A076412 Number of n's in A076411.

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, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100

Offset

0,2

Comments

Equals {1} union A053289. - Tom Verhoeff, Jan 06 2008

Further comments from Tom Verhoeff, Jan 06 2008: (Start)

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

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

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

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

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

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

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

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

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)

First difference of perfect powers: A001597 prepended by 1. - Robert G. Wilson v, May 21 2009

Question: Does every number appear at least once? See the comment in A053289. - Robert G. Wilson v, May 21 2009

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..10488

Example

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

Mathematica

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

Crossrefs

Cf. A001597, A076411.

Cf. A053289.

Sequence in context: A262216 A076446 A053289 * A053707 A075052 A111516

Adjacent sequences: A076409 A076410 A076411 * A076413 A076414 A076415

Keyword

nonn

Author

Zak Seidov, Oct 09 2002

Extensions

a(19)-a(71) from Robert G. Wilson v, May 21 2009

Status

approved