A289144 The difference between the second divisor of n and the penultimate divisor of n.

1, 2, 0, 4, -1, 6, -2, 0, -3, 10, -4, 12, -5, -2, -6, 16, -7, 18, -8, -4, -9, 22, -10, 0, -11, -6, -12, 28, -13, 30, -14, -8, -15, -2, -16, 36, -17, -10, -18, 40, -19, 42, -20, -12, -21, 46, -22, 0, -23, -14, -24, 52, -25, -6, -26, -16, -27, 58, -28

Offset

2,2

Comments

a(1) is not defined since 1 has fewer than 2 divisors.

n is a prime number if and only if a(n) > 0.

n is the square of a prime number if and only if a(n) = 0.

n is composite and is not the square of a prime number if and only if a(n) < 0.

If n is prime, then a(n) = n - 1,

else, if 2 divides n, then a(n) = 2 - (n / 2),

else, if 3 divides n, then a(n) = 3 - (n / 3),

else, if 5 divides n, then a(n) = 5 - (n / 5),

and so on, with infinitely many statements of the

"else, if p divides n, then a(n) = p - (n / p)," kind, over all p, prime numbers.

Links

Michel Marcus, Table of n, a(n) for n = 2..1000

Example

The divisors of 7 are { 1, 7 }. Then a(7) = 7 - 1 = 6.
The divisors of 25 are { 1, 5, 25 }. Then a(25) = 5 - 5 = 0.
The divisors of 221 are { 1, 13, 17, 221 }. Then a(221) = 13 - 17 = -4.

Prog

(PARI) a(n)={my(T=divisors(n)); T[2]-T[#T-1]}

Crossrefs

Cf. A027750.

Sequence in context: A369657 A338824 A056737 * A350576 A008797 A239004

Adjacent sequences: A289141 A289142 A289143 * A289145 A289146 A289147

Keyword

sign

Author

Luc Rousseau, Jun 26 2017

Status

approved