A129283 -id:A129283 - cp's OEIS frontend

A347084 Dirichlet inverse of A129283, n + A003415(n).

1, -3, -4, 1, -6, 13, -8, 1, 1, 19, -12, -6, -14, 25, 25, 1, -18, -5, -20, -8, 33, 37, -24, -5, 1, 43, 2, -10, -30, -87, -32, 1, 49, 55, 49, 6, -38, 61, 57, -7, -42, -113, -44, -14, -8, 73, -48, -4, 1, -5, 73, -16, -54, -9, 73, -9, 81, 91, -60, 51, -62, 97, -10, 1, 85, -165, -68, -20, 97, -163, -72, 2, -74, 115

Offset: 1

Antti Karttunen, Aug 17 2021

sign

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

Cf. A003415, A129283, A347082, A347085, A347086, A348995 (positions of 1's).

Cf. also A346241, A348976.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA129283(n/d).
a(n) = A347085(n) - A129283(n).
a(n) = A347082(n) - A347086(n).

A348970 a(n) = A003959(n) - A129283(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 7, 1, 1, 0, 8, 0, 1, 1, 33, 0, 9, 0, 10, 1, 1, 0, 40, 1, 1, 10, 12, 0, 11, 0, 131, 1, 1, 1, 48, 0, 1, 1, 54, 0, 13, 0, 16, 12, 1, 0, 164, 1, 13, 1, 18, 0, 57, 1, 68, 1, 1, 0, 64, 0, 1, 14, 473, 1, 17, 0, 22, 1, 15, 0, 204, 0, 1, 14, 24, 1, 19, 0, 230, 67, 1, 0, 80, 1, 1, 1, 96, 0, 75, 1, 28

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

There are no negative terms. We prove this by induction over the prime factorization of n, showing that A348507(n) >= A003415(n) for all values of n >= 1. At n=1, both sequences have value 0, and at the primes both sequences obtain the value 1, so the base cases hold. We know that A348507(n)-(n/p) = (p+1)*A348507(n/p) for all prime factors p of n (see comment in A348507). With the arithmetic derivative we obtain respectively that A003415(n) = A003415(p*(n/p)) = A003415(p)*(n/p) + p*A003415(n/p) = (n/p) + p*A003415(n/p), for any prime factor p of n. Now A348507(p*(n/p)) >= A003415(p*(n/p)) iff A348507(p*(n/p)) - (n/p) >= A003415(p*(n/p)) - (n/p), that is, iff (p+1)*A348507(n/p) >= p*A003415(n/p), which indeed follows by the induction hypothesis, which assumes that A348507(x) >= A003415(x) for all proper divisors x of n.

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, A003959, A129283, A211991, A348029, A348507, A348971, A348972, A348973, A348974.

Cf. A008578 (positions of zeros), A001358 (positions of ones).

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n - d[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348970(n) = (A003959(n) - (n+A003415(n)));

a(n) = A003959(n) - A129283(n) = A003959(n) - (n+A003415(n)).
a(n) = A348029(n) - A211991(n).
a(n) = A348507(n) - A003415(n).
For all n >= 1, a(A001358(n)) = 1.

A347086 Difference between the Dirichlet inverse of -A168036, n - A003415(n) and the Dirichlet inverse of A129283, n + A003415(n), where A003415 is the Arithmetic derivative of n.

Original entry on oeis.org

0, 2, 2, 0, 2, -10, 2, 2, 0, -14, 2, 6, 2, -18, -16, 8, 2, 6, 2, 6, -20, -26, 2, 6, 0, -30, 2, 6, 2, 74, 2, 26, -28, -38, -24, 0, 2, -42, -32, 2, 2, 94, 2, 6, 6, -50, 2, 10, 0, 6, -40, 6, 2, 14, -32, -2, -44, -62, 2, -48, 2, -66, 6, 80, -36, 134, 2, 6, -52, 130, 2, 20, 2, -78, 6, 6, -36, 154, 2, -6, 12, -86, 2, -60

Offset: 1

Antti Karttunen, Aug 17 2021

sign

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

Cf. A003415, A129283, A168036, A347082, A347084.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347082 = DirInverseCorrect(vector(up_to,n,n-A003415(n)));
A347082(n) = v347082[n];
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];
A347086(n) = (A347082(n)-A347084(n));

a(n) = A347082(n) - A347084(n).

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).

A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, -2, 0, 0, 0, 6, 0, -3, 0, -2, 0, 0, 0, 0, 5, 0, 6, -2, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, -6, 7, -5, 0, -2, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, -3, 22, 0, 0, 0, -2, 0, 0, 0, -5, 0, 0, -5, -2, 0, 0, 0, -6, 21, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, -2, 0, 0, 0, -4, 0, -7, -3, 7

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Dirichlet convolution of this sequence with A349338 is A348976.

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

Cf. A003415, A129283, A230593, A349337, A349435 (Dirichlet inverse), A349436 (sum with it).

Cf. also A348976, A349338.

s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, sinv[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(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));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];
A349434(n) = sumdiv(n,d,A129283(n/d)*A349337(d));

a(n) = Sum_{d|n} A129283(n/d) * A349337(d).

A348972 a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 3, 4, 1, 6, 1, 8, 1, 1, 1, 12, 4, 14, 1, 1, 3, 18, 3, 20, 2, 1, 1, 24, 4, 1, 1, 2, 12, 30, 1, 32, 1, 1, 1, 1, 48, 38, 1, 1, 54, 42, 1, 44, 4, 12, 1, 48, 4, 1, 1, 1, 18, 54, 3, 1, 4, 1, 1, 60, 8, 62, 1, 2, 1, 1, 1, 68, 2, 1, 3, 72, 12, 74, 1, 2, 12, 1, 1, 80, 2, 1, 1, 84, 16, 1, 1, 1, 12, 90, 3, 1, 4, 1, 1, 1, 4, 98

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Cf. A003415, A003959, A129283, A348970, A348973, A348974.

