A322979 -id:A322979 - cp's OEIS frontend

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 2, 1, 2, 1, 10, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 8, 1

Offset: 1

David W. Wilson

nonn

a(A046642(n)) = 1.
First occurrence of k: 1, 2, 9, 8, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 384, .... Conjecture: each k is present. - Robert G. Wilson v, Mar 27 2013
Conjecture is true. See David A. Corneth's comment in A324553. - Antti Karttunen, Mar 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Cf. A000005, A009194, A009195, A009205, A009213, A009230, A049820, A125168, A138010, A286540, A303781, A318459, A319337, A322979, A322980, A323073.

Cf. A046642 (positions of ones), A324553 (position of the first occurrence of each n).

a009191 n = gcd n $ a000005 n
-- Reinhard Zumkeller, May 09 2013, Aug 14 2011

f[n_] := GCD[n, DivisorSigma[0, n]]; Array[f, 105] (* Robert G. Wilson v, Mar 27 2013 *)

a(n)=gcd(numdiv(n),n) \\ Charles R Greathouse IV, Mar 26 2013

a(n) = gcd(n, A000005(n)) = gcd(n, A049820(n)). - Antti Karttunen, Sep 25 2018

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

Original entry on oeis.org

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).

A334490 a(n) = Sum_{d|n} gcd(d, sigma(d)).

Original entry on oeis.org

1, 2, 2, 3, 2, 9, 2, 4, 3, 5, 2, 14, 2, 5, 6, 5, 2, 13, 2, 8, 4, 5, 2, 27, 3, 5, 4, 34, 2, 21, 2, 6, 6, 5, 4, 19, 2, 5, 4, 19, 2, 19, 2, 10, 10, 5, 2, 32, 3, 7, 6, 8, 2, 20, 4, 43, 4, 5, 2, 40, 2, 5, 6, 7, 4, 21, 2, 8, 6, 11, 2, 35, 2, 5, 8, 10, 4, 19, 2, 22

Offset: 1

Jaroslav Krizek, May 03 2020

nonn

			a(6) = gcd(1, sigma(1)) + gcd(2, sigma(2)) + gcd(3, sigma(3)) + gcd(6, sigma(6)) = gcd(1, 1) + gcd(2, 3) + gcd(3, 4) + gcd(6, 12) = 1 + 1 + 1 + 6 = 9.

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))), A000203 (Sum_{d|n} gcd(d, pod(d)) = sigma(n)).
Cf. A000040, A007955 (pod(n)), A334491 (for product instead of sum).
Inverse Möbius transform of A009194.

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

a[n_] := DivisorSum[n, GCD[#, DivisorSigma[1, #]] &]; Array[a, 80] (* Amiram Eldar, May 03 2020 *)

a(n) = sumdiv(n, d, gcd(d, sigma(d))); \\ Michel Marcus, May 03 2020

a(p) = 2 for p = primes (A000040).

a(n) = Sum_{d|n} A009194(d). - Antti Karttunen, May 09 2020

A334664 a(n) = Product_{d|n} gcd(d, tau(d)).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 8, 3, 4, 1, 24, 1, 4, 1, 8, 1, 72, 1, 8, 1, 4, 1, 768, 1, 4, 3, 8, 1, 16, 1, 16, 1, 4, 1, 3888, 1, 4, 1, 256, 1, 16, 1, 8, 9, 4, 1, 1536, 1, 8, 1, 8, 1, 144, 1, 256, 1, 4, 1, 2304, 1, 4, 9, 16, 1, 16, 1, 8, 1, 16, 1, 1492992, 1, 4, 3, 8, 1

Offset: 1

Jaroslav Krizek, May 07 2020

nonn

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

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

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009191 (gcd(n, tau(n))).

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

Table[Times@@GCD[Divisors[n],DivisorSigma[0,Divisors[n]]],{n,80}] (* Harvey P. Dale, Mar 30 2024 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(d[k], numdiv(d[k]))); \\ Michel Marcus, May 08-11 2020

a(p) = 1 for p = odd primes (A065091).

A334782 a(n) = Sum_{d|n} lcm(d, tau(d)).

Original entry on oeis.org

1, 3, 7, 15, 11, 21, 15, 23, 16, 33, 23, 45, 27, 45, 77, 103, 35, 48, 39, 105, 105, 69, 47, 77, 86, 81, 124, 141, 59, 231, 63, 199, 161, 105, 165, 108, 75, 117, 189, 153, 83, 315, 87, 213, 176, 141, 95, 397, 162, 258, 245, 249, 107, 372, 253, 205, 273, 177

Offset: 1

Jaroslav Krizek, May 10 2020

nonn

			a(6) = lcm(1, tau(1)) + lcm(2, tau(2)) + lcm(3, tau(3)) + lcm(6, tau(6)) = lcm(1, 1) + lcm(2, 2) + lcm(3, 2) + lcm(6, 4) = 1 + 2 + 6 + 12 = 21.

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334783 (Sum_{d|n} lcm(d, sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009230 (lcm(n, tau(n))).

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

a[n_] := DivisorSum[n, LCM[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)

a(n) = sumdiv(n, d, lcm(d, numdiv(d))); \\ Michel Marcus, May 10 2020

a(p) = 2p + 1 for p = odd primes (A065091).

cp's OEIS Frontend

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

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

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A334490 a(n) = Sum_{d|n} gcd(d, sigma(d)).

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A334664 a(n) = Product_{d|n} gcd(d, tau(d)).

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A334782 a(n) = Sum_{d|n} lcm(d, tau(d)).

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