cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A300251 Möbius transform of arithmetic derivative (A003415).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 08 2018

Keywords

Links

Crossrefs

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

Programs

  • PARI
    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));

Formula

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

Views

Author

Antti Karttunen, Mar 08 2018

Keywords

Links

Crossrefs

Cf. A003415, A300245, A300251, A300253.
Cf. A001248 (seems to give the positions of 1's), A006881 (seems to give the positions of 2's).

Programs

  • PARI
    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));

Formula

a(n) = A003415(n) - A300251(n).
a(n) = -Sum_{d|n, dA008683(n/d)*A003415(d).

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

Views

Author

Antti Karttunen, Mar 08 2018

Keywords

Links

Crossrefs

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

Programs

Formula

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

A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 9, 4, 4, 2, 10, 3, 4, 11, 12, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 16, 2, 17, 2, 18, 19, 4, 2, 20, 3, 21, 4, 22, 2, 23, 4, 24, 4, 4, 2, 25, 2, 4, 26, 27, 4, 28, 2, 29, 4, 30, 2, 31, 2, 4, 32, 33, 4, 34, 2, 35, 36, 4, 2, 37, 4, 4, 4, 38, 2, 39, 4, 40, 4, 4, 4, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Comments

Restricted growth sequence transform of A319356.
The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.
For all i, j:
a(i) = a(j) => A000005(i) = A000005(j),
a(i) = a(j) => A319683(i) = A319683(j),
a(i) = a(j) => A319686(i) = A319686(j),
a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

Examples

			Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

Links

Crossrefs

Cf. A003415, A319356.
Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
Cf. also A300245, A300249, A319353, A319693.

Programs

  • PARI
    up_to = 65537;
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; };
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
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to,n,A319356(n)));
A319357(n) = v319357[n];

A328767 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where for n>1, f(n) = [A003415(i), A328382(i)], and f(1) = 1.

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, 14, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 22, 34, 35, 36, 2, 37, 38, 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, 60, 61, 62, 63, 2, 64, 65, 66, 67, 68, 69, 70, 2, 71, 72, 73, 2, 74, 2
Offset: 1

Views

Author

Antti Karttunen, Oct 28 2019

Keywords

Comments

For all i, j:
A305800(i) = A305800(j) => a(i) = a(j),
a(i) = a(j) => A327858(i) = A327858(j),
a(i) = a(j) => A328098(i) = A328098(j),

Links

Crossrefs

Cf. A003415, A305800, A327858, A328098, A328382.
Cf. also A300245, A300249, A327970.

Programs

  • PARI
    up_to = 65537;
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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328382(n) = (A276086(n)%A003415(n));
Aux328767(n) = if(1==n,1,[A003415(n), A328382(n)]);
v328767 = rgs_transform(vector(up_to, n, Aux328767(n)));
A328767(n) = v328767[n];
Showing 1-5 of 5 results.

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