A300252 -id:A300252 - cp's OEIS frontend

A300251 Möbius transform of arithmetic derivative (A003415).

0, 1, 1, 3, 1, 3, 1, 8, 5, 5, 1, 8, 1, 7, 6, 20, 1, 11, 1, 14, 8, 11, 1, 20, 9, 13, 21, 20, 1, 14, 1, 48, 12, 17, 10, 28, 1, 19, 14, 36, 1, 20, 1, 32, 26, 23, 1, 48, 13, 29, 18, 38, 1, 39, 14, 52, 20, 29, 1, 36, 1, 31, 36, 112, 16, 32, 1, 50, 24, 34, 1, 68, 1, 37, 38, 56, 16, 38, 1, 88, 81, 41, 1, 52, 20, 43, 30, 84, 1, 50

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A008683, A300245, A300252, A300253.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A300251(n) = sumdiv(n,d,moebius(n/d)*A003415(d));

a(n) = Sum_{d|n} A008683(n/d)*A003415(d).

a(n) = A003415(n) - A300252(n).

A319684 Sum of A003415(d) over the divisors d of n, where A003415 is arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 5, 1, 7, 1, 17, 7, 9, 1, 27, 1, 11, 10, 49, 1, 34, 1, 37, 12, 15, 1, 83, 11, 17, 34, 47, 1, 54, 1, 129, 16, 21, 14, 114, 1, 23, 18, 117, 1, 68, 1, 67, 55, 27, 1, 227, 15, 64, 22, 77, 1, 142, 18, 151, 24, 33, 1, 190, 1, 35, 69, 321, 20, 96, 1, 97, 28, 90, 1, 326, 1, 41, 75, 107, 20, 110, 1, 325, 142, 45, 1, 244, 24, 47, 34, 219, 1, 243, 22

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Inverse Möbius transform of A003415.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A319683.

Cf. also A300251, A300252, A305809, A317835.

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] &, 91]] (* Michael De Vlieger, May 24 2021 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319684(n) = sumdiv(n,d,A003415(d));

a(n) = Sum_{d|n} A003415(d).

a(n) = A319683(n) + A003415(n).

A305809 Dirichlet convolution of A003415 (Arithmetic derivative) with itself.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 8, 1, 2, 0, 18, 0, 2, 2, 40, 0, 22, 0, 22, 2, 2, 0, 96, 1, 2, 12, 26, 0, 40, 0, 160, 2, 2, 2, 147, 0, 2, 2, 128, 0, 48, 0, 34, 28, 2, 0, 400, 1, 34, 2, 38, 0, 156, 2, 160, 2, 2, 0, 276, 0, 2, 32, 560, 2, 64, 0, 46, 2, 56, 0, 680, 0, 2, 36, 50, 2, 72, 0, 560, 90, 2, 0, 348, 2, 2, 2, 224, 0, 346, 2, 58, 2, 2, 2, 1440, 0, 46, 40, 267, 0, 88

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A300251, A300252.

Cf. A008578 (positions of 0's), A001248 (of 1's), A006881 (of 2's).

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A305809(n) = sumdiv(n,d,A003415(n/d)*A003415(d));

a(n) = Sum_{d|n} A003415(n/d)*A003415(d).

A300245 Filter sequence combining arithmetic derivative (A003415) with its Möbius transform (A300251).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 11, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 27, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 40, 60, 61, 62, 2, 63, 36, 64, 65, 66

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

Restricted growth sequence transform of ordered pair [A003415(n), A300251(n)].

			a(16) = a(28) (= 11) because both A003415(16) = A003415(28) = 32 and A300251(16) = A300251(28) = 20.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A300249, A300251, A300252, A300253.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A300251(n) = sumdiv(n,d,moebius(n/d)*A003415(d));
Aux300245(n) = [A003415(n), A300251(n)];
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300245(n))),"b300245.txt");

A319683 Sum of A003415(d) over the proper divisors d of n, where A003415 is arithmetic derivative.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 5, 1, 2, 0, 11, 0, 2, 2, 17, 0, 13, 0, 13, 2, 2, 0, 39, 1, 2, 7, 15, 0, 23, 0, 49, 2, 2, 2, 54, 0, 2, 2, 49, 0, 27, 0, 19, 16, 2, 0, 115, 1, 19, 2, 21, 0, 61, 2, 59, 2, 2, 0, 98, 0, 2, 18, 129, 2, 35, 0, 25, 2, 31, 0, 170, 0, 2, 20, 27, 2, 39, 0, 149, 34, 2, 0, 120, 2, 2, 2, 79, 0, 120, 2, 31, 2, 2, 2, 307, 0, 25, 22, 92, 0

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A319684.

Cf. also A300251, A300252, A305809, A317835.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319683(n) = sumdiv(n,d,(dA003415(d));

a(n) = Sum_{d|n, dA003415(d).

a(n) = A319684(n) - A003415(n).

A300253 GCD of arithmetic derivative (A003415) and its Möbius transform (A300251).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 2, 4, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 1, 1, 1, 16, 2, 1, 2, 4, 1, 1, 2, 4, 1, 1, 1, 16, 13, 1, 1, 16, 1, 1, 2, 2, 1, 3, 2, 4, 2, 1, 1, 4, 1, 1, 3, 16, 2, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 8, 2, 1, 1, 88, 27, 1, 1, 4, 2, 1, 2, 28, 1, 1, 2, 4, 2, 1, 2, 16, 1, 11, 1, 2, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A085731, A300245, A300251, A300252.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A300251(n) = sumdiv(n,d,moebius(n/d)*A003415(d));
A300253(n) = gcd(A003415(n), A300251(n));

a(n) = gcd(A003415(n), A300251(n)) = gcd(A003415(n), A300252(n)).

A317835 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A003415 (arithmetic derivative of n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 15, 1, 9, 1, 81, 23, 13, 1, 95, 1, 17, 15, 1499, 1, 127, 1, 151, 19, 25, 1, 393, 39, 29, 193, 207, 1, 87, 1, 6311, 27, 37, 23, 969, 1, 41, 31, 661, 1, 119, 1, 319, 259, 49, 1, 5499, 55, 295, 39, 375, 1, 769, 31, 929, 43, 61, 1, 593, 1, 65, 347, 50075, 35, 183, 1, 487, 51, 183, 1, 2751, 1, 77, 371, 543, 35, 215, 1, 9643, 5611, 85, 1

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(240) = -5067.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A063524, A046644 (denominators).

Cf. also A300251, A300252, A305809.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A317835aux(n) = if(1==n,n,(A003415(n)-sumdiv(n,d,if((d>1)&&(dA317835aux(d)*A317835aux(n/d),0)))/2);
A317835(n) = numerator(A317835aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003415(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

A300251 Möbius transform of arithmetic derivative (A003415).

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A319684 Sum of A003415(d) over the divisors d of n, where A003415 is arithmetic derivative.

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A305809 Dirichlet convolution of A003415 (Arithmetic derivative) with itself.

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A300245 Filter sequence combining arithmetic derivative (A003415) with its Möbius transform (A300251).

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A319683 Sum of A003415(d) over the proper divisors d of n, where A003415 is arithmetic derivative.

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A300253 GCD of arithmetic derivative (A003415) and its Möbius transform (A300251).

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A317835 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A003415 (arithmetic derivative of n) + A063524 (1, 0, 0, 0, ...).

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