A095960 -id:A095960 - cp's OEIS frontend

A372720 a(n) = A000005(n) - A008479(n).

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

Offset: 1

Michael De Vlieger, May 13 2024

sign

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

			Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
  n  b(n) c(n) a(n)
 ------------------
  1    1    1    0
  2    2    1    1
  3    2    1    1
  4    3    2    1
  5    2    1    1
  6    4    1    3
  7    2    1    1
  8    4    3    1
  9    3    2    1
 10    4    1    3
 11    2    1    1
 12    6    2    4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.
a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).
a(n) = A183093(n) - A355432(n).

A073180 Number of divisors of n which are not greater than the squarefree kernel of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 4, 2, 5, 4, 4, 2, 5, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 5, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 5, 2, 4, 4, 5, 2, 4, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 5, 2, 4, 4, 5, 4, 8, 2, 6, 2, 4, 2, 11, 4, 4, 4, 6, 2, 10, 4, 5, 4, 4, 4, 5, 2, 4, 5, 5, 2, 8, 2

Offset: 1

Reinhard Zumkeller, Jul 19 2002

nonn

			The squarefree kernel of 56 = 7*2^3 is 14 = 7*2 and the divisors <= 14 of 56 are {1, 2, 4, 7, 8, 14}, therefore a(56) = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A007947, A034444, A073181, A073182, A095960.

a073180 n = length [x | x <- a027750_row n, x <= a007947 n]
-- Reinhard Zumkeller, Sep 10 2013

a[n_] := DivisorSum[n, 1 &, # <= Times @@ FactorInteger[n][[;;, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

a(n) >= A034444(n).

a(n) = A095960(n) + 1. - Reinhard Zumkeller, Sep 10 2013

cp's OEIS Frontend

A372720 a(n) = A000005(n) - A008479(n).

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