A141767 -id:A141767

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A141766

A positive integer n is included if both (p-1) and (p+1) divide n for every prime p that divides n.

+10
4

1, 12, 24, 36, 48, 60, 72, 96, 108, 120, 144, 168, 180, 192, 216, 240, 288, 300, 324, 336, 360, 384, 432, 480, 504, 540, 576, 600, 648, 660, 672, 720, 768, 840, 864, 900, 960, 972, 1008, 1080, 1152, 1176, 1200, 1296, 1320, 1344, 1440, 1500, 1512, 1536, 1620

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Every term is a multiple of 12.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

EXAMPLE

120 has the prime factorization of 2^3 * 3^1 * 5^1. The distinct primes dividing 120 are therefore 2,3,5. 2-1=1, 3-1=2 and 5-1=4 all divide 120. Also, 2+1=3, 3+1=4 and 5+1=6 all divide 120. So 120 is included in the sequence.

MATHEMATICA

Select[Range[2, 1620], Function[n, AllTrue[FactorInteger[n][[All, 1]], AllTrue[# + {-1, 1}, Divisible[n, #] &] &]]] (* Michael De Vlieger, Sep 22 2017 *)

PROG

(Haskell)

a141766 n = a141766_list !! (n-1)

a141766_list = filter f [1..] where

f x = all (== 0) $ map (mod x) $ (map pred ps) ++ (map succ ps)

where ps = a027748_row x

-- Reinhard Zumkeller, Aug 27 2013

CROSSREFS

Cf. A140470, A141767, A124240.

Cf. A027748.

KEYWORD

nonn

AUTHOR

Leroy Quet, Jul 02 2008

EXTENSIONS

a(12)-a(50) from Donovan Johnson, Sep 27 2008

a(1)=1 prepended by Max Alekseyev, Aug 27 2013

STATUS

approved

A160665

Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

+10
2

1, 3, 24, 48, 72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 406, 432, 480, 576, 600, 648, 672, 720, 768, 864, 936, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2160, 2208, 2304

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers n such that sum(i=0..n, A000032(i))/(n+1) is an integer. - Robert G. Wilson v, May 25 2009

Why do the terms in A141767 so closely correspond to A160665? Except for n = 1, 3, 406, 44758, 341446, 1413286, 3170242, 4861698, 7912534, ..., n == 0 (mod 24). - Robert G. Wilson v, May 25 2009

LINKS

Table of n, a(n) for n=1..48.

FORMULA

{n: n | A001610(n)}. - R. J. Mathar, May 25 2009

MAPLE

A000032 := proc(n) option remember ; if n <= 1 then 2-n; else procname(n-1)+procname(n-2) ; fi; end: A001610 := proc(n) add(A000032(i), i=0..n-1) ; end: for n from 1 to 3000 do if A001610(n) mod n = 0 then printf("%d, ", n) ; fi; od: # R. J. Mathar, May 25 2009

MATHEMATICA

lst = {}; a = 2; b = 1; s = 3; n = 3; While[n < 2447, c = a + b; s = s + c; If[Mod[c, n] == 0, AppendTo[lst, n]]; a = b; b = c; n++ ]; lst (* Robert G. Wilson v, May 25 2009 *)

CROSSREFS

Cf. A000032, A111058, A140826, A140971, A140972.

KEYWORD

nonn

AUTHOR

Ctibor O. Zizka, May 22 2009

EXTENSIONS

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

STATUS

approved

A254141

The average of a(n) consecutive Fibonacci numbers is never an integer.

+10
2

8, 16, 21, 28, 32, 40, 52, 55, 56, 64, 65, 68, 69, 80, 84, 85, 87, 88, 92, 93, 99, 104, 105, 112, 117, 119, 128, 132, 133, 136, 140, 141, 145, 148, 152, 153, 155, 156, 160, 161, 164, 165, 171, 172, 176, 184, 187, 188, 196, 200, 203, 204, 205, 207, 208, 209, 212

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Subset of A033949 and A175594 (essentially the same sequence).

Numbers of the form 2^k, with k>=3, appear to be part of the sequence.

The file "List of indexes and steps (k, x, y)" (see Links) for k = 1, 2, 3, 4, ... consecutive Fibonacci numbers gives the minimum index to start to calculate the average ( x ) and the step to add to get all the other averages ( y ).

e.g.: for k = 7 we have 7, 6, 8. This means that we must start from the 6th Fibonacci number to add 7 consecutive Fibonacci numbers and get an average that is integer. Fibonacci(6) + Fibonacci(7) + ... + Fibonacci(12) = 8 + 13 + 21 + 34 + 55 + 89 + 144 = 364 and 364 / 7 = 52.

Then 6 + 1*8 = 14, 6 + 2*8 = 22, 6 + 3*8 = 30, etc. are the other indexes:

Fibonacci(14) + Fibonacci (15) + ... + Fibonacci(20) = 377 + 610 + 987 + 1597 + 2584 + 4181 + 6765 = 17101 and 17101 / 7 = 2443;

Fibonacci(22) + Fibonacci(23) + ... + Fibonacci(28) = 17711 + 28657 + 46368 + 75025 + 121393 + 196418 + 317811 = 803383 and 803383 / 7 = 114769;

Fibonacci(30) + Fibonacci(31) + ... + Fibonacci(36) = 832040 + 1346269 + 2178309 + 3524578 + 5702887 + 9227465 + 14930352 = 37741900 and 37741900 / 7 = 5391700; etc.

In particular we note that:

x = 0 is A219612; x = 1 is A124456; x = 0 and y = k - 1 is A106535;

y = 1 is A141767; x = k - 1 and y = k + 1 is A000057;

x = y - 1 or y|k is A023172; y = k is A000351;

x = y - k + 1 appears to give only prime numbers: 3,11,19,31,59,71,79,131,179,191,239,251,271,311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971, etc.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..200

Paolo P. Lava, List of indexes and steps (k, x, y)

MAPLE

with(numtheory); with(combinat):P:=proc(q) local a, b, k, j, n, ok;

for j from 1 to q do b:=0; ok:=1;

for n from 0 to q do a:=add(fibonacci(n+k), k=0..j-1)/j;

if type(a, integer) then ok:=0; break; fi; od;

if ok=1 then print(j); fi; od; end: P(20000);

CROSSREFS

Cf. A000045, A000057, A000071, A023172, A033949, A106535, A124456, A141767, A175594, A219612.

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Jan 26 2015

STATUS

approved

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