Cf. also A343226.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n])), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348972(n) = gcd(A003959(n),(n+A003415(n)));

a(n) = gcd(A003959(n), A129283(n)) = gcd(A003959(n), n+A003415(n)).
a(n) = gcd(A003959(n), A348970(n)) = gcd(A129283(n), A348970(n)).
a(n) = A129283(n) / A348973(n) = A003959(n) / A348974(n).

A348973 Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 8, 1, 11, 1, 20, 15, 17, 1, 7, 1, 23, 23, 16, 1, 13, 1, 22, 31, 35, 1, 17, 35, 41, 27, 5, 1, 61, 1, 112, 47, 53, 47, 2, 1, 59, 55, 2, 1, 83, 1, 23, 7, 71, 1, 40, 63, 95, 71, 6, 1, 45, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 70, 95, 43, 1, 19, 1, 113, 65, 13, 95, 149, 1, 128, 189, 125, 1, 13, 107

Offset: 1

Antti Karttunen, Nov 06 2021

It is known that A129283(n) <= A003959(n) for all n (see A348970 for a proof), which implies that each ratio a(n)/A348974(n) is at most 1: 1/1, 1/1, 1/1, 8/9, 1/1, 11/12, 1/1, 20/27, 15/16, 17/18, 1/1, 7/9, 1/1, 23/24, 23/24, 16/27, 1/1, 13/16, 1/1, 22/27, 31/32, 35/36, 1/1, 17/27, 35/36, 41/42, 27/32, 5/6, 1/1, 61/72, 1/1, 112/243, etc.

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

Cf. A003415, A003959, A129283, A348970, A348972, A348974 (denominators).

Cf. also A345059.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Numerator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348973(n) = { my(u=n+A003415(n)); (u/gcd(A003959(n),u)); };

a(n) = A129283(n) / A348972(n) = A129283(n) / gcd(A003959(n), A129283(n)).

A348974 Denominator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 9, 1, 12, 1, 27, 16, 18, 1, 9, 1, 24, 24, 27, 1, 16, 1, 27, 32, 36, 1, 27, 36, 42, 32, 6, 1, 72, 1, 243, 48, 54, 48, 3, 1, 60, 56, 3, 1, 96, 1, 27, 8, 72, 1, 81, 64, 108, 72, 7, 1, 64, 72, 54, 80, 90, 1, 27, 1, 96, 64, 729, 84, 144, 1, 81, 96, 48, 1, 36, 1, 114, 72, 15, 96, 168, 1, 243, 256, 126, 1, 18, 108

Offset: 1

Antti Karttunen, Nov 06 2021

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

Cf. A003415, A003959, A129283, A348970, A348972, A348973 (numerators).

Cf. also A343227.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Denominator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348974(n) = { my(s=A003959(n)); (s/gcd(s,(n+A003415(n)))); };

a(n) = A003959(n) / A348972(n) = A003959(n) / gcd(A003959(n), A129283(n)).

A347085 Sum of A129283 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 21, 16, 36, 0, 22, 0, 48, 48, 49, 0, 34, 0, 36, 64, 72, 0, 63, 36, 84, 56, 50, 0, -26, 0, 113, 96, 108, 96, 102, 0, 120, 112, 101, 0, -30, 0, 78, 76, 144, 0, 156, 64, 90, 144, 92, 0, 126, 144, 139, 160, 180, 0, 203, 0, 192, 104, 257, 168, -38, 0, 120, 192, -34, 0, 230, 0, 228, 124, 134, 192

Offset: 1

Antti Karttunen, Aug 17 2021

sign

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

Cf. A003415, A129283, A347083, A347084, A347086.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(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));
v347084 = DirInverseCorrect(vector(up_to,n,A129283(n)));
A347084(n) = v347084[n];
A347085(n) = (A129283(n)+A347084(n));

a(n) = A129283(n) + A347084(n).

For n > 1, a(n) = -Sum_{d|n, 1A129283(d) * A347084(n/d).

A345059 a(n) = A129283(n) / gcd(sigma(n), A129283(n)), where A129283(n) is the sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 8, 1, 11, 1, 4, 15, 17, 1, 1, 1, 23, 23, 48, 1, 1, 1, 22, 31, 35, 1, 17, 35, 41, 27, 15, 1, 61, 1, 16, 47, 53, 47, 96, 1, 59, 55, 6, 1, 83, 1, 23, 14, 71, 1, 40, 21, 95, 71, 54, 1, 9, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 10, 95, 43, 1, 76, 1, 113, 65, 39, 95, 149, 1, 128, 189, 125, 1, 13, 107

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12800
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A129283, A211991, A343226, A343227.

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));
A345059(n) = { my(u=A129283(n)); (u/gcd(u,sigma(n))); };

a(n) = A129283(n) / A343226(n) = A129283(n) / gcd(A000203, A129283(n)).

cp's OEIS Frontend

A347084 Dirichlet inverse of A129283, n + A003415(n).

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A348970 a(n) = A003959(n) - A129283(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

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A347086 Difference between the Dirichlet inverse of -A168036, n - A003415(n) and the Dirichlet inverse of A129283, n + A003415(n), where A003415 is the Arithmetic derivative of n.

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

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A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).

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A348972 a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

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A348973 Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

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A348974 Denominator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

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