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A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n. 1
0, 1, 1, 3, 1, 4, 1, 7, 5, 10, 1, 11, 1, 16, 7, 15, 1, 6, 1, 31, 13, 28, 1, 36, 9, 34, 19, 31, 1, -20, 1, 31, 25, 46, 7, 47, 1, 52, 31, 106, 1, -62, 1, 31, 21, 64, 1, 151, 13, 66, 43, 31, 1, -34, 19, 8, 49, 82, 1, 727, 1, 88, 71, 63, 25, -6, 1, 31, 61, 148, 1, 12, 1, 106, 11, 31, 13, 22, 1, 439, 65, 118, 1, 1541 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

From David A. Corneth, May 20 2016: (Start)

The bottom of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):

a

. . b-a

b . . . . c-2b+a

. . c-b . . . . . d-3c+3b-a

c . . . . d-2c+b . . . . . . e-4d+6c-4b+a

. . d-c . . . . . e-3d+3c-b

d . . . . e-2d+c

. . e-d

e

From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.

(End)

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = n - A187202(n).

a(n) = 1, if n is prime.

a(2^k) = 2^k - 1 = A000225(k), k >= 0.

EXAMPLE

For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is:

1 . 2 . 3 . 6 . 9 . 18

. 1 . 1 . 3 . 3 . 9

. . 0 . 2 . 0 . 6

. . . 2 .-2 . 6

. . . .-4 . 8

. . . . . 12

The bottom entry is 12, so a(18) = 18 - 12 = 6.

MATHEMATICA

Array[# - First@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &, 84] (* Michael De Vlieger, May 20 2016 *)

PROG

(Sage)

def A273133(n):

    D = divisors(n)

    T = matrix(ZZ, len(D))

    for (m, d) in enumerate(D):

        T[0, m] = d

        for k in range(m-1, -1, -1) :

            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]

    return n - T[len(D)-1, 0]

print [A273133(n) for n in range(1, 85)] # Peter Luschny, May 18 2016

(PARI) a(n) = my(d=divisors(n)); n-sum(i=1, #d, binomial(#d-1, i-1)*(-1)^(#d-i)*d[i]) \\ David A. Corneth, May 20 2016

CROSSREFS

Cf. A000005, A000040, A000225, A007318, A187202, A273102, A273103, A273109.

Sequence in context: A006022 A078896 A322582 * A318841 A300238 A180062

Adjacent sequences:  A273130 A273131 A273132 * A273134 A273135 A273136

KEYWORD

sign

AUTHOR

Omar E. Pol, May 17 2016

STATUS

approved

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Last modified February 18 09:41 EST 2019. Contains 320249 sequences. (Running on oeis4.)