A055155 -id:A055155 - cp's OEIS frontend

A347088 a(n) = A055155(n) - d(n), where A055155(n) = Sum_{d|n} gcd(d, n/d) and d(n) gives the number of divisors of n.

0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 5, 0, 4, 0, 2, 0, 0, 0, 4, 4, 0, 4, 2, 0, 0, 0, 8, 0, 0, 0, 11, 0, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 10, 6, 8, 0, 2, 0, 8, 0, 4, 0, 0, 0, 4, 0, 0, 4, 15, 0, 0, 0, 2, 0, 0, 0, 18, 0, 0, 8, 2, 0, 0, 0, 10, 12, 0, 0, 4, 0, 0, 0, 4, 0, 8, 0, 2, 0, 0, 0, 16, 0, 12, 4, 19, 0, 0, 0, 4, 0

Offset: 1

Antti Karttunen, Aug 17 2021

nonn

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

Cf. A000005, A005117 (positions of zeros), A055155, A347089.

A055155(n) = sumdiv(n, d, gcd(d, n/d)); \\ From A055155
A347088(n) = (A055155(n)-numdiv(n));

from sympy import gcd, divisors, divisor_count
def A347088(n): return sum(gcd(d,n//d) for d in divisors(n,generator=True)) - divisor_count(n) # Chai Wah Wu, Aug 19 2021

a(n) = A055155(n) - A000005(n).

A347089 a(n) = gcd(A055155(n), d(n)), where A055155(n) = Sum_{d|n} gcd(d, n/d) and d(n) gives the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 5, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 10, 3, 2, 4, 2, 2, 8, 4, 4, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 6, 2, 4, 2, 2, 4, 8, 2, 10, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 4, 2, 6, 2, 1, 2, 8, 2, 4, 8

Offset: 1

Antti Karttunen, Aug 17 2021

nonn

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

Cf. A000005, A055155, A347088.

A055155(n) = sumdiv(n, d, gcd(d, n/d)); \\ From A055155
A347089(n) = gcd(A055155(n),numdiv(n));

from sympy import gcd, divisors, divisor_count
def A347089(n): return gcd(divisor_count(n),sum(gcd(d,n//d) for d in divisors(n,generator=True))) # Chai Wah Wu, Aug 19 2021

a(n) = gcd(A000005(n), A055155(n)).

a(n) = gcd(A000005(n), A347088(n)) = gcd(A055155(n), A347088(n)).

A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 4, 4, 2, 10, 11, 4, 12, 7, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 10, 2, 13, 2, 7, 9, 4, 2, 16, 17, 18, 4, 7, 2, 19, 4, 10, 4, 4, 2, 20, 2, 4, 9, 21, 4, 13, 2, 7, 4, 13, 2, 22, 2, 4, 18, 7, 4, 13, 2, 16, 23, 4, 2, 20, 4, 4, 4, 10, 2, 24, 4, 7, 4, 4, 4, 25, 2, 26, 9, 27, 2, 13, 2, 10, 13, 4, 2, 28, 2, 13

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Restricted growth sequence transform of A291757, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291757(i) = A291757(j) <=> A003557(i) = A003557(j) and A046523(i) = A046523(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
Also the restricted growth sequence transform of A294876, Product_{d|n, d>1} prime(gcd(d,n/d)). (This was the original definition).
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A319347(i) = A319347(j) => a(i) = a(j),
  a(i) = a(j) => A055155(i) = A055155(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. also A291751, A291757, A294876, A295300, A295888, A319347, A322021.

Cf. A000188, A055155, A294897, A295666, A322020 (a few of the matched sequences).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };
v294877 = rgs_transform(vector(up_to,n,A294876(n)));
A294877(n) = v294877[n];

A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v294877 = rgs_transform(vector(up_to,n,[A003557(n),A046523(n)]));
A294877(n) = v294877[n]; \\ Antti Karttunen, Nov 28 2018

Name changed and comments added by Antti Karttunen, Nov 28 2018

A057670 a(n) = Sum_{k|n} lcm(k, n/k).

