A300251 -id:A300251 - cp's OEIS frontend

A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 2, 1, 7, 6, 1, 1, 1, 1, 4, 8, 11, 1, 2, 1, 13, 1, 6, 1, 14, 1, 1, 12, 17, 10, 0, 1, 19, 14, 4, 1, 20, 1, 10, 4, 23, 1, 2, 1, 1, 18, 12, 1, 1, 14, 6, 20, 29, 1, 8, 1, 31, 6, 1, 16, 32, 1, 16, 24, 34, 1, 0, 1, 37, 2, 18, 16, 38, 1, 4, 1, 41, 1, 12, 20, 43, 30, 10, 1, 4, 18, 22, 32, 47

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Conjecture 1: After the initial zero, the positions of other zeros is given by A036785.

Conjecture 2: No negative terms. Checked up to n = 2^24.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A008683, A036785, A342001, A347232 [= a(A276086(n))].

Cf. also A300251, A300717, A347234, A347235, A347395.

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));
A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));

a(n) = Sum_{d|n} A008683(n/d) * A342001(d).
Dirichlet g.f.: Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 08 2022
Sum_{k=1..n} a(k) ~ c * A065464 * n^2 / 2, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023

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).

A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 6, 1, 2, 0, 16, 0, 2, 2, 24, 0, 19, 0, 20, 2, 2, 0, 72, 1, 2, 9, 24, 0, 40, 0, 80, 2, 2, 2, 111, 0, 2, 2, 96, 0, 48, 0, 32, 25, 2, 0, 256, 1, 29, 2, 36, 0, 117, 2, 120, 2, 2, 0, 244, 0, 2, 29, 240, 2, 64, 0, 44, 2, 56, 0, 446, 0, 2, 31, 48, 2, 72, 0, 352, 54, 2, 0, 308, 2, 2, 2, 168, 0, 304, 2, 56

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A003415 (arithmetic derivative) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A347131.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001221, A003415, A069359, A347131.

Cf. also A300251, A305809, A347132, A347134, A347135.

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
A347133(n) = sumdiv(n,d,A003415(n/d)*A069359(d));

a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A347131(d).

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.

A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 14, 6, 6, 1, 14, 1, 8, 7, 46, 1, 18, 1, 22, 9, 12, 1, 46, 10, 14, 30, 30, 1, 22, 1, 146, 13, 18, 11, 60, 1, 20, 15, 74, 1, 30, 1, 46, 36, 24, 1, 146, 14, 40, 19, 54, 1, 78, 15, 102, 21, 30, 1, 74, 1, 32, 48, 454, 17, 46, 1, 70, 25, 46, 1, 192, 1, 38, 50, 78, 17, 54, 1, 238, 138, 42, 1, 102, 21

Offset: 1

Antti Karttunen, Nov 05 2021

Möbius transform of A348507.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003959, A003968, A008683, A300251, A348507, A348970.

Cf. A005596, A104141.

f1[p_, e_] := p*(p + 1)^(e - 1); f2[p_, e_] := (p - 1)*p^(e - 1); a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A348971(n) = { my(f=factor(n),m1=1,m2=1,p); for(i=1, #f~, p = f[i, 1]; m1 *= p*(p+1)^(f[i, 2]-1); m2 *= (p-1)*p^(f[i, 2]-1)); (m1-m2); };

A348971(n) = { my(f=factor(n),p); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f)-eulerphi(n); }

a(n) = A003968(n) - A000010(n).
a(n) = Sum_{d|n} A008683(n/d) * A348507(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A104141 * (1/A005596 - 1) = 0.5088692487... . - Amiram Eldar, Oct 05 2023

