A306695 -id:A306695 - cp's OEIS frontend

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of squarefree divisors of n.

(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

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

Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
Cf. also A348734, A348735.
Cf. A059956, A065463, A082020, A094387, A339076.

a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332880(n) = numerator(A001615(n)/n);

Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = numerator of Sum_{d|n} mu(d)^2/d.
a(n) = numerator of psi(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
From Amiram Eldar, Nov 24 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)

A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 2, 17, 1, 19, 5, 21, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 5, 23, 47, 1, 7, 5, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of squarefree divisors of n.

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

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

Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.

a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332881(n) = denominator(A001615(n)/n);

Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = denominator of Sum_{d|n} mu(d)^2/d.
a(n) = denominator of psi(n)/n.
a(p) = p, where p is prime.
a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021

A327979 a(n) = gcd(n, A002322(n)), where A002322 is Carmichael lambda, also known as psi.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 6, 1, 4, 3, 2, 1, 2, 5, 2, 9, 2, 1, 2, 1, 8, 1, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 1, 4, 7, 10, 1, 4, 1, 18, 5, 2, 3, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 5, 2, 1, 6, 1, 4, 27, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 8, 1, 14, 3, 20, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Oct 04 2019

nonn

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

Cf. A002322.

Cf. also A009191, A009194, A009195, A306695.

Array[GCD[#, CarmichaelLambda[#]] &, 100] (* Amiram Eldar, Oct 04 2019 *)

PARI
```
A327979(n) = gcd(n, lcm(znstar(n)[2]));
```

a(n) = gcd(n, A002322(n)).

A306528 Numbers k such that gcd(k, phi(k)) = gcd(k, psi(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 43, 44, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 92, 94, 97, 98, 100

Offset: 1

Torlach Rush, Feb 21 2019

nonn

Here phi(n) is Euler's totient function A000010 and psi(n) is Dedekind's psi function A001615.
This sequence contains all prime powers p^k where phi(p^k) and psi(p^k) are equidistant from p^k, and gcd(p^k, phi(p^k)) = gcd(p^k, psi(p^k)) = p^(k - 1). For the prime numbers themselves this is trivial since phi(p) and psi(p) differ from p by 1 and 1^0 = 1.
If prime p|k, then p*k is in the sequence if and only if k is in the sequence. - Robert Israel, Mar 05 2019

			1 is a term because gcd(1, 1) = gcd(1, 1) = 1.
2 is a term because gcd(2, 1) = gcd(2, 3) = 1.
3 is a term because gcd(3, 2) = gcd(3, 4) = 1.
4 is a term because gcd(4, 2) = gcd(4, 6) = 2.
5 is a term because gcd(5, 4) = gcd(5, 6) = 1.
6 is not a term because gcd(6, 2) <> gcd(6, 12).
7 is a term because gcd(7, 6) = gcd(7, 8) = 1.

Cf. A000010, A001615, A009195, A306695.

filter:= proc(n) local p,F;
  F:= numtheory:-factorset(n);
  igcd(n, n*mul(1-1/p, p=F)) = igcd(n, n*mul(1+1/p,p=F))
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 05 2019

dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(k) = gcd(k, eulerphi(k)) == gcd(k, dpsi(k)); \\ Michel Marcus, Feb 27 2019

A306711 Numbers k such that gcd(k, phi(k)) <> gcd(k, psi(k)).

Original entry on oeis.org

6, 12, 15, 18, 21, 24, 30, 33, 36, 39, 45, 48, 51, 54, 55, 57, 60, 63, 66, 69, 72, 75, 87, 90, 91, 93, 95, 96, 99, 102, 108, 110, 111, 117, 120, 123, 129, 132, 135, 138, 141, 144, 145, 147, 150, 153, 155, 159, 162, 165, 171, 174, 177, 180, 182, 183, 189, 190, 192, 198, 201

Offset: 1

Torlach Rush, Mar 05 2019

nonn

Numbers m such that A306695(m) = m are terms.

			6 is a term because gcd(6,2) <> gcd(6,12).
12 is a term because gcd(12,4) <> gcd(12, 24).
13 is not a term because gcd(13,12) = gcd(13, 14).
14 is not a term because gcd(14,6) = gcd(14, 24).

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

Cf. A000010 (Euler totient function), A001615 (Dedekind psi function).
Complement of A306528.
Cf. A306695.

psi:= k -> mul((t+1)/t, t=numtheory:-factorset(k))*k:
select(t -> igcd(t, psi(t)) <> igcd(t, numtheory:-phi(t)), [$1..1000]); # Robert Israel, Apr 28 2019

dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(k) = gcd(k, eulerphi(k)) != gcd(k, dpsi(k)); \\ Michel Marcus, Mar 21 2019

cp's OEIS Frontend

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

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A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

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A327979 a(n) = gcd(n, A002322(n)), where A002322 is Carmichael lambda, also known as psi.

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A306528 Numbers k such that gcd(k, phi(k)) = gcd(k, psi(k)).

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A306711 Numbers k such that gcd(k, phi(k)) <> gcd(k, psi(k)).

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Maple

PARI