, 54, 1, 57, 16, 21, 14, 87, 1, 23, 18, 81, 1, 68, 1, 55, 43, 27, 1, 135, 9, 52, 22, 63, 1, 94, 18, 103, 24, 33, 1, 166, 1, 35, 53, 120, 20, 96, 1, 79, 28, 90, 1, 218, 1, 41, 59, 87, 20, 110, 1, 187, 58, 45, 1, 212, 24, 47, 34, 147, 1, 207, 22

Offset: 1

Antti Karttunen, Nov 12 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

Cf. A001222.

Cf. also A323599, A329371, A329372, A329373.

A329347(n) = sumdiv(n,d,(n/d)*bigomega(d));

a(n) = Sum_{d|n} d * A001222(n/d).

A329373 Dirichlet convolution of the identity function with A322993.

Original entry on oeis.org

0, 1, 1, 5, 1, 10, 1, 17, 6, 16, 1, 40, 1, 26, 13, 49, 1, 49, 1, 66, 19, 46, 1, 124, 8, 80, 25, 108, 1, 114, 1, 129, 31, 148, 17, 185, 1, 278, 49, 206, 1, 182, 1, 192, 65, 538, 1, 340, 10, 111, 85, 330, 1, 190, 25, 336, 151, 1056, 1, 428, 1, 2082, 97, 321, 35, 318, 1, 606, 283, 258, 1, 557, 1, 4136, 87, 1128, 23, 530, 1, 566, 90, 8236, 1, 684, 55, 16430

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Equally, Dirichlet convolution of sigma (A000203) with A322994 (Möbius transform of A322993).

Antti Karttunen, Table of n, a(n) for n = 1..2001
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..32768
Index entries for sequences computed from indices in prime factorization

Cf. A000203, A000265, A156552, A322993, A322994, A324542, A329372, A329374.

A000265(n) = (n/2^valuation(n, 2));
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A322993(n) = if(1==n,0,A000265(A156552(n)));
A329373(n) = sumdiv(n,d,(n/d)*A322993(d));

a(n) = Sum_{d|n} d * A322993(n/d).

a(n) = Sum_{d|n} A000203(n/d) * A322994(d).

A329371 Dirichlet convolution of the identity function with A246277.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 12, 1, 28, 1, 16, 11, 32, 1, 37, 1, 44, 15, 24, 1, 80, 7, 28, 19, 60, 1, 82, 1, 80, 21, 36, 15, 128, 1, 40, 27, 128, 1, 114, 1, 92, 49, 48, 1, 208, 9, 89, 33, 108, 1, 146, 21, 176, 39, 60, 1, 284, 1, 64, 69, 192, 25, 174, 1, 140, 45, 170, 1, 364, 1, 76, 70, 156, 21, 210, 1, 336, 65, 84, 1, 396, 33, 88, 55, 272, 1, 368, 25, 188, 63

Offset: 1

Antti Karttunen, Nov 12 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 computed from indices in prime factorization

Cf. A246277.

Cf. also A305796, A323599, A329347, A329372, A329373.

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A329371(n) = sumdiv(n,d,(n/d)*A246277(d));

a(n) = Sum_{d|n} d * A246277(n/d).

cp's OEIS Frontend

A329374 a(1) = 0; for n > 1, a(n) = A000265(A329372(n)), where A329372 is Dirichlet convolution of the identity function with A156552.

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A329347 Dirichlet convolution of the identity function with bigomega.

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A329373 Dirichlet convolution of the identity function with A322993.

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PARI

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A329371 Dirichlet convolution of the identity function with A246277.

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Author

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PARI

Formula