A193672 -id:A193672 - cp's OEIS frontend

A187202 The bottom entry in the difference table of the divisors of n.

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, -2, 8, 1, 16, 12, 18, -11, 8, -6, 22, -12, 16, -8, 8, -3, 28, 50, 30, 1, 8, -12, 28, -11, 36, -14, 8, -66, 40, 104, 42, 13, 24, -18, 46, -103, 36, -16, 8, 21, 52, 88, 36, 48, 8, -24, 58, -667, 60, -26, -8, 1, 40, 72

Offset: 1

Omar E. Pol, Aug 01 2011

Note that if n is prime then a(n) = n - 1.
Note that if n is a power of 2 then a(n) = 1.
a(A193671(n)) > 0; a(A187204(n)) = 0; a(A193672(n)) < 0. [Reinhard Zumkeller, Aug 02 2011]
First differs from A187203 at a(14). - Omar E. Pol, May 14 2016
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)

			a(18) = 12 because 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
with bottom entry a(18) = 12.
Note that A187203(18) = 4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A007318, A027750, A187203, A273102.

a187202 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

f:=proc(n) local k,d,lis; lis:=divisors(n); d:=nops(lis);
add( (-1)^k*binomial(d-1,k)*lis[d-k], k=0..d-1); end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, May 01 2016

Table[d = Divisors[n]; Differences[d, Length[d] - 1][[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)

A187202(n)={ for(i=2,#n=divisors(n), n=vecextract(n,"^1")-vecextract(n,"^-1")); n[1]}  \\ M. F. Hasler, Aug 01 2011

a(n) = Sum_{k=0..d-1} (-1)^k*binomial(d-1,k)*D[d-k], where D is a sorted list of the d = A000005(n) divisors of n. - N. J. A. Sloane, May 01 2016

a(2^k) = 1.

Edited by N. J. A. Sloane, May 01 2016

A187204 Numbers n such that the bottom entry in the difference table of the divisors of n is 0.

Original entry on oeis.org

10, 171, 1947, 2619, 265105, 478834027, 974622397, 11373118351

Offset: 1

Omar E. Pol, Aug 01 2011

Numbers n such that A187202(n) = 0.
11373118351 and 1756410942451 are also in the sequence (not necessarily the next two terms). - Donovan Johnson, Aug 05 2011
For every integer m, does there exist a prime p such that abs(A187202(r * m)) > abs(A187202(q * m)) and sign(A187202(r * m)) = sign(A187202(q * m)), and q >= p is prime and prime r > q? - David A. Corneth, Apr 08 2017
No other terms up to 3*10^9. - Michel Marcus, Apr 09 2017
a(9) > 6*10^10. 138662735650982521 and 168248347462416481 are also terms. - Giovanni Resta, Apr 12 2017

			10 has divisors 1, 2, 5, 10. The third difference of these numbers is 0.  This is the only possible number having 2 prime factors of the form p*q. The other terms have factorization 171 = 3^2*19, 1947 = 3*11*59, 2619 = 3^3*97, and 265105 = 5*37*1433.

Cf. A027750, A187202, A187203, A193671, A193672.

import Data.List (elemIndices)
a187204 n = a187204_list !! (n-1)
a187204_list = map (+ 1) $ elemIndices 0 $ map a187202 [1..]
-- Reinhard Zumkeller, Aug 02 2011

t = {}; Do[d = Divisors[n]; If[Differences[d, Length[d]-1] == {0}, AppendTo[t, n]], {n, 10^4}]; t (* T. D. Noe, Aug 01 2011 *)

is(n) = my(d=divisors(n)); !sum(i=1, #d, binomial(#d-1,i-1)*d[i]*(-1)^i) \\ David A. Corneth, Apr 08 2017

Suggested by T. D. Noe in the "history" of A187203.
a(6)-a(7) from Donovan Johnson, Aug 03 2011
a(8) from Giovanni Resta, Apr 11 2017

A193671 Numbers such that the last of the differences of divisors is > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 44, 45, 47, 49, 51, 52, 53, 54, 55, 56, 57, 59, 61, 64, 65, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 87, 88, 89, 90, 91

Offset: 1

Reinhard Zumkeller, Aug 02 2011

nonn

A187202(a(n)) > 0.

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

Cf. A187204, A193672.

import Data.List (findIndices)
a193671 n = a193671_list !! (n-1)
a193671_list = map (+ 1) $ findIndices (> 0) $ map a187202 [1..]

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A187202 The bottom entry in the difference table of the divisors of n.

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A187204 Numbers n such that the bottom entry in the difference table of the divisors of n is 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A193671 Numbers such that the last of the differences of divisors is > 0.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell