A348983 -id:A348983 - 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

A348982 a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 6, 1, 11, 1, 22, 9, 15, 1, 52, 1, 19, 17, 66, 1, 69, 1, 76, 21, 27, 1, 176, 15, 31, 51, 100, 1, 145, 1, 178, 29, 39, 25, 288, 1, 43, 33, 264, 1, 189, 1, 148, 123, 51, 1, 508, 21, 145, 41, 172, 1, 339, 33, 352, 45, 63, 1, 632, 1, 67, 159, 450, 37, 277, 1, 220, 53, 265, 1, 924, 1, 79, 175, 244, 37, 321

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A001615 with A322582.

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

Cf. A001615, A003958, A322582, A327251, A348980, A348981, A348982, A348983, A349132.

Cf. also A347132, A349142.

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
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348982(n) = sumdiv(n,d,A001615(n/d)*A322582(d));

a(n) = Sum_{d|n} A001615(n/d) * A322582(d).
For all n >= 1, a(n) <= A347132(n) <= A349142(n).
a(n) = A327251(n) - A349132(n). - Antti Karttunen, Nov 14 2021

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A349140.
Dirichlet convolution of sigma (A000203) with A349141.
Dirichlet convolution of A060640 with A348971.

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

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.

Cf. also A349123, A348983.

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

A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
a(n) = Sum_{d|n} d * A349140(n/d).
a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
For all n >= 1, a(n) >= A349123(n) >= A348983(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).

cp's OEIS Frontend

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

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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).

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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).

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A348982 a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

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A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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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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