A100715 - OEIS

A100715

Pseudoperfect (or semiperfect) k-brilliant numbers for some k>1: some set of proper divisors of a(n) sums to a(n) and a(n) = p(1)p(2)...p(m) for primes all with the same number of digits.

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 210, 216, 224, 240, 252

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OFFSET

1,1

COMMENTS

Since every multiple of a semiperfect number is semiperfect, there are an infinite number of values in this sequence and also an infinite number of values in the complement (pseudoperfect or semiperfect numbers which are not k-brilliant numbers).

REFERENCES

Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.

Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 177, 1992.

LINKS

Table of n, a(n) for n=1..39.

Eric Weisstein's World of Mathematics, Semiperfect Number

Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.

FORMULA

a(n) is an element in the intersection of A005835 and A078972. a(n) in A005835 and a(n) is a semiprime with the same number of digits in each prime factor.

EXAMPLE

6 = 2 * 3 is 2-brilliant.

12 = 2 * 2 * 3 is 3-brilliant.

18 = 2 * 3 * 3 is 3-brilliant.

20 = 2 * 2 * 5 is 3-brilliant.

24 = 2 * 2 * 2 * 3 is 4-brilliant.

28 = 2 * 2 * 7 is 3-brilliant.

30 = 2 * 3 * 5 is 3-brilliant.

36 = 2 * 2 * 3 * 3 is 4-brilliant.

40 = 2 * 2 * 2 * 5 is 4-brilliant.

264 is not in the sequence because it is pseudoperfect but 264 = 2 * 2 * 2 * 3 * 11 and 11 has more digits than 2.

CROSSREFS