generalised divisor power mean
Ctibor.ZIZKA
Ctibor.ZIZKA at seznam.cz
Tue Jul 15 15:06:14 CEST 2008
sigma_i (1,...,n) is the divisor function defined for integer n as the sum of i-th powers of divisors of n.
s is from < 0 ; r-1 >
sigma_i (1,...,n) is the generalised harmonic mean of i-th powers of divisors of n.
s is from < r+1 ; 0 >
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Subject: Re: request for help with obscure sequence
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p[n - 6]]];p[ -6] = 1; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2]
= 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
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From: Richard Mathar <mathar at strw.leidenuniv.nl>
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To: seqfan at ext.jussieu.fr
Subject: Re: A005002 inconsistent with wikipedia entry about Stirling numbers of the second kind?
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combinat[bell](n) ;
add(combinat[stirling2](n,i),i=1..k) ;
if k= 1 then
A005001(n) ;
elif k= n then
1 ;
else
k*beta(n-1,k)+A000110(n-1)-A102661(n-1,k-2) ;
fi ;
beta(n,2) ;
beta(n,3) ;
seq(A005002(n),n=2..30) ;
seq(A005003(n),n=3..30) ;
Conference on Combinatorial Mathematics, New York, April 4-7, 1978.
Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy
of Sciences, 319, 1979.
Conference on Combinatorial Mathematics, New York, April 4-7, 1978.
Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy
of Sciences, 319, 1979.
Conference on Combinatorial Mathematics, New York, April 4-7, 1978.
Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy
of Sciences, 319, 1979.
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To: maximilian.hasler at gmail.com, njas at research.att.com
Subject: Re: A005002 inconsistent with wikipedia entry about Stirling numbers of the second kind?
Cc: mathar at strw.leidenuniv.nl, seqfan at ext.jussieu.fr
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Me: you are forgetting what the OEIS is for: if you come across
self-contained definition. That will have to wait
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From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
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Subject: Binary Palindromic Sequence
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I just submitted these 3 sequences:
%S A143014 1,3,9,27,189
%N A143014 a(1) = 1. a(n) = the smallest multiple of a(n-1), a(n) > a(n-1), such that a(n) in binary is a palindrome.
%C A143014 All terms are odd.
There are an infinite number of terms. Proof: (2^m + 1)*a(n) is a palindrome, where m is >= the number of binary digits in a(n). So a(n+1) <= (2^m + 1)*a(n).
%Y A143014 A143015,A143016
%O A143014 1
%K A143014 ,base,more,nonn,
%S A143015 1,3,3,3,7
%N A143015 a(1) = 1. a(n) = the smallest integer >=2 such that (product{k=1 to n} a(k)) in binary is a palindrome.
%C A143015 All terms are odd.
There are an infinite number of terms.
%F A143015 For n>=2, a(n) = A143014(n)/A143014(n-1).
%Y A143015 A143014,A143016
%O A143015 1
%K A143015 ,base,more,nonn,
%S A143016 1,11,1001,11011,10111101
%N A143016 a(n) = A143014(n) written in binary.
%C A143016 Each term is a palindrome.
%Y A143016 A143014,A143015
%O A143016 1
%K A143016 ,base,more,nonn,
First, a stupid question. Are all terms of A143015 palindromes in base 2?
(I admit that I haven't thought too hard about this.)
Second, maybe someone would like to calculate and submit similar sequences for other bases.
(A143014's base 10 analog is already in the database: sequence A068664.)
Third, hopefully A143015 isn't already in the database. Is it? (Lots of hits came up for "3,3,3,7", but nothing came up for "3,3,3,7 palindrome" or "3,3,3,7 palindromic".)
Thanks,
Leroy Quet
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