Bounds for odd \(k\)-perfect numbers. (English) Zbl 1246.11010
The authors prove that the size of the set of \(k\)-perfect numbers with \(r\) distinct prime factors is bounded above by \((k-1)4^{r^{3}}\), generalizing Paul Pollack’s bound in the case when \(k=2\). The proof follows Pollack’s methods. The new key step involves a reduction to “primitive” \(\alpha\)-perfect numbers.
It is interesting to note that this bound on the number of \(k\)-perfect integers with \(r\) distinct prime factors is significantly smaller than the best known bound on the size of any such number.
It is interesting to note that this bound on the number of \(k\)-perfect integers with \(r\) distinct prime factors is significantly smaller than the best known bound on the size of any such number.
Reviewer: Pace Nielsen (Provo)
MSC:
11A25 | Arithmetic functions; related numbers; inversion formulas |
11A51 | Factorization; primality |
11N25 | Distribution of integers with specified multiplicative constraints |
References:
[1] | DOI: 10.1007/BF01362422 · Zbl 0324.10001 · doi:10.1007/BF01362422 |
[2] | DOI: 10.4169/amer.math.monthly.118.02.161 · Zbl 1225.11009 · doi:10.4169/amer.math.monthly.118.02.161 |
[3] | DOI: 10.1090/S0025-5718-07-01990-4 · Zbl 1142.11086 · doi:10.1090/S0025-5718-07-01990-4 |
[4] | Nielsen, Integers 3 (2003) |
[5] | DOI: 10.1007/BF01360967 · Zbl 0084.04205 · doi:10.1007/BF01360967 |
[6] | DOI: 10.2307/2370405 · JFM 44.0220.02 · doi:10.2307/2370405 |
[7] | Cook, Number Theory (Ottawa, ON, 1996) pp 67– (1999) |
[8] | Cohen, Math. Chronicle 9 pp 120– (1980) |
[9] | Cohen, Bull. Malays. Math. Soc. 8 pp 23– (1985) |
[10] | DOI: 10.1017/S0305004100072030 · Zbl 0805.11005 · doi:10.1017/S0305004100072030 |
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