A300245 -id:A300245

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A300251

Möbius transform of arithmetic derivative (A003415).

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

FORMULA

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

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

PROG

(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));

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 08 2018

STATUS

approved

A300252

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

FORMULA

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

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

PROG

(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, (d<n)*moebius(n/d)*A003415(d));

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).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 08 2018

STATUS

approved

A300253

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

LINKS

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

FORMULA

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

PROG

(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));

A300253(n) = gcd(A003415(n), A300251(n));

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 08 2018

STATUS

approved

A319357

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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]

LINKS

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

EXAMPLE

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).

PROG

(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(d<n, m *= prime(1+A003415(d)))); (m); };

v319357 = rgs_transform(vector(up_to, n, A319356(n)));

A319357(n) = v319357[n];

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 02 2018

STATUS

approved

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.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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

PROG

(PARI)

up_to = 65537;

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];

CROSSREFS

Cf. A003415, A305800, A327858, A328098, A328382.

Cf. also A300245, A300249, A327970.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 28 2019

STATUS

approved

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