A348976 Möbius transform of A129283, which is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 2, 3, 5, 5, 5, 7, 12, 11, 9, 11, 12, 13, 13, 14, 28, 17, 17, 19, 22, 20, 21, 23, 28, 29, 25, 39, 32, 29, 22, 31, 64, 32, 33, 34, 40, 37, 37, 38, 52, 41, 32, 43, 52, 50, 45, 47, 64, 55, 49, 50, 62, 53, 57, 54, 76, 56, 57, 59, 52, 61, 61, 72, 144, 64, 52, 67, 82, 68, 58, 71, 92, 73, 73, 78, 92, 76, 62, 79, 120

Offset: 1

Antti Karttunen, Nov 09 2021

nonn

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

Cf. A000010, A003415, A008683, A129283, A300251, A347084, A348970.

f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, MoebiusMu[#]*d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A348976(n) = sumdiv(n,d,moebius(n/d)*A129283(d));

a(n) = Sum_{d|n} A008683(n/d) * A129283(d).

a(n) = A000010(n) + A300251(n).

A349123 a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 8, 1, 15, 1, 40, 12, 21, 1, 96, 1, 27, 24, 160, 1, 126, 1, 144, 30, 39, 1, 440, 20, 45, 90, 192, 1, 279, 1, 560, 42, 57, 36, 720, 1, 63, 48, 680, 1, 369, 1, 288, 234, 75, 1, 1680, 28, 270, 60, 336, 1, 810, 48, 920, 66, 93, 1, 1656, 1, 99, 306, 1792, 54, 549, 1, 432, 78, 531, 1, 3120, 1, 117, 330, 480, 54, 639

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

This sequence is the Dirichlet convolution of at least the following pairs of sequences:
  - A003415 (the arithmetic derivative) with A038040,
  - A000027 (the identity function) with A347130,
  - A000203 (sigma) with A347131,
  - A018804 with A319684,
  - A060640 with A300251.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000027, A000203, A003415, A003557, A018804, A060640, A300251, A319684, A347130, A349124.

Cf. also A348983, A349143.

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A038040(n) = (n*numdiv(n));
A349123(n) = sumdiv(n,d,A038040(d)*A003415(n/d));

a(n) = Sum_{d|n} A038040(d) * A003415(n/d).
a(n) = Sum_{d|n} d * A347130(n/d).
a(n) = Sum_{d|n} A000203(d) * A347131(n/d).
a(n) = Sum_{d|n} A018804(d) * A319684(n/d).
a(n) = Sum_{d|n} A060640(d) * A300251(n/d).
For all n >= 1, A348983(n) <= a(n) <= A349143(n).
a(n) = A003557(n) * A349124(n).

A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 8, 4, 3, 1, 5, 1, 4, 3, 20, 1, 6, 1, 8, 4, 6, 1, 12, 7, 7, 14, 11, 1, 3, 1, 48, 6, 9, 5, 14, 1, 10, 7, 20, 1, 4, 1, 17, 8, 12, 1, 28, 10, 13, 9, 20, 1, 18, 7, 28, 10, 15, 1, 6, 1, 16, 11, 112, 8, 6, 1, 26, 12, 5, 1, 32, 1, 19, 11, 29, 8, 7, 1, 48, 46, 21, 1, 8, 10, 22, 15, 44, 1, 6, 9, 35, 16

Offset: 1

Antti Karttunen, Nov 21 2021

Dirichlet convolution of A349394 with A349432.

Dirichlet convolution with A349136 gives A300251.

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

Cf. A003415, A003602, A349134, A349394, A349432.

Cf. also A300251, A347955, A349136, A349431.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349380(n) = sumdiv(n,d,A003415(d)*A349134(n/d));

a(n) = Sum_{d|n} A003415(n/d) * A349134(d).

a(n) = Sum_{d|n} A349394(n/d) * A349432(d).

cp's OEIS Frontend

A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

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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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A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

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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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A348971 a(n) = Product(p*(p+1)^(e-1)) - Product((p-1)*p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

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A348976 Möbius transform of A129283, which is sum of n and its arithmetic derivative.

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A349123 a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.

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A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

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A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.