A300253 -id:A300253 - 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).

A300252 Difference between arithmetic derivative (A003415) and its Möbius transform (A300251).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 8, 0, 2, 2, 12, 0, 10, 0, 10, 2, 2, 0, 24, 1, 2, 6, 12, 0, 17, 0, 32, 2, 2, 2, 32, 0, 2, 2, 32, 0, 21, 0, 16, 13, 2, 0, 64, 1, 16, 2, 18, 0, 42, 2, 40, 2, 2, 0, 56, 0, 2, 15, 80, 2, 29, 0, 22, 2, 25, 0, 88, 0, 2, 17, 24, 2, 33, 0, 88, 27, 2, 0, 72, 2, 2, 2, 56, 0, 73, 2, 28, 2, 2, 2, 160, 0, 22, 19, 62, 0, 41, 0, 64, 27

Offset: 1

Antti Karttunen, Mar 08 2018

nonn

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

Cf. A003415, A300245, A300251, A300253.

Cf. A001248 (seems to give the positions of 1's), A006881 (seems to give the positions 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
A300252(n) = -sumdiv(n,d,(dA003415(d));

a(n) = A003415(n) - A300251(n).

a(n) = -Sum_{d|n, dA008683(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");

A340070 a(n) = gcd(A003415(n), A069359(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 2, 1, 9, 8, 8, 1, 3, 1, 2, 10, 13, 1, 4, 5, 15, 9, 2, 1, 31, 1, 16, 14, 19, 12, 30, 1, 21, 16, 4, 1, 41, 1, 2, 3, 25, 1, 8, 7, 5, 20, 2, 1, 9, 16, 4, 22, 31, 1, 2, 1, 33, 3, 32, 18, 61, 1, 2, 26, 59, 1, 12, 1, 39, 5, 2, 18, 71, 1, 8, 27, 43, 1, 2, 22, 45, 32, 4, 1, 3, 20, 2, 34, 49, 24

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

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

Cf. A003415, A005117, A069359, A329039.

Cf. also A085731, A300253, A327858.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A340070(n) = gcd(A003415(n), A069359(n));

a(n) = gcd(A003415(n), A069359(n)) = gcd(A003415(n), A329039(n)).

For all squarefree k, a(k) = A003415(k) = A069359(k).

cp's OEIS Frontend

A300251 Möbius transform of arithmetic derivative (A003415).

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A300252 Difference between arithmetic derivative (A003415) and its Möbius transform (A300251).

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

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A340070 a(n) = gcd(A003415(n), A069359(n)).

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