A322376 Positive integers k that are the sum of divisors of some positive integer but there exists no divisor d where 1 < d, k/d of k such that d and k/d have this property.

1, 3, 4, 6, 7, 8, 13, 14, 15, 20, 30, 31, 38, 40, 44, 57, 62, 63, 68, 74, 102, 110, 121, 127, 133, 138, 150, 158, 164, 174, 183, 194, 198, 200, 212, 230, 242, 255, 258, 278, 282, 284, 307, 314, 318, 332, 338, 348, 350, 354, 368, 374, 380, 398, 402, 410, 422, 458

Offset

1,2

Comments

Also number k such that k is in A002191 but there is no divisor d where 1 < d, k/d of k such that both d and k/d are in A002191. A002191 is closed under multiplication with terms in this sequence as primitive terms.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

4 is in A002191 as sigma(3) = 4 but no divisor of 4 as described above exists. The only candidate is 2 but 2 isn't in A002191.

Maple

N:= 1000: # for terms <= N
S:= select(`<=`, map(NumberTheory:-sigma, {$1..N-1}), N):
filter:= proc(n)
   andmap(t -> not member(t, S) or not member(n/t, S), NumberTheory:-Divisors(n) minus {1, n})
end proc:
sort(convert(select(filter, S), list)); # Robert Israel, Mar 31 2026

Prog

(PARI) upto(n) = my(u = List(), t, res=List()); for(i=1, n, c=sigma(i); if(c<=n, listput(u, c))); listsort(u, 1); u=Vec(u); for(i=1, #u, t=1; d=divisors(u[i]); for(j=2, (#d + 1)\2, if(vecsearch(u, d[j]) > 0 && vecsearch(u, u[i]/d[j]) > 0, t=0; next(1))); if(t==1, listput(res, u[i]))); res

Crossrefs

Cf. A000203, A002191.

Sequence in context: A395508 A239458 A007370 * A376424 A044813 A154661

Adjacent sequences: A322373 A322374 A322375 * A322377 A322378 A322379

Keyword

nonn

Author

David A. Corneth, Jan 24 2019

Status

approved