A348029 -id:A348029

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

A348970

a(n) = 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.

+10
9

0, 0, 0, 1, 0, 1, 0, 7, 1, 1, 0, 8, 0, 1, 1, 33, 0, 9, 0, 10, 1, 1, 0, 40, 1, 1, 10, 12, 0, 11, 0, 131, 1, 1, 1, 48, 0, 1, 1, 54, 0, 13, 0, 16, 12, 1, 0, 164, 1, 13, 1, 18, 0, 57, 1, 68, 1, 1, 0, 64, 0, 1, 14, 473, 1, 17, 0, 22, 1, 15, 0, 204, 0, 1, 14, 24, 1, 19, 0, 230, 67, 1, 0, 80, 1, 1, 1, 96, 0, 75, 1, 28

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

OFFSET

1,8

COMMENTS

There are no negative terms. We prove this by induction over the prime factorization of n, showing that A348507(n) >= A003415(n) for all values of n >= 1. At n=1, both sequences have value 0, and at the primes both sequences obtain the value 1, so the base cases hold. We know that A348507(n)-(n/p) = (p+1)*A348507(n/p) for all prime factors p of n (see comment in A348507). With the arithmetic derivative we obtain respectively that A003415(n) = A003415(p*(n/p)) = A003415(p)*(n/p) + p*A003415(n/p) = (n/p) + p*A003415(n/p), for any prime factor p of n. Now A348507(p*(n/p)) >= A003415(p*(n/p)) iff A348507(p*(n/p)) - (n/p) >= A003415(p*(n/p)) - (n/p), that is, iff (p+1)*A348507(n/p) >= p*A003415(n/p), which indeed follows by the induction hypothesis, which assumes that A348507(x) >= A003415(x) for all proper divisors x of n.

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) - A129283(n) = A003959(n) - (n+A003415(n)).

a(n) = A348029(n) - A211991(n).

a(n) = A348507(n) - A003415(n).

For all n >= 1, a(A001358(n)) = 1.

MATHEMATICA

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n - d[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

PROG

(PARI)

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); };

A348970(n) = (A003959(n) - (n+A003415(n)));

CROSSREFS

Cf. A003415, A003959, A129283, A211991, A348029, A348507, A348971, A348972, A348973, A348974.

Cf. A008578 (positions of zeros), A001358 (positions of ones).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 05 2021

STATUS

approved

A348047

a(n) = gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

+10
7

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 8, 8, 30, 72, 32, 9, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 24, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32

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

OFFSET

1,2

COMMENTS

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 196 = 4*49, where a(196) = 3, although a(4) = 1 and a(49) = 4.

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = gcd(A000203(n), A003959(n)).

a(n) = gcd(A000203(n), A348029(n)) = gcd(A003959(n), A348029(n)).

a(n) = A000203(n)/ A348048(n) = A003959(n) / A348049(n).

MATHEMATICA

f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[Times @@ f @@@ FactorInteger[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)

PROG

(PARI)

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

A348047(n) = gcd(sigma(n), A003959(n));

CROSSREFS

Cf. A000203, A003959, A348029, A348048, A348049.

Cf. also A344695.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 21 2021

STATUS

approved

A348048

a(n) = sigma(n) / gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

+10
5

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 5, 7, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 7, 1, 57, 13

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

OFFSET

1,4

COMMENTS

Not multiplicative. For example, a(196) = 133 != a(4) * a(49).

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A000203(n) / A348047(n) = A000203(n) / gcd(A000203(n), A003959(n)).

MATHEMATICA

f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := (s = DivisorSigma[1, n]) / GCD[s, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)

PROG

(PARI)

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

A348048(n) = { my(u=sigma(n)); (u/gcd(u, A003959(n))); };

CROSSREFS

Cf. A000203, A003959, A348029, A348047, A348049.

Cf. also A344696.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 21 2021

STATUS

approved

A348945

a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

+10
5

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 0, 0, 75, 0, 0, 0, 6, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 18, 0, 24, 0, 0, 0, 0, 0, 0, 0, 270, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 108, 48, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 300, 0, 0, 0, 10

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

OFFSET

1,8

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A348944(n) - A000203(n) = ((1/2) * (A003959(n)+A034448(n))) - A000203(n).

a(n) = (1/2) * (A348029(n)-A048146(n)).

