A247334 Highly abundant numbers which are not abundant.

1, 2, 3, 4, 6, 8, 10, 16

Offset: 1

Author

Andrew Rodland, Sep 13 2014

Keywords

• fini
• full
• nonn

Comments

A number n is called "abundant" if sigma(n) > 2n, and "highly abundant" if sigma(n) > sigma(m) for all m < n. With these definitions, it's possible for a number to be highly abundant but not abundant. (A similar situation occurs with 2 being prime and highly composite.)

Fischer shows that all highly abundant numbers greater than 20 are multiples of 6. Since 6 is perfect and multiples of perfect numbers are abundant, this list is finite and complete.

Examples

			10 is in the sequence because sigma(10) > sigma(m) for m = 1 to 9, yet sigma(10) = 17 < 20.
		

Links

• Daniel Fischer, Prove that if Fn is highly abundant, then so is n, Mathematics Stack Exchange, Aug 13 2013

Crossrefs

Members of A002093 not in A005101. Members of A002093 in (A000396 union A005100).

Programs

PARI

for(n=1, 1000, if((sum(i=1, n-1, sign(sigma(n)-sigma(i))) == n-1) && (sigma(n) <= 2*n), print1(n, ", "))) \\ Michel Marcus, Sep 21 2014

PARI

is_A247334(n)={!for(i=2,n-1, sigma(n)>sigma(i)||return) && sigma(n)<=2*n} \\ M. F. Hasler, Oct 15 2014