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A064599
Numbers k such that the sum of the nonunitary divisors of k is a divisor of k; i.e., sigma(k) - usigma(k) divides k.
3
4, 9, 18, 24, 25, 49, 112, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 1984, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569
OFFSET
1,1
COMMENTS
The sequence consists of the nonunitary perfect numbers (A064591), squares of primes (A001248) and 18.
Ligh and Wall (1987) called the terms that are not nonunitary perfect numbers (i.e., the numbers k such that A048146(k) is a proper divisor of k) "non-unitary subperfect numbers". - Amiram Eldar, Jan 19 2026
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 333-338.
MATHEMATICA
nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, v=nusigma[ n ]; If[ v>0&&Mod[ n, v ]==0, Print[ n ] ] ]
PROG
(PARI) usigma(n)= { local(f, s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
isok(m) = {my(v=sigma(m) - usigma(m)); v > 0 && m%v == 0; } \\ Harry J. Smith, Sep 19 2009
CROSSREFS
Cf. A000203 (sigma), A034448 (usigma), A001248, A048146, A064591.
Sequence in context: A313357 A313358 A263351 * A368780 A062952 A344405
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Sep 25 2001
STATUS
approved