A055615 -id:A055615 - cp's OEIS frontend

A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).

0, 1, 1, 2, 1, 0, 1, 4, 3, 0, 1, 0, 1, 0, 0, 8, 1, 0, 1, 0, 0, 0, 1, 0, 5, 0, 9, 0, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 7, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 32, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 27, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

nonn

Dirichlet convolution of this sequence with Euler phi (A000010) is A300251.
Convolving this sequence with sigma (A000203) produces A319684.
With a(1) = 1 instead of 0, this would be the Dirichlet convolution of A129283 (A003415(n)+n) with A055615. Thus when we subtract A063524 from that convolution, we get this sequence. (See also A349434). Compare also to the convolution of A069359 (sequence agreeing with A003415 on squarefree numbers) with A055615, which is the characteristic function of primes, A010051. - Antti Karttunen, Nov 20 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
P. Haukkanen, J. K. Merikoski and T. Tossavainen, Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative, Mathematical Communications 25 (2020), 107-115.

Cf. A003415, A003557, A055615, A063524, A069513, A100995, A120007, A129283, A246655.

Cf. also A000010, A000203, A069359, A300251, A319684, A327564, A349340, A349396, A349434, A349618, A349619, A349620, A349621.

import Math.NumberTheory.Primes
a n = case factorise n of
    [(p,e)] -> unPrime p^(e-1) :: Int
     -> 0 -- _Sebastian Karlsson, Nov 19 2021

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A055615(n) = (n*moebius(n));
A349394(n) = sumdiv(n,d,A003415(n/d)*A055615(d));

A349394(n) = { my(p=0,e); if((e=isprimepower(n,&p)),p^(e-1),0); }; \\ (After Sebastian Karlsson's new formula) - Antti Karttunen, Nov 20 2021

a(n) = Sum_{d|n} A003415(n/d) * A055615(d).
a(n) = 0 unless n is a prime power (A246655), in which case a(p^e) = p^(e-1). - Sebastian Karlsson, Nov 19 2021
a(n) = A003557(n) * A069513(n). [From above] - Antti Karttunen, Nov 20 2021
Dirichlet g.f.: Sum_{p prime} 1/(p^s-p) [Follows from the D.g.f. of A003415 proved by Haukkanen et al.]. - Sebastian Karlsson, Nov 25 2021
Sum_{k=1..n} a(k) has an average value c*n, where c = A137245 = Sum_{primes p} 1/(p*log(p)) = 1.63661632335... - Vaclav Kotesovec, Mar 03 2023

Added Sebastian Karlsson's formula as the new primary definition - Antti Karttunen, Nov 20 2021

A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, -1, -1, -1, -2, 1, -3, -1, -1, 2, -5, 1, -6, 3, 4, -1, -8, 1, -9, 2, 6, 5, -11, 1, -2, 6, -1, 3, -14, -4, -15, -1, 10, 8, 12, 1, -18, 9, 12, 2, -20, -6, -21, 5, 4, 11, -23, 1, -3, 2, 16, 6, -26, 1, 20, 3, 18, 14, -29, -4, -30, 15, 6, -1, 24, -10, -33, 8, 22, -12, -35, 1, -36, 18, 4, 9, 30, -12, -39, 2, -1, 20

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Dirichlet convolution of this sequence with A000010 gives A349136, which also proves the formula involving A023900.

Convolution with A000203 gives A349371.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Sequence A297381 negated.
Cf. A003602, A023900, A055615, A297381, A349432 (Dirichlet inverse), A349433 (sum with it).
Cf. also A000010, A000203, A349136, A349371, and also A349444, A349447.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, # * MoebiusMu [#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349431(n) = sumdiv(n,d,A003602(n/d)*A055615(d));

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349431(n) = if(!bitand(n,n-1),A023900(n),A023900(n)/2);

a(n) = Sum_{d|n} A003602(n/d) * A055615(d).
a(n) = A023900(n) when n is a power of 2, and a(n) = A023900(n)/2 for all other numbers.
a(n) = -A297381(n).

A349387 Dirichlet convolution of A003961 with A055615 (Dirichlet inverse of n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 4, 9, 10, 2, 2, 6, 4, 4, 4, 27, 2, 10, 4, 6, 8, 2, 6, 18, 14, 4, 50, 12, 2, 4, 6, 81, 4, 2, 8, 30, 4, 4, 8, 18, 2, 8, 4, 6, 20, 6, 6, 54, 44, 14, 4, 12, 6, 50, 4, 36, 8, 2, 2, 12, 6, 6, 40, 243, 8, 4, 4, 6, 12, 8, 2, 90, 6, 4, 28, 12, 8, 8, 4, 54, 250, 2, 6, 24, 4, 4, 4, 18, 8, 20, 16, 18, 12, 6

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because A003961 and A055615 are.
Convolving this with A000010 gives A003972, and convolving this with A000203 gives A003973.
Multiplicative with a(p^e) = nextprime(p)^e - p * nextprime(p)^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

Cf. A000040, A001223, A003961, A055615, A151800, A349388 (Dirichlet inverse), A349389 (sum with it), A378606 (Möbius transform).

