A122036 Odd abundant numbers (A005231) which are not in A136446, i.e., not sum of some of their proper divisors > 1.

351351

Offset: 1

Author

N. J. A. Sloane, Apr 11 2008, following correspondence from R. K. Guy, M. F. Hasler and others

Keywords

• nonn
• bref
• more
• nice
• hard

Comments

It is conjectured that there are no odd weird numbers (A006037), i.e., that all odd abundant numbers (A005231) are pseudoperfect (A005835); this sequence lists those which are not equal to the sum of a subset of proper divisors > 1.

No second term in the range <= 53850001. - R. J. Mathar, Mar 21 2011

No other terms congruent to 21 (mod 30) below 10^9. - M. F. Hasler, Jul 16 2016

a(2) > 10^16. - Wenjie Fang, Jul 17 2017

Examples

			a(1) = 351351 = 3^3 * 7 * 11 * 13^2 is the sum of all its 47 proper divisors (including 1) except 7 and 11, but it is not possible to get the same sum without using the trivial divisor 1: The sum of all proper divisors *larger than 1* yields 351351 + 7 + 11 - 1 = 351351 + 17, and it is not possible to get 17 as sum of a subset of {3, 7, 9, 11, 13, 21, ...}. Thus, 351351 is not in A136446, and therefore in this sequence. - _M. F. Hasler_, Jul 16 2016, edited Mar 15 2021
		

Links

• Index entries for one-term sequences

Crossrefs

Cf. A005231, A005835, A006037, A136446.

Programs

PARI

is_A122036(n)={n>351350 && !is_A005835(n,n=divisors(n)[2..-2]) && n && vecsum(n)>=n[1]*n[#n] && n[1]>2} \\ (Checking for abundant & odd after is_A005835() rather than before, to make it faster when operating on candidates known to satisfy these conditions.) Updated for current PARI syntax by M. F. Hasler, Jul 16 2016, further edits Jan 31 2020
forstep(n=1,10^7,2, is_A122036(n) && print1(n","))

Extensions

Comments and PARI code from M. F. Hasler, Apr 12 2008

Edited by M. F. Hasler, Jul 16 2016, Mar 15 2021