A009205 -id:A009205 - cp's OEIS frontend

A334579 a(n) = Sum_{d|n} gcd(tau(d), sigma(d)).

1, 2, 3, 3, 3, 8, 3, 4, 4, 6, 3, 11, 3, 8, 9, 5, 3, 12, 3, 13, 9, 8, 3, 16, 4, 6, 8, 11, 3, 24, 3, 8, 9, 6, 9, 16, 3, 8, 9, 16, 3, 26, 3, 15, 16, 8, 3, 19, 6, 10, 9, 9, 3, 24, 9, 20, 9, 6, 3, 45, 3, 8, 12, 9, 9, 26, 3, 13, 9, 24, 3, 24, 3, 6, 12, 11, 9, 24, 3

Offset: 1

Jaroslav Krizek, May 06 2020

nonn

Inverse Möbius transform of A009205. - Antti Karttunen, May 19 2020

			a(6) = gcd(tau(1), sigma(1)) + gcd(tau(2), sigma(2)) + gcd(tau(3), sigma(3)) + gcd(tau(6), sigma(6)) = gcd(1, 1) + gcd(2, 3) + gcd(2, 4) + gcd(4, 12) = 1 + 1 + 2 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334490 (Sum_{d|n} gcd(d, sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).
Cf. A334729 (with product, instead of sum).

[&+[GCD(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]

a[n_] := DivisorSum[n, GCD[DivisorSigma[0, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 07 2020 *)

a(n) = sumdiv(n, d, gcd(numdiv(d), sigma(d))); \\ Michel Marcus, May 07 2020

a(p) = 3 for p = odd primes (A065091).

A334729 a(n) = Product_{d|n} gcd(tau(d), sigma(d)).

Original entry on oeis.org

1, 1, 2, 1, 2, 8, 2, 1, 2, 4, 2, 16, 2, 8, 16, 1, 2, 24, 2, 24, 16, 8, 2, 64, 2, 4, 8, 16, 2, 1024, 2, 3, 16, 4, 16, 48, 2, 8, 16, 48, 2, 2048, 2, 48, 96, 8, 2, 128, 6, 12, 16, 8, 2, 768, 16, 128, 16, 4, 2, 147456, 2, 8, 32, 3, 16, 2048, 2, 24, 16, 1024, 2

Offset: 1

Jaroslav Krizek, May 09 2020

nonn

			a(6) = gcd(tau(1), sigma(1)) * gcd(tau(2), sigma(2)) * gcd(tau(3), sigma(3)) * gcd(tau(6), sigma(6)) = gcd(1, 1) * gcd(2, 3) * gcd(2, 4) * gcd(4, 12) = 1 * 1 * 2 * 4 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A334491 (Product_{d|n} gcd(d, sigma(d))), A334579 (Sum_{d|n} gcd(tau(d), sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).

[&*[GCD(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]

g:= proc(d) option remember; igcd(numtheory:-tau(d), numtheory:-sigma(d)) end proc:
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, May 11 2020

a[n_] := Product[GCD[DivisorSigma[0, d], DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), sigma(d[k]))); \\ Michel Marcus, May 09-11 2020

a(p) = 2 for p = odd primes (A065091).

A296081 a(n) = gcd(tau(n)-1, sigma(n)-1), where tau = A000005 and sigma = A000203.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

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

Cf. A032741, A039653, A065608, A284288, A296082, A296083.

Cf. also A009205.

Array[GCD[DivisorSigma[0,#]-1,DivisorSigma[1,#]-1]&,120] (* Harvey P. Dale, Oct 18 2022 *)

(define (A296081 n) (gcd (A039653 n) (A032741 n)))

a(n) = gcd(A032741(n), A039653(n)).

a(n) = gcd(A039653(n), A065608(n)).

A087801 Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.

Original entry on oeis.org

1, 1, 2, 1, 4, 16, 2, 1, 1, 2, 2, 2, 4, 4, 4, 1, 4, 9, 2, 12, 4, 8, 2, 4, 1, 2, 4, 2, 4, 16, 2, 3, 4, 2, 4, 1, 4, 16, 4, 2, 4, 8, 2, 18, 12, 4, 2, 2, 3, 9, 4, 4, 4, 64, 4, 64, 4, 2, 2, 36, 4, 4, 2, 1, 8, 8, 2, 12, 4, 8, 2, 9, 4, 2, 2, 2, 4, 16, 2, 4, 1, 2, 2, 4, 16, 8, 4, 4, 4, 6, 4, 6, 4, 4, 4, 24

Offset: 1

Reinhard Zumkeller, Oct 11 2003

nonn

a(n) = GCD(A007503(n), A064840(n)).

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

Cf. A009191, A009194, A009205, A007503, A064840, A286360.

GCD[Total[#],Times@@#]&/@Table[{DivisorSigma[0,n],DivisorSigma[1,n]},{n,100}] (* Harvey P. Dale, Jul 17 2018 *)

A087801(n) = gcd(sigma(n)+numdiv(n), sigma(n)*numdiv(n)); \\ Antti Karttunen, May 22 2017

A324554 a(n) = the smallest number m such that gcd(tau(m), sigma(m)) = n where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 3, 18, 6, 648, 20, 2916, 30, 288, 304, 82944, 60, 36864, 832, 16200, 168, 5509980288, 612, 31719424, 432, 23328, 44032, 247669456896, 420, 9487368, 258048, 14112, 2496, 31581162962944, 4176, 26843545600, 840, 4064256, 4390912, 42693156, 1980, 151801324109824

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

a(n) = the smallest number m such that A009205(m) = n.

a(p) = q^(c*p-1) * k for p prime, where q is some prime, c and k are positive integers. - David A. Corneth, Mar 07 2019

			For n=3; a(3) = 18 because gcd(tau(18), sigma(18)) = gcd (6, 39) = 3 and 18 is the smallest.

Cf. A000005, A007955, A009205.

Cf. also A324553, A324555.

[Min([n: n in[1..10^5] | GCD(NumberOfDivisors(n), SumOfDivisors(n)) eq k]): k in [1..16]]

Array[Block[{m = 1}, While[GCD @@ DivisorSigma[{0, 1}, m] != #, m++]; m] &, 16] (* Michael De Vlieger, Mar 24 2019 *)

A324554search_and_print(searchlimit) = { my(m = Map(), k); for(n=1,searchlimit,k=gcd(sigma(n),numdiv(n)); if(!mapisdefined(m,k), mapput(m,k,n))); for(k=1, oo, if(!mapisdefined(m,k), break, print1(mapget(m,k), ", "))); }; \\ Antti Karttunen, Mar 06 2019

a(17)-a(37) from Jon E. Schoenfield, Mar 06 2019

cp's OEIS Frontend

A334579 a(n) = Sum_{d|n} gcd(tau(d), sigma(d)).

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A334729 a(n) = Product_{d|n} gcd(tau(d), sigma(d)).

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A296081 a(n) = gcd(tau(n)-1, sigma(n)-1), where tau = A000005 and sigma = A000203.

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A087801 Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.

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A324554 a(n) = the smallest number m such that gcd(tau(m), sigma(m)) = n where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

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