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A076412 Number of n's in A076411. 2

%I

%S 1,3,4,1,7,9,2,5,4,13,15,17,19,21,4,3,16,25,27,20,9,18,13,33,35,19,18,

%T 39,41,43,28,17,47,49,51,53,55,57,59,61,39,24,65,67,69,71,35,38,75,77,

%U 79,81,47,36,85,87,89,23,68,71,10,12,95,97,99,101,103,40,65,107,109,100

%N Number of n's in A076411.

%C Equals {1} union A053289. - _Tom Verhoeff_, Jan 06 2008

%C Further comments from _Tom Verhoeff_, Jan 06 2008: (Start)

%C In general, for any nonnegative increasing sequence A (offset 1), i.e., with 0 <= A(i) < A(i+1), define

%C F = 'first differences of A' (offset 1), i.e., F(n) = A(n+1) - A(n)

%C L = 'number of A(i) less than n' (offset 1)

%C M = 'number of values at most n in L' (offset 0; auxiiliary sequence)

%C N = 'number of n's in L' (offset 0). Then M = A, i.e. M(k) = A(k+1), N = [ A(1) ] union F.

%C Proof: Observe that L is nonnegative and ascending: 0 <= L(i) <= L(i+1).

%C M(0) = N(0) = number of 0's in L = number of i >= 0 such that no A(j) < i = min A = A(1)

%C For k > 0, M(k) = number of values at most k in L = A(k+1)

%C N(k) = number of k's in L = number i >= 0 such that exactly k A(j) < i = M(k) - M(k-1) = A(k+1) - A(k) = F(k). QED (End)

%C First difference of perfect powers: A001597 prepended by 1. - _Robert G. Wilson v_, May 21 2009

%C Question: Does every number appear at least once? See the comment in A053289. - _Robert G. Wilson v_, May 21 2009

%H Robert G. Wilson v, <a href="/A076412/b076412.txt">Table of n, a(n) for n = 0..10488</a>

%e a(9)=13 because 9 appears 13 times in A076411.

%t t = Join[{0, 1}, Select[ Range@ 3600, GCD @@ Last /@ FactorInteger@# > 1 &]]; Rest@t - Most@t (* _Robert G. Wilson v_, May 21 2009 *)

%Y Cf. A001597, A076411.

%Y Cf. A053289.

%K nonn

%O 0,2

%A _Zak Seidov_, Oct 09 2002

%E a(19)-a(71) from _Robert G. Wilson v_, May 21 2009

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Last modified June 19 20:12 EDT 2021. Contains 345144 sequences. (Running on oeis4.)