A141767 -id:A141767 - cp's OEIS frontend

A141766 A positive integer n is included if both (p-1) and (p+1) divide n for every prime p that divides n.

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

Offset: 1

Leroy Quet, Jul 02 2008

nonn

Every term is a multiple of 12.

			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.

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

Cf. A140470, A141767, A124240.

Cf. A027748.

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

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

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

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

A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

Original entry on oeis.org

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

Offset: 1

Ctibor O. Zizka, May 22 2009

nonn

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000032, A001610, A111058, A140826, A140971, A140972.

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

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 *)

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

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

A254141 The average of a(n) consecutive Fibonacci numbers is never an integer.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Jan 26 2015

nonn

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 an 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.

Paolo P. Lava, Table of n, a(n) for n = 1..200
Paolo P. Lava, List of indexes and steps (k, x, y)

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

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);

cp's OEIS Frontend

A141766 A positive integer n is included if both (p-1) and (p+1) divide n for every prime p that divides n.

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A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

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Maple

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A254141 The average of a(n) consecutive Fibonacci numbers is never an integer.

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Maple