A348982 -id:A348982 - cp's OEIS frontend

A322582 a(n) = n - A003958(n), where A003958 is fully multiplicative with a(p) = (p-1).

0, 1, 1, 3, 1, 4, 1, 7, 5, 6, 1, 10, 1, 8, 7, 15, 1, 14, 1, 16, 9, 12, 1, 22, 9, 14, 19, 22, 1, 22, 1, 31, 13, 18, 11, 32, 1, 20, 15, 36, 1, 30, 1, 34, 29, 24, 1, 46, 13, 34, 19, 40, 1, 46, 15, 50, 21, 30, 1, 52, 1, 32, 39, 63, 17, 46, 1, 52, 25, 46, 1, 68, 1, 38, 43, 58, 17, 54, 1, 76, 65, 42, 1, 72, 21, 44, 31, 78, 1

Offset: 1

Antti Karttunen, Dec 17 2018

nonn

a(p*(n/p)) - (n/p) = (p-1)*a(n/p) holds for all prime divisors p of n, which can be seen by expanding the left hand side as p*(n/p) - A003958(p*(n/p)) - (n/p) = (p-1)*(n/p) - (p-1)*A003958(n/p) = (p-1)*((n/p) - A003958(n/p)) = (p-1)*a(n/p). This shows that this sequence gives a lower limit for arithmetic derivative (A003415) in the same way as A348507 gives an upper limit for it. - Antti Karttunen, Nov 07 2021

With n = Product_{i=1..k} p_i the prime factorization of n, if one constructs for each i a test with a probability of success equal to 1/p_i, and if the tests are independent, then a(n)/n is the probability that at least one of the k tests succeeds. - Luc Rousseau, Jan 14 2023

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

Cf. A003415, A003958, A322581, A348507, A348928 [= gcd(n,a(n))], A348975 (difference from the arithmetic derivative).
Cf. A349139, A348980, A348981, A348982, A348983 (Dirichlet convolutions with other sequences).
Cf. A168065 (gives the arithmetic mean of this and A348507), A168066.

a[1] = 0; a[n_] := n - Times @@ ((First[#] - 1)^Last[#] & /@ FactorInteger[n]); Array[a, 60] (* Amiram Eldar, Dec 17 2018 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));

A020639(n) = if(1==n, n, (factor(n)[1, 1]));
A322582(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (spf-1)); (s); }; \\ (Compare to the similar programs given in A003415 and A348507) - Antti Karttunen, Nov 07 2021

a(n) = n - A003958(n).
From Antti Karttunen, Nov 07 2021: (Start)
a(n) = A003415(n) - A348975(n).
For all n >= 1, a(n) <= A003415(n) <= A348507(n).
For n > 1, a(n) = a(A032742(n))*(A020639(n)-1) + A032742(n). [See the comment above and compare with Reinhard Zumkeller's May 09 2011 formula for A003415]
(End)

A348980 a(n) = Sum_{d|n} d * A322582(n/d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 5, 1, 9, 1, 17, 8, 13, 1, 37, 1, 17, 15, 49, 1, 51, 1, 57, 19, 25, 1, 117, 14, 29, 43, 77, 1, 105, 1, 129, 27, 37, 23, 191, 1, 41, 31, 185, 1, 141, 1, 117, 99, 49, 1, 325, 20, 117, 39, 137, 1, 237, 31, 253, 43, 61, 1, 405, 1, 65, 131, 321, 35, 213, 1, 177, 51, 209, 1, 579, 1, 77, 145, 197, 35, 249, 1, 521

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A322582 with the identity function, A000027.

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

Cf. A000027, A003958, A038040, A322582, A348981 (Möbius transform), A348982, A348983, A349130.

Cf. also A347130, A349140.

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*(n/# - s[n/#]) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348980(n) = sumdiv(n,d,d*A322582(n/d));

a(n) = Sum_{d|n} d * A322582(n/d).
For all n >= 1, a(n) <= A347130(n) <= A349140(n).
a(n) = A038040(n) - A349130(n). - Antti Karttunen, Nov 14 2021

A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 12, 7, 11, 1, 24, 1, 15, 13, 32, 1, 35, 1, 40, 17, 23, 1, 68, 13, 27, 35, 56, 1, 71, 1, 80, 25, 35, 21, 112, 1, 39, 29, 116, 1, 99, 1, 88, 77, 47, 1, 176, 19, 91, 37, 104, 1, 151, 29, 164, 41, 59, 1, 232, 1, 63, 105, 192, 33, 155, 1, 136, 49, 159, 1, 308, 1, 75, 117, 152, 33, 183, 1, 304, 151

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of Euler phi (A000010) with A322582.

