A279436 Number of nonprimes less than or equal to n that do not divide n.

0, 0, 0, 0, 1, 1, 2, 1, 3, 4, 5, 3, 6, 6, 7, 6, 9, 7, 10, 8, 11, 12, 13, 9, 14, 15, 15, 15, 18, 15, 19, 16, 20, 21, 22, 18, 24, 24, 25, 22, 27, 24, 28, 26, 27, 30, 31, 25, 32, 31, 34, 33, 36, 32, 37, 34, 39, 40, 41, 34, 42, 42, 41, 40, 45, 43, 47, 45, 48, 46, 50, 42, 51, 51, 50, 51, 54, 52, 56, 50, 55, 58, 59, 52, 60, 61, 62, 59, 64, 57, 65, 64, 67, 68, 69, 62, 71, 69, 70, 68

Offset

1,7

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

G.f.: A(x) = B(x) + C(x) - D(x), where B(x) = Sum_{k>=1} x^(2*k+1)/((1 - x^k)*(1 - x^(k+1))), C(x) = Sum_{k>=1} x^prime(k)/(1 - x^prime(k)), D(x) = Sum_{k>=1} x^prime(k)/(1 - x).

a(n) = n - A000720(n) - A000005(n) + A001221(n).

a(n) = A062298(n) - A033273(n).

a(n) = A049820(n) - A048865(n).

a(n) = A229109(n) - A082514(n).

a(A000040(n)) = A065890(n).

a(A000040(n)) + 1 = A014689(n).

A000040(n) - a(A000040(n)) = n + 1.

Example

a(10) = 4 because 10 has 4 divisors {1,2,5,10} therefore 6 non-divisors {3,4,6,7,8,9} out of which 4 are nonprimes {4,6,8,9}.

Mathematica

Table[n - PrimePi[n] - DivisorSigma[0, n] + PrimeNu[n], {n, 1, 100}]

Prog

(PARI) for(n=1, 50, print1(n - primepi(n) - numdiv(n) + omega(n), ", ")) \\ G. C. Greubel, May 22 2017
(PARI) first(n)=my(v=vector(n), pp); forfactored(k=1, n, if(k[2][, 2]==[1]~, pp++); v[k[1]]=k[1] - pp - numdiv(k) + omega(k)); v \\ Charles R Greathouse IV, May 23 2017
(Python)
from sympy import primepi, divisor_count, primefactors
def a(n): return 0 if n==1 else n - primepi(n) - divisor_count(n) + len(primefactors(n)) # Indranil Ghosh, May 23 2017

Crossrefs

Cf. A000005, A000040, A000720, A001221, A014689, A033273, A048865, A049820, A062298, A065890, A082514, A229109.

Sequence in context: A340383 A117407 A232095 * A082470 A101204 A169808

Adjacent sequences: A279433 A279434 A279435 * A279437 A279438 A279439

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 12 2016

Status

approved