A353361 -id:A353361 - cp's OEIS frontend

A353362 Number of divisors d of n for which A156552(d) is a multiple of 3.

1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 2, 2, 1, 2, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 1, 2, 4, 1, 2, 1, 3, 2, 2, 1, 3, 2, 3, 1, 2, 1, 2, 2, 2, 2, 1, 1, 4, 1, 2, 3, 4, 1, 3, 1, 3, 1, 2, 1, 4, 1, 1, 2, 2, 1, 2, 1, 5, 3, 2, 1, 4, 2, 1, 2, 4, 1, 5, 2, 3, 1, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 2, 3

Offset: 1

Antti Karttunen, Apr 15 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Inverse Möbius transform of A353269.
Cf. A000005, A003961, A156552, A348717, A353303, A353304, A353361.
Cf. also A353352.
Differs from A353332 for the first time at n=30, where a(30) = 3, while A353332(30) = 2.

A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353362(n) = sumdiv(n,d,A353269(d));

a(n) = Sum_{d|n} A353269(d).
a(n) = A000005(n) - A353361(n).
a(p) = 1 for all primes p.
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

A353351 Number of divisors d of n for which A048675(d) is not a multiple of 3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 15 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A003961, A048675, A332820, A348717, A353328, A353329, A353350, A353352, A353354.

Cf. also A353361.

f[p_, e_] := e*2^(PrimePi[p] - 1); q[1] = False; q[n_] := ! Divisible[Plus @@ f @@@ FactorInteger[n], 3]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Apr 15 2022 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A353350(n) = (0==(A048675(n)%3));
A353351(n) = sumdiv(n,d,!A353350(d));

a(n) = Sum_{d|n} (1-A353350(d)).
a(n) = A000005(n) - A353352(n).
a(p) = 1 for all primes p.
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
a(n) = A353328(n) + A353329(n).

A353364 Inverse Möbius transform of A332814.

Original entry on oeis.org

0, 1, -1, 1, 1, -1, -1, 2, -1, 2, 1, -2, -1, -1, 1, 2, 1, 0, -1, 3, -2, 2, 1, -2, 1, -1, -2, -2, -1, 1, 1, 3, 1, 2, -1, -1, -1, -1, -2, 4, 1, -2, -1, 3, 2, 2, 1, -3, -1, 3, 1, -2, -1, -2, 2, -2, -2, -1, 1, 2, -1, 2, -3, 3, -1, 1, 1, 3, 1, -2, -1, 0, 1, -1, 0, -2, 1, -2, -1, 5, -2, 2, 1, -4, 2, -1, -2, 4, -1, 3, -2, 3

Offset: 1

Antti Karttunen, Apr 15 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A156552, A332814.

Cf. also A353354, A353361, A353362.

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332814(n) = { my(u=A156552(n)%3); if(2==u,-1,u); };
A353364(n) = sumdiv(n,d,A332814(d));

a(n) = Sum_{d|n} A332814(d).

For all n >= 1, a(A003961(n)) = -a(n), a(A000040(n)) = ((-1)^(n-1)).

cp's OEIS Frontend

A353362 Number of divisors d of n for which A156552(d) is a multiple of 3.

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A353351 Number of divisors d of n for which A048675(d) is not a multiple of 3.

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A353364 Inverse Möbius transform of A332814.

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