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A293575 Difference between the number of proper divisors of n and the number of squares dividing n. 2
-1, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 3, 0, 2, 2, 1, 0, 3, 0, 3, 2, 2, 0, 5, 0, 2, 1, 3, 0, 6, 0, 2, 2, 2, 2, 4, 0, 2, 2, 5, 0, 6, 0, 3, 3, 2, 0, 6, 0, 3, 2, 3, 0, 5, 2, 5, 2, 2, 0, 9, 0, 2, 3, 2, 2, 6, 0, 3, 2, 6, 0, 7, 0, 2, 3, 3, 2, 6, 0, 6, 1, 2, 0, 9, 2, 2, 2, 5, 0, 9, 2, 3, 2, 2, 2, 8, 0, 3, 3, 4, 0, 6, 0, 5, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

The difference between the number of ways of writing n = m + k and the number of ways of writing n = r*s, where m|k and r|s.

First occurrence of k beginning with k=-1: 1, 2, 8, 6, 12, 36, 24, 30, 72, 96, 60, 2097152, 216, 576, 120, 210, 1152, 240, 864, etc. - Robert G. Wilson v, Nov 28 2017

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A032741(n) - A046951(n).

a(n) = A056595(n) - 1. - Antti Karttunen, Oct 30 2017

a(n) = 0 iff n is a prime or a square of a prime, A000430. - Robert G. Wilson v, Nov 28 2017

EXAMPLE

a(6) = 2 because 2 is difference of number of ways of writing n = 1 + 5 = 2 + 4 = 3 + 3 where 1|5, 2|4, 3|3 and number of ways of writing n = 1*6 where 1|6.

MATHEMATICA

f[n_] := Block[{d = Divisors@ n}, Length@ d - Length[ Select[ d, IntegerQ@ Sqrt@# &]] - 1];; Array[f, 105] (* Robert G. Wilson v, Nov 28 2017 *)

CROSSREFS

One less than A056595.

Cf. A000430, A032741, A046951, A166684, A080257.

Sequence in context: A208459 A144764 A084929 * A054014 A158945 A156667

Adjacent sequences:  A293572 A293573 A293574 * A293576 A293577 A293578

KEYWORD

sign

AUTHOR

Juri-Stepan Gerasimov, Oct 14 2017

STATUS

approved

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Last modified August 5 22:11 EDT 2020. Contains 336214 sequences. (Running on oeis4.)