A355828 -id:A355828 - cp's OEIS frontend

A366384 Lexicographically earliest infinite sequence such that a(i) = a(j) => A355828(i) = A355828(j) for all i, j >= 1, where A355828 is Dirichlet inverse of A342671, the greatest common divisor of sigma(n) and A003961(n).

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

Offset: 1

Antti Karttunen, Oct 12 2023

nonn

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

Cf. A000203, A003961, A342671, A355828.

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; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
v366384 = rgs_transform(DirInverseCorrect(vector(up_to,n,A342671(n))));
A366384(n) = v366384[n];

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

A355829 Dirichlet inverse of A009194, the greatest common divisor of sigma(n) and n, where sigma is the sum of divisors function.

Original entry on oeis.org

1, -1, -1, 0, -1, -4, -1, 0, 0, 0, -1, 7, -1, 0, -1, 0, -1, 8, -1, 1, 1, 0, -1, -10, 0, 0, 0, -25, -1, 10, -1, 0, -1, 0, 1, 15, -1, 0, 1, -8, -1, 6, -1, -1, 2, 0, -1, 16, 0, 2, -1, 1, -1, -6, 1, 46, 1, 0, -1, -9, -1, 0, 0, 0, 1, 10, -1, 1, -1, 2, -1, -29, -1, 0, 4, -1, 1, 6, -1, 16, 0, 0, -1, 29, 1, 0, -1, 2, -1, -8

Offset: 1

Antti Karttunen, Jul 20 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A009194.

Cf. also A355828.

s[n_] := GCD[n, DivisorSigma[1, n]]; a[1] = 1; a[n_] := - DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 20 2022 *)

A009194(n) = gcd(n, sigma(n));
memoA355829 = Map();
A355829(n) = if(1==n,1,my(v); if(mapisdefined(memoA355829,n,&v), v, v = -sumdiv(n,d,if(dA009194(n/d)*A355829(d),0)); mapput(memoA355829,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA009194(n/d) * a(d).

cp's OEIS Frontend

A366384 Lexicographically earliest infinite sequence such that a(i) = a(j) => A355828(i) = A355828(j) for all i, j >= 1, where A355828 is Dirichlet inverse of A342671, the greatest common divisor of sigma(n) and A003961(n).

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A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

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A355829 Dirichlet inverse of A009194, the greatest common divisor of sigma(n) and n, where sigma is the sum of divisors function.

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