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

1,4

Comments

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

Links

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

Formula

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)

Mathematica

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

Prog

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

Crossrefs

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.

Sequence in context: A006022 A326065 A078896 * A273133 A318841 A300238

Adjacent sequences: A322579 A322580 A322581 * A322583 A322584 A322585

Keyword

nonn

Author

Antti Karttunen, Dec 17 2018

Status

approved