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 24, 21, 40, 22, 60, 26, 56, 60, 52, 34, 84, 38, 100, 84, 88, 46, 144, 55, 104, 72, 140, 58, 240, 62, 112, 132, 136, 140, 210, 74, 152, 156, 240, 82, 336, 86, 220, 210, 184, 94, 312, 105, 220, 204, 260, 106, 288, 220, 336, 228, 232, 118, 600

Offset: 1

Leroy Quet, Oct 18 2000

			a(8) = lcm(1,8) + lcm(2,4) + lcm(4,2) + lcm(8,1) = 8 + 4 + 4 + 8 = 24.

G. C. Greubel, Table of n, a(n) for n = 1..10000
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A001620, A055155, A107661.

Table[DivisorSum[n, LCM[#, n/#] &], {n, 59}] (* Michael De Vlieger, Dec 11 2017 *)
f[p_, e_] := (2*p^(e + 1) - p^Ceiling[(e + 1)/2] - p^Floor[(e + 1)/2])/(p - 1); f[p_, 1] := 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 27 2023 *)

a(n) = sumdiv(n, d, lcm(d, n/d)); \\ Michel Marcus, May 19 2014

Multiplicative with a(p) = 2*p, a(p^k) = (2*p^(k+1) - p^ceiling((k+1)/2) - p^floor((k+1)/2)) / (p-1). a(n) is odd iff n is an odd square. - Henry Bottomley, May 16 2005
Multiplicative with a(p^e) = Sum_{k=0..e} p^max(k, e-k),  (cf. A107661). - Mitch Harris, May 18 2005
Dirichlet g.f.: (zeta(s-1))^2*zeta(2s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
Sum_{k=1..n} a(k) ~ 3*zeta(3)*n^2 / (2*Pi^2) * (2*log(n) - 24*zeta'(2)/Pi^2 - 1 + 4*gamma + 4*zeta'(3)/zeta(3)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 01 2019

A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 18, 10, 8, 2, 72, 2, 8, 8, 126, 2, 200, 2, 72, 8, 8, 2, 648, 22, 8, 50, 72, 2, 128, 2, 882, 8, 8, 8, 23400, 2, 8, 8, 648, 2, 128, 2, 72, 200, 8, 2, 31752, 34, 968, 8, 72, 2, 5000, 8, 648, 8, 8, 2, 10368, 2, 8, 200, 16758, 8, 128, 2, 72, 8, 128, 2, 2737800, 2, 8, 968, 72, 8, 128, 2, 31752, 1150, 8, 2, 10368, 8, 8, 8, 648, 2, 80000, 8, 72

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

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

Cf. A294877 (rgs-version of this filter).
Cf. A000040, A000188, A034444, A055155.
Cf. also A293442, A293514, A293524.

A294876[n_] := Product[Prime[GCD[d, n/d]], {d, Rest[Divisors[n]]}];
Array[A294876, 100] (* Paolo Xausa, Feb 22 2024 *)

A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };

a(n) = Product_{d|n, d>1} A000040(gcd(d,n/d)).
Other identities. For all n >= 1:
1+A007814(a(n)) = A034444(n).
1+A056239(a(n)) = A055155(n).
For n > 1, A061395(a(n)) = A000188(n).

A345266 a(n) = Sum_{p|n, p prime} gcd(p,n/p).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 1, 3, 5, 2, 3, 3, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 2, 3, 1, 3, 1, 3, 4, 2, 1, 3, 7, 6, 2, 3, 1, 4, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 6, 3, 2, 3, 1, 3, 3, 2, 1, 4, 2, 2, 2, 3, 1, 5, 2, 3, 2, 2, 2, 3, 1, 8, 4, 7, 1, 3, 1, 3, 3

Offset: 1

Wesley Ivan Hurt, Jun 13 2021

			a(18) = Sum_{p|18} gcd(p,18/p) = gcd(2,9) + gcd(3,6) = 1 + 3 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to gcd's.

Cf. A001221 (omega), A007947 (rad), A008472 (sopf), A345302.

Cf. A055155, A056169, A063958.

