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A094960 Positive integers n such that the derivative of n-th Bernoulli polynomial B(n,x) contains only integer coefficients. 0
1, 2, 4, 6, 10, 12, 28, 30, 36, 60 (list; graph; refs; listen; history; text; internal format)



No other terms below 10^9. [From Max Alekseyev, Dec 08 2011]

Integer n belongs to this sequence if n*binomial(n-1,k)*bernoulli(k) is integer for each k=0,1,...,n-1. [From Max Alekseyev, Dec 08 2011]

If for a prime p>=3, n ends with base-p digits a,b with a+b>=p, then for k=(a+1)(p-1), the number n*binomial(n-1,k)*bernoulli(k) is not integer (contains p in the denominator). For p=3, this implies that n == 5, 7, or 8 (mod 9) are not in this sequence; for p=5, this implies that n = 9, 13, 14, 17, 18, 19, 21, 22, 23, or 24 (mod 25) are not in this sequence; and so on. [From Max Alekseyev, Jun 04 2012]

Conjecture: for every integer n>78 there exists a prime p such that the sum of last two base-p digits of n is at least p. This conjecture would imply that this sequence is finite and 60 is the last term. The conjecture is true for n such that n+1 is not a prime or a power of 2. [From Max Alekseyev, Jun 04 2012]


Table of n, a(n) for n=1..10.


B(6,x) = x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42 so B'(6,x) contains only integer coefficients and 6 is in the sequence.


p:=proc(n) if denom(diff(bernoulli(n, x), x))=1 then n else fi end:seq(p(n), n=1..100); (Deutsch)


Sequence in context: A045963 A128169 A095923 * A100195 A032396 A271884

Adjacent sequences:  A094957 A094958 A094959 * A094961 A094962 A094963




Benoit Cloitre, Jun 19 2004



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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)