Cf. also A000010, A000203, A003972, A003973, A347236, A349381.

f[p_,e_] := (q = NextPrime[p])^e - p * q^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A055615(n) = (n*moebius(n));
A349387(n) = sumdiv(n,d,A003961(n/d)*A055615(d));

a(n) = Sum_{d|n} A003961(n/d) * A055615(d).

For all n >= 1, a(A000040(n)) = A001223(n).

A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, -1, 0, 1, -2, 1, 0, 0, -2, 1, -6, 1, -2, 0, 0, 1, -2, -3, 0, -3, -2, 1, 0, 1, -3, 0, 0, 0, 2, 1, 0, 0, -2, 1, 0, 1, -2, -6, 0, 1, -2, -5, -20, 0, -2, 1, -6, 0, -2, 0, 0, 1, 0, 1, 0, -6, -4, 0, 0, 1, -2, 0, 0, 1, 8, 1, 0, -20, -2, 0, 0, 1, -2, -5, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A346485.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A003557, A055615, A342001.
Cf. A346485, A347234,  A347235, A347395, A347954, A347959, A347961, A347963 for Dirichlet convolutions of A342001 with other sequences.
Cf. also A349394.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A055615(n) = (n*moebius(n));
A349396(n) = sumdiv(n,d,A342001(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(d) * A342001(n/d).

A261869 First differences of A055615.

Original entry on oeis.org

-3, -1, 3, -5, 11, -13, 7, 0, 10, -21, 11, -13, 27, 1, -15, -17, 17, -19, 19, 21, 1, -45, 23, 0, 26, -26, 0, -29, -1, -1, 31, 33, 1, 1, -35, -37, 75, 1, -39, -41, -1, -1, 43, 0, 46, -93, 47, 0, 0, 51, -51, -53, 53, 55, -55, 57, 1, -117, 59, -61, 123, -62, 0

Offset: 1

Reinhard Zumkeller, Sep 04 2015

sign

a(n) = (n+1)*mu(n+1) - n*mu(n), where mu = A008683 (Moebius function);
a(A068781(n)) = 0;
abs(a(n)) <= 2*n + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008683, A055615, A068781, A261890 (first differences).

import Data.List (genericIndex)
a261869 n = genericIndex a261869_list (n - 1)
a261869_list = zipWith (-) (tail a055615_list) a055615_list

#[[2]]-#[[1]]&/@Partition[Table[n*MoebiusMu[n],{n,70}],2,1] (* Harvey P. Dale, Apr 17 2016 *)

A261890 Second differences of A055615, first differences of A261869.

Original entry on oeis.org

2, 4, -8, 16, -24, 20, -7, 10, -31, 32, -24, 40, -26, -16, -2, 34, -36, 38, 2, -20, -46, 68, -23, 26, -52, 26, -29, 28, 0, 32, 2, -32, 0, -36, -2, 112, -74, -40, -2, 40, 0, 44, -43, 46, -139, 140, -47, 0, 51, -102, -2, 106, 2, -110, 112, -56, -118, 176, -120

Offset: 1

Reinhard Zumkeller, Sep 05 2015

sign

a(n) = (n+2)*mu(n+2) - 2*(n+1)*mu(n+1) + n*mu(n), where mu = A008683 (Moebius function);
a(A114180(n)) = 0;
abs(a(n)) <= 4*(n+1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008683, A055615, A261869, A114180.

import Data.List (genericIndex)
a261890 n = genericIndex a261890_list (n - 1)
a261890_list = zipWith (-) (tail a261869_list) a261869_list

Differences[Table[n MoebiusMu[n],{n,80}],2] (* Harvey P. Dale, Mar 12 2023 *)

a(n) = A055615(n+2) - 2*A055615(n+1) + A055615(n).

