A295884 Number of exponents larger than 3 in the prime factorization of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,1296

Comments

a(1296) is the first term greater than 1, and a(810000) is the first term greater than 2. - Harvey P. Dale, Dec 22 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

Formula

Additive with a(p^e) = 1 when e > 3, 0 otherwise.

a(n) = A295659(n) - A295883(n).

a(n) = A056170(A062378(n)) = A056170(A003557(A003557(n))) = A001221(A003557^3(n)).

a(n) = A001221(A053164(n)) = A001221(A008835(n)).

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^4 = 0.076993... (A085964). - Amiram Eldar, Nov 01 2020

Example

For n = 16 = 2^4, there is one exponent and it is larger than 3, thus a(16) = 1.
For n = 96 = 2^5 * 3^1, there are two exponents, and the other one is larger than 3, thus a(96) = 1.
For n = 1296 = 2^4 * 3^4, there are two exponents larger than 3, thus a(1296) = 2.

Mathematica

Array[Total@ Map[Boole[# > 3] &, FactorInteger[#][[All, -1]]] &, 120] (* Michael De Vlieger, Nov 29 2017 *)
Table[Count[FactorInteger[n][[All, 2]], _?(#>3&)], {n, 130}] (* Harvey P. Dale, Dec 22 2017 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A295884 n) (if (= 1 n) 0 (+ (if (> (A067029 n) 3) 1 0) (A295884 (A028234 n)))))

Crossrefs

Cf. A000188, A001221, A003557, A008835, A053164, A056170, A062378, A085964, A295659, A295883.

Sequence in context: A277164 A011730 A358261 * A101637 A337380 A011729

Adjacent sequences: A295881 A295882 A295883 * A295885 A295886 A295887

Keyword

nonn

Author

Antti Karttunen, Nov 29 2017

Status

approved