A131377 -id:A131377 - cp's OEIS frontend

A131807 Partial sums of A131377.

1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 22, 22, 23, 24, 25, 26, 26, 26, 26, 26, 26, 26, 27, 28, 29, 30, 31, 32, 32, 32, 33, 34, 35, 36, 37, 38, 38, 38, 38, 38, 39, 40, 40

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2007

Cf. A131377.

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

a(n) = sum(m=0, n, (primepi(m) % 2) == 0) \\ Michel Marcus, Aug 15 2023

A131378 Starting with 0, the sequence a(n) changes from 0 to 1 or back when the next number n is a prime.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 04 2007

Zero together with A071986. - Omar E. Pol, Feb 19 2011

Parity of A230980. - Omar E. Pol, Jul 19 2019

			n = 0, 1, 2, 3, 4, 5, etc.
a(n)= 0, 0, 1, 0, 0, 1, etc.
Starting with 0 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. A131377.

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

P:=proc(n) local i,k; k:=0; 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);

nxt[{n_,a_}]:={n+1,Which[a==0&&PrimeQ[n+1],1,a==1&&PrimeQ[n+1],0,True,a]}; NestList[nxt,{0,0},100][[All,2]] (* Harvey P. Dale, Jul 19 2019 *)

A345220 Number of divisors d of n with an even number of primes not exceeding d.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Jun 11 2021

nonn

Inverse Möbius transform of (pi(n)+1) mod 2 = A131377(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 even values of pi(d).

Cf. A000005 (tau), A000720 (pi), A131377, A345219.

Table[Sum[Mod[PrimePi[d] + 1, 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)+1) mod 2).

a(n) = A000005(n) - A345219(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

A131807 Partial sums of A131377.

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A131378 Starting with 0, the sequence a(n) changes from 0 to 1 or back when the next number n is a prime.

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A345220 Number of divisors d of n with an even 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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Programs

Mathematica

Formula