A349437 Dirichlet convolution of A252463 with A055615 (Dirichlet inverse of n), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, -1, -1, 0, -2, 2, -2, 0, -2, 4, -4, 0, -2, 4, 2, 0, -4, 4, -2, 0, 2, 8, -4, 0, -6, 4, -4, 0, -6, -4, -2, 0, 4, 8, 4, 0, -6, 4, 2, 0, -4, -4, -2, 0, 4, 8, -4, 0, -10, 12, 4, 0, -6, 8, 8, 0, 2, 12, -6, 0, -2, 4, 4, 0, 4, -8, -6, 0, 4, -8, -4, 0, -2, 12, 6, 0, 8, -4, -6, 0, -8, 8, -4, 0, 8, 4, 6, 0, -6, -8, 4, 0, 2

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Dirichlet convolution of this sequence with Euler phi (A000010) is A348045.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A055615, A064989, A252463, A349438 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A348045.

f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A055615(n) = (n*moebius(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A349437(n) = sumdiv(n,d,A055615(n/d)*A252463(d));

a(n) = Sum_{d|n} A055615(n/d) * A252463(d).

A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, -1, -2, 2, -4, 2, -6, 0, 6, 4, -10, -4, -12, 6, 8, 0, -16, -6, -18, -8, 12, 10, -22, 0, 20, 12, 0, -12, -28, -8, -30, 0, 20, 16, 24, 12, -36, 18, 24, 0, -40, -12, -42, -20, -24, 22, -46, 0, 42, -20, 32, -24, -52, 0, 40, 0, 36, 28, -58, 16, -60, 30, -36, 0, 48, -20, -66, -32, 44, -24, -70, 0, -72, 36, -40, -36

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative because A055615 and A057521 are.

Convolving this with Euler phi (A000010) produces A349379.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A055615, A057521, A349442 (Dirichlet inverse), A349443 (sum with it).

Cf. also A097945, A349379.

f[p_, e_] := Which[e > 2, 0, e == 2, p^2 - p, e == 1, 1 - p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A055615(n) = (n*moebius(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349441(n) = sumdiv(n,d,A057521(n/d)*A055615(d));

a(n) = Sum_{d|n} A057521(n/d) * A055615(d).

Multiplicative with a(p^e) = 1 - p is e = 1, p^2 - p if e = 2, and 0 otherwise. - Amiram Eldar, Nov 19 2021

A349617 Dirichlet convolution of A064664 (the inverse permutation of EKG-permutation) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 2, -1, 5, -6, 7, 2, -9, -11, 9, 2, 15, -15, -29, 1, 16, 18, 18, 5, -41, -21, 20, -4, -26, -29, 4, 7, 28, 64, 30, -3, -61, -34, -80, 9, 30, -38, -81, -6, 33, 92, 38, 14, 51, -44, 42, 10, -48, 53, -99, 6, 47, 4, -102, -17, -111, -58, 48, -4, 54, -62, 69, 2, -151, 146, 61, 18, -131, 157, 63, -3, 65, -68, 92, 18

Offset: 1

Antti Karttunen, Nov 23 2021

Dirichlet convolution of this sequence with A000010 (Euler phi) is A304526 (Möbius transform of the inverse permutation of EKG-sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A055615, A064413, A064664, A349616 (Dirichlet inverse).

Cf. also A000010, A304526, A349614.

A055615(n) = (n*moebius(n));
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
A349617(n) = sumdiv(n,d,A064664(d)*A055615(n/d));

a(n) = Sum_{d|n} A064664(d) * A055615(n/d).

A349624 Dirichlet convolution of A326042 with A055615 (Dirichlet inverse of n), where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

1, -1, -1, 9, -4, 1, -5, -19, 23, 4, -6, -9, -9, 5, 4, 43, -14, -23, -17, -36, 5, 6, -17, 19, 29, 9, -65, -45, -28, -4, -14, -43, 6, 14, 20, 207, -27, 17, 9, 76, -34, -5, -41, -54, -92, 17, -39, -43, 71, -29, 14, -81, -47, 65, 24, 95, 17, 28, -30, 36, -48, 14, -115, 981, 36, -6, -63, -126, 17, -20, -40, -437, -70, 27

Offset: 1

Antti Karttunen, Nov 26 2021

Multiplicative because A055615 and A326042 are.

Cf. A000203, A003961, A055615, A064989, A326042, A349625 (Dirichlet inverse), A349626.

Cf. also A348736, A349573.

f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055615(n) = (n*moebius(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A349624(n) = sumdiv(n,d,A055615(n/d)*A326042(d));

a(n) = Sum_{d|n} A055615(d) * A326042(n/d).

For all n >= 1, Sum_{d|n, dA326042(n) - n = -A348736(n).

cp's OEIS Frontend

A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).

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A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

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A349387 Dirichlet convolution of A003961 with A055615 (Dirichlet inverse of n), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

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A261869 First differences of A055615.

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A261890 Second differences of A055615, first differences of A261869.

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A349437 Dirichlet convolution of A252463 with A055615 (Dirichlet inverse of n), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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