A280363 a(n) = floor(log_p(n)) where p = A020639(n), i.e., the least prime factor of n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 4, 1, 4, 2, 4, 1, 4, 2, 4, 3, 4, 1, 4, 1, 5, 3, 5, 2, 5, 1, 5, 3, 5, 1, 5, 1, 5, 3, 5, 1, 5, 2, 5, 3, 5, 1, 5, 2, 5, 3, 5, 1, 5, 1, 5, 3, 6, 2, 6, 1, 6, 3, 6, 1, 6, 1, 6, 3, 6, 2, 6, 1, 6, 4, 6, 1, 6, 2, 6, 4, 6, 1, 6, 2, 6, 4, 6, 2, 6, 1, 6, 4, 6, 1, 6, 1, 6, 4, 6, 1, 6, 1, 6, 4, 6, 1, 6, 2, 6, 4, 6, 2, 6

Offset: 1

Author

Michael De Vlieger, Jan 01 2017

Keywords

• nonn
• easy

Comments

a(1) = 0 since 1 is the empty product.

a(p) = 1 since the exponent e of the largest power p^e of the prime divisor p is p^1 (i.e., p itself).

a(p^m) = m since the largest power p^e of the prime divisor p is p^m, (p^m itself), i.e., e = m.

a(n) is the greatest value of the power e of p^e across the prime divisors p of n such that p^e <= n.

Consider integers 1<=r<=n with all prime divisors p of r also dividing n. Let m be the smallest power n^m | r, and let e be the largest value of m across 1<=r<=n. This is A280274(n). This sequence underlies A280274: A280274(1) = 0, A280274(n) = 1 with n having omega(n) = 1. A280274(n) = a(n) for squarefree n. A280274(n) for all other n is ceiling(a(n)/k), with k being the multiplicity of p = A020639(n) in the prime decomposition of n.

Examples

			a(10) = 3, because 2^3 = 8 and 5^1 = 5 are less than 10 = 2*5, and of the multiplicities of these numbers, 3 is the greatest.
a(12) = 3, because 2^3 = 8 and 3^2 = 9 are less than 12 = 2*2*3, and of the multiplicities of these numbers, 3 is the greatest.
a(16) = 4, because 2^4 = 16 = n, and is the largest power of the distinct prime divisor 2 of 16.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Eric W. Weisstein World of Mathematics, Least Prime Factor

Crossrefs

Cf. A020639, A007947, A280274.

Programs

Mathematica

Table[If[n == 1, 0, Floor[Log[FactorInteger[n][[1, 1]], n]]], {n, 120}]

PARI

a(n) = if (n==1, 0, logint(n, vecmin(factor(n)[,1]))); \\ Michel Marcus, Jan 01 2017