Table[Sum[GCD[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

a(n) = my(f=factor(n), p); sum(k=1, #f~, p=f[k, 1]; gcd(p,n/p)); \\ Michel Marcus, Jun 16 2021

A345266(n) = vecsum(apply(p->gcd(p,n/p), factor(n)[,1])); \\ Antti Karttunen, Nov 13 2021

a(p) = 1 for p prime.
From Wesley Ivan Hurt, Nov 21 2021: (Start)
a(n) = A056169(n) + A063958(n).
If n is squarefree, then a(n) = omega(n).
a(p^k) = p for primes p and k >= 2. (End)

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

A068976 a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 10, 11, 4, 2, 12, 2, 4, 4, 26, 2, 22, 2, 12, 4, 4, 2, 20, 27, 4, 20, 12, 2, 8, 2, 42, 4, 4, 4, 66, 2, 4, 4, 20, 2, 8, 2, 12, 22, 4, 2, 52, 51, 54, 4, 12, 2, 40, 4, 20, 4, 4, 2, 24, 2, 4, 22, 106, 4, 8, 2, 12, 4, 8, 2, 110, 2, 4, 54, 12, 4, 8, 2, 52, 101, 4, 2, 24, 4, 4, 4

Offset: 1

Benoit Cloitre, Apr 06 2002

More generally, a(n,m) = Sum_{d divides n} gcd(d,n/d)^m is multiplicative with a(p^e,m) = (p^(m*e/2)*(p^m+1)-2)/(p^m-1) if e is even else 2*(p^(m*(e+1)/2)-1)/(p^m-1). - Vladeta Jovovic, May 30 2003

Robert Israel, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio
R. Sivaramakrishnan, A. Somayajulu, H. Scheid and E. A. Bender, A Number-Theoretic Identity (Advanced Problem 5446 and solutions), American Mathematical Monthly 74 (1967), 1274-1276.

Cf. A055155 (m=1).

Cf. A000010, A000203, A001222, A008833, A010052, A034444, A061547.

R:= proc(n) uses numtheory; local K,k;
  K:= select(k -> (n mod k^2 = 0), divisors(n));
  add(k^2*2^nops(factorset(n/k^2)),k=K);
end proc:
seq(R(n),n=1..100); # Robert Israel, Oct 18 2015

a[n_]:=Total[GCD[#, n/#]^2 & /@ Divisors[n]]; Table[a[n], {n, 1, 87}] (* Jean-François Alcover, Jul 26 2011 *)
f[p_, e_] := If[OddQ[e], 2*(p^(e+1)-1)/(p^2-1), (p^(e+2)+p^e-2)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 03 2020 *)

a(n) = Sum_{d divides n} gcd(d, n/d)^2. Multiplicative with a(p^e) = (p^(e+2)+p^e-2)/(p^2-1) if e is even else 2*(p^(e+1)-1)/(p^2-1). - Vladeta Jovovic, May 30 2003
Dirichlet g.f.: zeta^2(s)*zeta(2s-2)/zeta(2s). Dirichlet convolution of A034444 and the sequence n*A010052(n). - R. J. Mathar, Apr 18 2011
Inverse Mobius transform of A008833. - R. J. Mathar, Oct 31 2011
a(n) = Sum_{d divides n} (-1)^A001222(d) * A000010(d) * A000203(n/d) = Sum_{k^2 divides n} k^2 * 2^A001221(n/k^2). - Robert Israel, Oct 18 2015
Sum_{k=1..n} a(k) ~ Zeta(3/2)^2 * n^(3/2) / (3*Zeta(3)) - (3*n*(log(n) - 1 + 2*gamma + 2*log(2*Pi) - 12*Zeta'(2)/Pi^2))/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 05 2019
a(2^k) = (3/2)*2^k + (1/6)*(-2)^k - 2/3 = A061547(k+2). - Amiram Eldar, Sep 03 2020

A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 11, 18, 12, 24, 14, 24, 24, 23, 18, 33, 20, 36, 32, 36, 24, 48, 27, 42, 32, 48, 30, 72, 32, 45, 48, 54, 48, 66, 38, 60, 56, 72, 42, 96, 44, 72, 66, 72, 48, 92, 51, 81, 72, 84, 54, 96, 72, 96, 80, 90, 60, 144, 62, 96, 88, 88, 84, 144, 68, 108, 96, 144

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to gcd's.
Index entries for sequences related to lcm's.

Cf. A055155, A057660, A057661, A057670, A332618.

