A329373 -id:A329373 - cp's OEIS frontend

A329347 Dirichlet convolution of the identity function with bigomega.

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).

A329372 Dirichlet convolution of the identity function with A156552.

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 8, 17, 12, 22, 16, 44, 32, 40, 32, 49, 64, 61, 128, 78, 56, 76, 256, 132, 32, 142, 50, 136, 512, 152, 1024, 129, 104, 274, 88, 209, 2048, 532, 188, 230, 4096, 256, 8192, 252, 148, 1048, 16384, 356, 80, 159, 356, 454, 32768, 240, 160, 392, 680, 2078, 65536, 504, 131072, 4128, 248, 321, 280, 464, 262144, 858, 1328, 400

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Equally, Dirichlet convolution of sigma (A000203) with A297112 (Möbius transform of A156552).

Antti Karttunen, Table of n, a(n) for n = 1..1024
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000203, A061395, A156552, A297112, A297167, A329374.

Cf. also A329371, A329373.

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
A329372(n) = sumdiv(n,d,(n/d)*A156552(d));

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));

a(n) = Sum_{d|n} d * A156552(n/d).
a(n) = Sum_{d|n} A000203(n/d) * A297112(d).
A000265(a(n)) = A329374(n).

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).

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

Original entry on oeis.org

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)).

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A329347 Dirichlet convolution of the identity function with bigomega.

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A329372 Dirichlet convolution of the identity function with A156552.

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A329371 Dirichlet convolution of the identity function with A246277.

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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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