Möbius transform of A348980.

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

Cf. A000010, A003958, A008683, A018804, A322582, A348980 (Inverse Möbius transform), A348981, A348982, A348983, A349131.

Cf. also A347131, A349141.

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#]) * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348981(n) = sumdiv(n,d,A322582(n/d)*eulerphi(d));

a(n) = Sum_{d|n} A000010(n/d) * A322582(d).
a(n) = Sum_{d|n} A008683(n/d) * A348980(d).
a(n) = Sum_{k=1..n} A322582(gcd(n,k)).
For all n >= 1, a(n) <= A347131(n) <= A349141(n).
a(n) = A018804(n) - A349131(n). - Antti Karttunen, Nov 14 2021

A348983 a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 7, 1, 14, 1, 31, 11, 20, 1, 80, 1, 26, 23, 111, 1, 109, 1, 122, 29, 38, 1, 328, 19, 44, 76, 164, 1, 250, 1, 351, 41, 56, 35, 565, 1, 62, 47, 514, 1, 334, 1, 248, 208, 74, 1, 1128, 27, 245, 59, 290, 1, 650, 47, 700, 65, 92, 1, 1336, 1, 98, 274, 1023, 53, 502, 1, 374, 77, 490, 1, 2213, 1, 116, 302, 416, 53

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A322582 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A348980.
Dirichlet convolution of sigma (A000203) with A348981.

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

Cf. A000005, A000027, A000203, A003958, A038040, A322582, A348980, A348981, A348982.

Cf. also A349123, A349143.

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A038040(n) = (n*numdiv(n));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348983(n) = sumdiv(n,d,A038040(n/d)*A322582(d));

a(n) = Sum_{d|n} A038040(n/d) * A322582(d).
a(n) = Sum_{d|n} d * A348980(n/d).
a(n) = Sum_{d|n} A000203(d) * A348981(n/d).
For all n >= 1, a(n) <= A349123(n) <= A349143(n).

A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 22, 24, 40, 22, 60, 26, 56, 60, 46, 34, 96, 38, 100, 84, 88, 46, 132, 70, 104, 84, 140, 58, 240, 62, 94, 132, 136, 140, 240, 74, 152, 156, 220, 82, 336, 86, 220, 240, 184, 94, 276, 140, 280, 204, 260, 106, 336, 220, 308, 228, 232, 118, 600, 122, 248, 336, 190, 260, 528, 134, 340, 276, 560

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with Dedekind psi function, A001615.

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

Cf. A001615, A003958, A327251, A348982, A349130, A349131, A349133, A349172.

f[p_, e_] := (p + 1)*p^e - p*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349132(n) = sumdiv(n,d,A001615(d)*A003958(n/d));

a(n) = Sum_{d|n} A001615(d) * A003958(n/d).
a(n) = A327251(n) - A348982(n).
For all n >= 1, a(n) <= A349172(n).
Multiplicative with a(p^e) = (p+1)*p^e - p*(p-1)^e. - Amiram Eldar, Nov 09 2021

A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 8, 1, 13, 1, 40, 11, 17, 1, 80, 1, 21, 19, 164, 1, 99, 1, 112, 23, 29, 1, 364, 17, 33, 77, 144, 1, 191, 1, 604, 31, 41, 27, 528, 1, 45, 35, 524, 1, 243, 1, 208, 165, 53, 1, 1424, 23, 187, 43, 240, 1, 597, 35, 684, 47, 65, 1, 1072, 1, 69, 209, 2084, 39, 347, 1, 304, 55, 327, 1, 2244, 1, 81, 221, 336, 39, 399

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A001615 with A348507.

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

Cf. A001615, A003959, A327251, A348507, A349140, A349141, A349143, A349172.

Cf. also A347132, A348982.

f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #)*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349142(n) = sumdiv(n,d,A001615(d)*A348507(n/d));

a(n) = Sum_{d|n} A001615(n/d) * A348507(d).
For all n >= 1, a(n) >= A347132(n) >= A348982(n).
a(n) = A349172(n) - A327251(n). - Antti Karttunen, Nov 14 2021

cp's OEIS Frontend

A322582 a(n) = n - A003958(n), where A003958 is fully multiplicative with a(p) = (p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A348980 a(n) = Sum_{d|n} d * A322582(n/d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348983 a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula