A349139 -id:A349139

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A322582

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 17 2018

STATUS

approved

A348507

a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

+10
14

0, 1, 1, 5, 1, 6, 1, 19, 7, 8, 1, 24, 1, 10, 9, 65, 1, 30, 1, 34, 11, 14, 1, 84, 11, 16, 37, 44, 1, 42, 1, 211, 15, 20, 13, 108, 1, 22, 17, 122, 1, 54, 1, 64, 51, 26, 1, 276, 15, 58, 21, 74, 1, 138, 17, 160, 23, 32, 1, 156, 1, 34, 65, 665, 19, 78, 1, 94, 27, 74, 1, 360, 1, 40, 69, 104, 19, 90, 1, 406, 175, 44, 1, 204

(list; graph; refs; listen; history; text; internal format)

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 (A003959(p*(n/p)) - (p*(n/p))) - (n/p) = (p+1)*A003959(n/p)-((p+1)*(n/p)) = (p+1)*(A003959(n/p)-(n/p)) = (p+1)*a(n/p). This implies that a(n) >= A003415(n) for all n. (See also comments in A348970). - Antti Karttunen, Nov 06 2021

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

FORMULA

a(n) = A003959(n) - n.

a(n) = A348508(n) + n.

a(n) = A001065(n) + A348029(n).

From Antti Karttunen, Nov 06 2021: (Start)

a(n) = Sum_{d|n} A348971(d).

a(n) = A003415(n) + A348970(n).

For all n >= 1, A322582(n) <= A003415(n) <= a(n).

For n > 1, a(n) = a(A032742(n))*(1+A020639(n)) + A032742(n). [See the comments above, and compare this with Reinhard Zumkeller's May 09 2011 recursive formula for A003415]

(End)

MATHEMATICA

f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)

PROG

(PARI)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

A348507(n) = (A003959(n) - n);

(PARI)

A020639(n) = if(1==n, n, (factor(n)[1, 1]));

A348507(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (1+spf)); (s); }; \\ (Compare this with similar programs given in A003415 and in A322582) - Antti Karttunen, Nov 06 2021

CROSSREFS

Cf. A001065, A003415, A003959, A020639, A032742, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970.

Cf. A348971 (Möbius transform) and A349139, A349140, A349141, A349142, A349143 (other Dirichlet convolutions).

Cf. also A168065 (the arithmetic mean of this and A322582), A168066.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 30 2021

STATUS

approved

A349129

a(n) = Sum_{d|n} A003958(d) * A003959(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and A003959 is fully multiplicative with a(p) = (p+1).

+10
6

1, 4, 6, 13, 10, 24, 14, 40, 28, 40, 22, 78, 26, 56, 60, 121, 34, 112, 38, 130, 84, 88, 46, 240, 76, 104, 120, 182, 58, 240, 62, 364, 132, 136, 140, 364, 74, 152, 156, 400, 82, 336, 86, 286, 280, 184, 94, 726, 148, 304, 204, 338, 106, 480, 220, 560, 228, 232, 118, 780, 122, 248, 392, 1093, 260, 528, 134, 442, 276

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Dirichlet convolution of A003958 with A003959.

LINKS

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

FORMULA

Multiplicative with a(p^e) = ((p+1)^(e+1) - (p-1)^(e+1))/2. - Amiram Eldar, Nov 09 2021

For all n >= 1, A349130(n) <= a(n) <= A349170(n).

MATHEMATICA

f[p_, e_] := ((p + 1)^(e + 1) - (p - 1)^(e + 1))/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

PROG

(PARI)

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

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

A349129(n) = sumdiv(n, d, A003958(d)*A003959(n/d));

CROSSREFS

Cf. A003958, A003959, A168066, A349139.

Cf. also A349130, A349131, A349132, A349133 and A349170, A349171, A349172, A349173.

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Nov 09 2021

STATUS

approved

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