MATHEMATICA

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 0; a[n_] := (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2 - Times @@ f1 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

PROG

(PARI)

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

A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };

A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

A348945(n) = (A348944(n)-sigma(n));

CROSSREFS

Cf. A000203, A003959, A034448, A048146, A348029, A348944, A348946.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 05 2021

STATUS

approved

A348030

a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

+10
4

0, 0, 0, 2, 0, 0, 0, 10, 3, 0, 0, 6, 0, 0, 0, 38, 0, 6, 0, 10, 0, 0, 0, 30, 5, 0, 21, 14, 0, 0, 0, 130, 0, 0, 0, 36, 0, 0, 0, 50, 0, 0, 0, 22, 15, 0, 0, 114, 7, 10, 0, 26, 0, 42, 0, 70, 0, 0, 0, 30, 0, 0, 21, 422, 0, 0, 0, 34, 0, 0, 0, 144, 0, 0, 15, 38, 0, 0, 0, 190, 111, 0, 0, 42, 0, 0, 0, 110, 0, 30, 0, 46

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

OFFSET

1,4

COMMENTS

Möbius transform of A348029(n), which is A003959(n) - sigma(n).

LINKS

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

FORMULA

a(n) = A003968(n) - n.

a(n) = Sum_{d|n} A008683(n/d) * A348029(d).

MATHEMATICA

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

PROG

(PARI)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }

A348030(n) = (A003968(n)-n);

CROSSREFS

Cf. A003959, A003968, A005117 (positions of zeros), A008683, A348029, A348036.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 19 2021

STATUS

approved

A348049

a(n) = A003959(n) / gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

+10
4

1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 9, 1, 1, 1, 81, 1, 16, 1, 9, 1, 1, 1, 9, 36, 1, 8, 9, 1, 1, 1, 27, 1, 1, 1, 144, 1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 81, 64, 36, 1, 9, 1, 8, 1, 9, 1, 1, 1, 9, 1, 1, 16, 729, 1, 1, 1, 9, 1, 1, 1, 144, 1, 1, 36, 9, 1, 1, 1, 81, 256, 1, 1, 9, 1, 1, 1, 9, 1, 16, 1, 9, 1, 1, 1, 27, 1, 64

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

OFFSET

1,4

COMMENTS

Not multiplicative. For example, a(196) = 192 != a(4) * a(49).

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A003959(n) / A348047(n) = A003959(n) / gcd(A000203(n), A003959(n)).

MATHEMATICA

f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := (m = Times @@ f @@@ FactorInteger[n]) / GCD[m, DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)

PROG

(PARI)

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

A348049(n) = { my(u=A003959(n)); (u/gcd(u, sigma(n))); };

CROSSREFS

Cf. A000203, A003959, A005117 (positions of 1's), A348029, A348047, A348048.

Cf. also A344697.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 21 2021

STATUS

approved

A348732

a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

+10
3

0, 0, 0, 4, 0, 0, 0, 18, 6, 0, 0, 16, 0, 0, 0, 64, 0, 18, 0, 24, 0, 0, 0, 72, 10, 0, 36, 32, 0, 0, 0, 210, 0, 0, 0, 94, 0, 0, 0, 108, 0, 0, 0, 48, 36, 0, 0, 256, 14, 30, 0, 56, 0, 108, 0, 144, 0, 0, 0, 96, 0, 0, 48, 664, 0, 0, 0, 72, 0, 0, 0, 342, 0, 0, 40, 80, 0, 0, 0, 384, 174, 0, 0, 128, 0, 0, 0, 216, 0, 108

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

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..21125

FORMULA

a(n) = A003959(n) - A034448(n).

a(n) = A348507(n) - A034460(n).

a(n) = A048146(n) + A348029(n).

MATHEMATICA

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

PROG

(PARI)

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

A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

A348732(n) = (A003959(n)-A034448(n));

CROSSREFS

Cf. A003959, A005117 (positions of zeros), A034448, A034460, A048146, A348029, A348507.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 31 2021

STATUS

approved

A348508

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

+10
2

-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

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

OFFSET

1,3

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI)

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

A348508(n) = (A003959(n) - 2*n);

CROSSREFS

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

KEYWORD

sign

AUTHOR

Antti Karttunen, Oct 30 2021

STATUS

approved