Cf. A013663, A013664.

a:= n-> add(d/igcd(d, n/d), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 17 2020

Table[Sum[LCM[d, n/d]/d, {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1) + e/2, (p^(e + 2) - p)/(p^2 - 1) + (e + 1)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

A332619(n) = sumdiv(n,d,lcm(d,n/d)/d); \\ Antti Karttunen, Nov 12 2021

a(n) = Sum_{d|n} d / gcd(d, n/d).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) + e/2 if e is even, and (p^(e+2)-p)/(p^2-1) + (e + 1)/2 if e is odd.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*zeta(6)/(8*zeta(5)) = 0.740543... . (End)

A337180 a(n) = Sum_{d|n} d * gcd(d,n/d).

Original entry on oeis.org

1, 3, 4, 9, 6, 12, 8, 21, 19, 18, 12, 36, 14, 24, 24, 53, 18, 57, 20, 54, 32, 36, 24, 84, 51, 42, 64, 72, 30, 72, 32, 117, 48, 54, 48, 171, 38, 60, 56, 126, 42, 96, 44, 108, 114, 72, 48, 212, 99, 153, 72, 126, 54, 192, 72, 168, 80, 90, 60, 216, 62, 96, 152, 277, 84, 144

Offset: 1

Wesley Ivan Hurt, Jan 28 2021

If p is prime, a(n) = Sum_{d|p} d * gcd(d,p/d) = 1*1 + p*1 = p + 1. - Wesley Ivan Hurt, May 21 2021

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

Cf. A002117, A013662, A055155.

Table[Sum[k*GCD[k, n/k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
(* Second program: *)
Table[DivisorSum[n, # GCD[#, n/#] &], {n, 100}] (* Michael De Vlieger, Nov 11 2021 *)

a(n) = sumdiv(n, d, d*gcd(d, n/d)); \\ Michel Marcus, Jan 29 2021

Multiplicative with a(p^e) = (p^(2*floor(e/2)+2)-1)/(p^2-1) + p^e*ceiling(e/2). - Sebastian Karlsson, Nov 11 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (5*zeta(4))/(4*zeta(3)) = 1.1254908... . - Amiram Eldar, Nov 18 2022

A332618 a(n) = Sum_{d|n} lcm(d, n/d) / gcd(d, n/d).

Original entry on oeis.org

1, 4, 6, 9, 10, 24, 14, 20, 19, 40, 22, 54, 26, 56, 60, 41, 34, 76, 38, 90, 84, 88, 46, 120, 51, 104, 60, 126, 58, 240, 62, 84, 132, 136, 140, 171, 74, 152, 156, 200, 82, 336, 86, 198, 190, 184, 94, 246, 99, 204, 204, 234, 106, 240, 220, 280, 228, 232, 118, 540

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to gcd's.
Index entries for sequences related to lcm's.

Cf. A007913, A055155, A056789, A057670, A332619, A337180.

a:= n-> n*add(1/igcd(d, n/d)^2, d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 17 2020

Table[Sum[LCM[d, n/d]/GCD[d, n/d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := If[EvenQ[e], (2*p^(e+2) - p^2 - 1)/(p^2 - 1), 2*(p^(e+2) - p)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

A332618(n) = sumdiv(n,d,lcm(d,n/d)/gcd(d,n/d)); \\ Antti Karttunen, Nov 12 2021

a(n) = n * Sum_{d|n} 1 / gcd(d, n/d)^2.
Multiplicative with a(p^e) = (2*p^(e+2) - p^2 - 1)/(p^2 - 1) if e is even, a(p^e) = 2*(p^(e+2) - p)/(p^2 - 1) if e is odd. - Sebastian Karlsson, May 07 2022
From Peter Bala, Jan 24 2024: (Start)
a(n) = Sum_{d divides n} A007913(d)*n/d.
Dirichlet g.f.: zeta(2*s)*zeta(s-1)^2/zeta(2*s-2). (End)

cp's OEIS Frontend

A347088 a(n) = A055155(n) - d(n), where A055155(n) = Sum_{d|n} gcd(d, n/d) and d(n) gives the number of divisors of n.

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A347089 a(n) = gcd(A055155(n), d(n)), where A055155(n) = Sum_{d|n} gcd(d, n/d) and d(n) gives the number of divisors of n.

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A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

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A057670 a(n) = Sum_{k|n} lcm(k, n/k).

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A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

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A345266 a(n) = Sum_{p|n, p prime} gcd(p,n/p).

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A068976 a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.

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A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

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