A348970 -id:A348970 - cp's OEIS frontend

A348507 a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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

Offset: 1

Antti Karttunen, Oct 30 2021

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 (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

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A003415, A003959, A020639, A032742, A065488, 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.

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

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);

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

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)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065488 - 1. - Amiram Eldar, Jun 01 2025

A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 14, 6, 6, 1, 14, 1, 8, 7, 46, 1, 18, 1, 22, 9, 12, 1, 46, 10, 14, 30, 30, 1, 22, 1, 146, 13, 18, 11, 60, 1, 20, 15, 74, 1, 30, 1, 46, 36, 24, 1, 146, 14, 40, 19, 54, 1, 78, 15, 102, 21, 30, 1, 74, 1, 32, 48, 454, 17, 46, 1, 70, 25, 46, 1, 192, 1, 38, 50, 78, 17, 54, 1, 238, 138, 42, 1, 102, 21

Offset: 1

Antti Karttunen, Nov 05 2021

Möbius transform of A348507.

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

Cf. A000010, A003959, A003968, A008683, A300251, A348507, A348970.

Cf. A005596, A104141.

f1[p_, e_] := p*(p + 1)^(e - 1); f2[p_, e_] := (p - 1)*p^(e - 1); a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A348971(n) = { my(f=factor(n),m1=1,m2=1,p); for(i=1, #f~, p = f[i, 1]; m1 *= p*(p+1)^(f[i, 2]-1); m2 *= (p-1)*p^(f[i, 2]-1)); (m1-m2); };

A348971(n) = { my(f=factor(n),p); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f)-eulerphi(n); }

a(n) = A003968(n) - A000010(n).
a(n) = Sum_{d|n} A008683(n/d) * A348507(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A104141 * (1/A005596 - 1) = 0.5088692487... . - Amiram Eldar, Oct 05 2023

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).

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)).

A348508 a(n) = A003959(n) - 2*n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

-1, -1, -2, 1, -4, 0, -6, 11, -2, -2, -10, 12, -12, -4, -6, 49, -16, 12, -18, 14, -10, -8, -22, 60, -14, -10, 10, 16, -28, 12, -30, 179, -18, -14, -22, 72, -36, -16, -22, 82, -40, 12, -42, 20, 6, -20, -46, 228, -34, 8, -30, 22, -52, 84, -38, 104, -34, -26, -58, 96, -60, -28, 2, 601, -46, 12, -66, 26, -42, 4, -70, 288

Offset: 1

Antti Karttunen, Oct 30 2021

sign

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

Cf. A003959, A033879, A138636, A168036, A252748, A348029, A348970.

f[p_, e_] := (p + 1)^e; a[1] = -1; a[n_] := Times @@ f @@@ FactorInteger[n] - 2*n; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348508(n) = (A003959(n) - 2*n);

a(n) = A003959(n) - 2*n.
a(n) = A348507(n) - n.
a(n) = A348029(n) - A033879(n).
From Antti Karttunen, Dec 05 2021: (Start)
a(n) = A168036(n) + A348970(n).
For all n >= 1, a(A138636(n)) = 12.
(End)
a(p) = 1 - p if p prime. - Bernard Schott, Feb 17 2022

A348975 a(n) = A003415(n) + A003958(n) - n, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 5, 1, 1, 0, 6, 0, 1, 1, 17, 0, 7, 0, 8, 1, 1, 0, 22, 1, 1, 8, 10, 0, 9, 0, 49, 1, 1, 1, 28, 0, 1, 1, 32, 0, 11, 0, 14, 10, 1, 0, 66, 1, 11, 1, 16, 0, 35, 1, 42, 1, 1, 0, 40, 0, 1, 12, 129, 1, 15, 0, 20, 1, 13, 0, 88, 0, 1, 12, 22, 1, 17, 0, 100, 43, 1, 0, 52, 1, 1, 1, 62, 0, 49, 1, 26, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 09 2021

No negative terms. See comments in A322582.

This is the difference between the arithmetic derivative of n [= A003415(n)] and its guaranteed lower bound A322582(n) [= n - A003958(n)].

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

Cf. A003415, A003958, A168036, A322582.

Cf. also A348970 for the corresponding difference from a guaranteed upper bound.

MapAt[# + 1 &, Array[If[# < 2, 0, # Total[#2/#1 & @@@ #2]] + Times @@ Map[(#1 - 1)^#2 & @@ # &, #2] - #1 & @@ {#, FactorInteger[#]} &, 95], 1] (* Michael De Vlieger, Mar 15 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348975(n) = (A003415(n) - A322582(n));

a(n) = A003415(n) - A322582(n).

a(n) = A003958(n) + A168036(n).

cp's OEIS Frontend

A348507 a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A348971 a(n) = Product(p*(p+1)^(e-1)) - Product((p-1)*p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

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

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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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A348508 a(n) = A003959(n) - 2*n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A348975 a(n) = A003415(n) + A003958(n) - n, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

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A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.