A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1

Offset

1,13

Comments

Length of second signature of A013929(n) (cf. A212172).

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Primefan, The First 2500 Integers Factored (1st of 5 pages).

Formula

a(n) = A056170(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^2)/(1-1/zeta(2)) = A085548 / A229099 = 1.15347789194214704903... . - Amiram Eldar, Oct 01 2023

Example

24 = 2^3*3 has 1 exponent of size 2 or greater in its prime factorization. Since 24 = A013929(8), a(8) = 1.

Mathematica

f[n_] := Module[{c = Count[FactorInteger[n][[;; , 2]], _?(# > 1&)]}, If[n > 1 && c > 0, c, Nothing]]; f[1] = 0; Array[f, 300] (* Amiram Eldar, Oct 01 2023 *)

Prog

(Haskell)
a212177 n = a212177_list !! (n-1)
a212177_list = filter (> 0) a056170_list
-- Reinhard Zumkeller, Dec 29 2012

Crossrefs

Cf. A013929, A212172, A212174.

Cf. A046028, A056170, A085548, A124010, A229099.

Sequence in context: A083235 A307609 A043287 * A345999 A249622 A287656

Adjacent sequences: A212174 A212175 A212176 * A212178 A212179 A212180

Keyword

nonn,easy

Author

Matthew Vandermast, Jun 03 2012

Status

approved