A333352 a(n) is the product of indices of the smallest and greatest prime factors of n.

1, 1, 4, 1, 9, 2, 16, 1, 4, 3, 25, 2, 36, 4, 6, 1, 49, 2, 64, 3, 8, 5, 81, 2, 9, 6, 4, 4, 100, 3, 121, 1, 10, 7, 12, 2, 144, 8, 12, 3, 169, 4, 196, 5, 6, 9, 225, 2, 16, 3, 14, 6, 256, 2, 15, 4, 16, 10, 289, 3, 324, 11, 8, 1, 18, 5, 361, 7, 18, 4, 400, 2, 441, 12, 6, 8, 20, 6, 484, 3

Offset: 1

Author

Ilya Gutkovskiy, Mar 15 2020

Keywords

• nonn

Examples

			a(315) = a(3^2 * 5 * 7) = a(prime(2)^2 * prime(3) * prime(4)) = 2 * 4 = 8.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000079 (positions of 1's), A000720, A002110, A006530, A020639, A033845 (positions of 2's), A055396, A061395, A066048, A156061, A243055.

Programs

Mathematica

a[1] = 1; a[n_] := PrimePi[FactorInteger[n] [[1, 1]]] PrimePi[ FactorInteger[ n] [[-1, 1]]]; Table[a[n], {n, 1, 80}]

PARI

a(n) = if (n==1, 1, my(f=factor(n)[,1]); primepi(vecmin(f))*primepi(vecmax(f))); \\ Michel Marcus, Mar 16 2020

Formula

If n = Product (p_j^k_j) then a(n) = min{pi(p_j)} * max{pi(p_j)}, where pi = A000720.

a(n) = A055396(n) * A061395(n) for n > 1.

a(2*n) = A061395(n) for n > 1.

a(n^k) = a(n) for k > 0

a(2*prime(n)^k) = n for k > 0.

a(prime(n)^k) = n^2 for k > 0.

a(n!) = pi(n) for n > 1.

a(A002110(n)) = n.