A345220 -id:A345220 - cp's OEIS frontend

A131377 a(n) = (pi(n)+1) mod 2.

1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 04 2007

Old name was: Starting with 1, the sequence a(n) changes from 1 to 0 or back when the next number n is a prime.

Möbius transform of A345220(n). - Wesley Ivan Hurt, Jul 05 2025

			n = 0, 1, 2, 3, 4, 5, etc..
a(n)= 1, 1, 0, 1, 1, 0, etc.
Starting with 1 the sequence changes when we move from 1 to 2 because 2 is prime, again from 2 to 3 because also 3 is prime, then from 4 to 5 being 5 prime and so on.

Cf. A000035 (n mod 2), A000720 (pi), A008683 (mu), A036234, A131378, A345220.

Cf. A071986. - Omar E. Pol, Feb 19 2011

P:=proc(n) local i,k; k:=1; for i from 0 by 1 to n do if isprime(i) then if k=1 then k:=0; else k:=1; fi; fi; print(k); od; end: P(100);

Table[Mod[PrimePi[n] + 1, 2], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 05 2025 *)

a(n) = 1 - A071986(n).
From Wesley Ivan Hurt, Jul 05 2025: (Start)
a(n) = A000035(A036234(n)).
a(n) = Sum_{d|n} A345220(d) * mu(n/d). (End)

New name from Wesley Ivan Hurt, Jul 05 2025

A345219 Number of divisors d of n with an odd number of primes not exceeding d.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Jun 11 2021

nonn

Inverse Möbius transform of pi(n) mod 2 = A071986(n). - Wesley Ivan Hurt, Jul 04 2025

			a(24) = 4; The divisors d of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 and the corresponding values of pi(d) are: 0, 1, 2, 2, 3, 4, 5, 9. There are 4 odd values of pi(d).

Cf. A000005 (tau), A000720 (pi), A071986, A345220.

Table[Sum[Mod[PrimePi[d], 2], {d, Divisors[n]}], {n, 100}]

a(n) = sumdiv(n, d, primepi(d) % 2); \\ Michel Marcus, Jun 11 2021

a(n) = Sum_{d|n} (pi(d) mod 2).

a(n) = A000005(n) - A345220(n). - Wesley Ivan Hurt, Jul 05 2025

A385628 Sum of the divisors d of n with an even number of primes not exceeding d.

Original entry on oeis.org

1, 1, 4, 5, 1, 4, 8, 13, 13, 11, 1, 8, 14, 22, 19, 29, 1, 13, 20, 35, 32, 23, 1, 16, 1, 14, 13, 26, 30, 59, 1, 29, 4, 1, 8, 17, 38, 58, 56, 83, 1, 46, 44, 71, 73, 47, 1, 32, 8, 11, 4, 18, 54, 67, 56, 90, 80, 88, 1, 83, 62, 63, 104, 93, 79, 92, 1, 5, 4, 32, 72, 97, 1, 38, 19, 62, 8, 56, 80, 179, 94, 83, 1, 50, 1, 44, 33, 79, 90, 203, 112, 143, 97, 95, 115

Offset: 1

Wesley Ivan Hurt, Jul 05 2025

Inverse Möbius transform of n * ((pi(n)+1) mod 2) = n * A131377(n).

			The sum of the divisors d of 16 such that pi(d) is even gives 1 + 4 + 8 + 16 = 29.

Cf. A000203 (sigma), A000720 (pi), A131377, A345220, A385625.

Table[Sum[d*Mod[PrimePi[d] + 1, 2], {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} d * ((pi(d)+1) mod 2).

a(n) = A000203(n) - A385625(n).

cp's OEIS Frontend

A131377 a(n) = (pi(n)+1) mod 2.

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A345219 Number of divisors d of n with an odd number of primes not exceeding d.

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A385628 Sum of the divisors d of n with an even number of primes not exceeding d.

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