A381214 a(n) is the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

1, 2, 0, 4, 2, 6, 1, 1, 4, 10, 2, 12, 6, 4, 2, 16, 2, 18, 4, 6, 10, 22, 2, 3, 12, 0, 6, 28, 4, 30, 3, 10, 16, 6, 1, 36, 18, 12, 4, 40, 6, 42, 10, 4, 22, 46, 3, 5, 4, 16, 12, 52, 2, 10, 6, 18, 28, 58, 4, 60, 30, 6, 4, 12, 10, 66, 16, 22, 6, 70, 1, 72, 36, 4, 18

Offset: 2

Author

Paolo Xausa, Feb 19 2025

Keywords

• nonn
• easy

Examples

			a(36) = 1 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and 3 - 2 = 1.
a(31500) = 6 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 7 - 1 = 6.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A046665, A081812, A381212.

Cf. A051674 (positions of zeros), A381215 (positions of ones).

Programs

Mathematica

A381214[n_] := Max[#] - Min[#] & [Flatten[FactorInteger[n]]];
Array[A381214, 100, 2]

PARI

a(n) = my(f=factor(n), s=setunion(Set(f[,1]), Set(f[,2]))); vecmax(s) - vecmin(s); \\ Michel Marcus, Feb 20 2025

Formula

a(n) = A081812(n) - A381212(n).