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

Author

Zak Seidov, Oct 09 2002

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A001597, A076411.

Cf. A053289.

Programs

Mathematica

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

Extensions

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