Issues in nonlinear hyperperfect numbers
HTML articles powered by AMS MathViewer
 by Daniel Minoli PDF
 Math. Comp. 34 (1980), 639645 Request permission
Abstract:
Hyperperfect numbers (HP) are a generalization of perfect numbers and as such share remarkably similar properties. In this note we show, among other things, that if $m = p_1^{{\alpha _1}}p_2^{{\alpha _2}}$ is 2HP then ${\alpha _2} = 1$, with ${p_1} = 3$, ${p_2} = {3^{{\alpha _1} + 1}}  2$; this is in agreement with the structure of the perfect case (1HP) stating that such a number is of the form $m = p_1^{{\alpha _1}}{p_2}$ with ${p_1} = 2$ and ${p_2} = {2^{{\alpha _1} + 1}}  1$.References

L. E. DICKSON, History of the Theory of Numbers, Vol. 1, Chelsea, New York, 1952.
 H. L. Abbott, C. E. Aull, Ezra Brown, and D. Suryanarayana, Quasiperfect numbers, Acta Arith. 22 (1973), 439–447. MR 316368, DOI 10.4064/aa224439447 M. KISHORE, “Odd almost perfect numbers,” Notices Amer. Math. Soc., v. 22, 1975, p. A380, Abstract #75FA92.
 R. P. Jerrard and Nicholas Temperley, Almost perfect numbers, Math. Mag. 46 (1973), 84–87. MR 376511, DOI 10.2307/2689036
 James T. Cross, A note on almost perfect numbers, Math. Mag. 47 (1974), 230–231. MR 354536, DOI 10.2307/2689220
 Peter Hagis Jr. and Graham Lord, Quasiamicable numbers, Math. Comp. 31 (1977), no. 138, 608–611. MR 434939, DOI 10.1090/S00255718197704349393
 Henri Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), 423–429. MR 271004, DOI 10.1090/S00255718197002710046
 Paul Bratley, Fred Lunnon, and John McKay, Amicable numbers and their distribution, Math. Comp. 24 (1970), 431–432. MR 271005, DOI 10.1090/S00255718197002710058
 Walter E. Beck and Rudolph M. Najar, Fixed points of certain arithmetic functions, Fibonacci Quart. 15 (1977), no. 4, 337–342. MR 491437
 Peter Hagis Jr., Lower bounds for relatively prime amicable numbers of opposite parity, Math. Comp. 24 (1970), 963–968. MR 276167, DOI 10.1090/S00255718197002761674
 Peter Hagis Jr., Unitary amicable numbers, Math. Comp. 25 (1971), 915–918. MR 299551, DOI 10.1090/S00255718197102995512
 Masao Kishore, Odd integers $N$ with five distinct prime factors for which $210^{12}<\sigma (N)/N<2+10^{12}$, Math. Comp. 32 (1978), no. 141, 303–309. MR 485658, DOI 10.1090/S0025571819780485658X
 Leonard Eugene Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with $n$ Distinct Prime Factors, Amer. J. Math. 35 (1913), no. 4, 413–422. MR 1506194, DOI 10.2307/2370405
 Peter Hagis Jr., A lower bound for the set of odd perfect numbers, Math. Comp. 27 (1973), 951–953. MR 325507, DOI 10.1090/S00255718197303255079
 Bryant Tuckerman, A search procedure and lower bound for odd perfect numbers, Math. Comp. 27 (1973), 943–949. MR 325506, DOI 10.1090/S00255718197303255067
 S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617–623. MR 347726, DOI 10.1090/S00255718197403477269 A. E. ZACHARIOV, “Perfect, semiperfect and Ore numbers,” Bull. Soc. Math. Grèce, v. 13, 1972, pp. 1222.
 Daniel Minoli and Robert Bear, Hyperperfect numbers, Pi Mu Epsilon J. 6 (1975), no. 3, 153–157. MR 389749
 Daniel Minoli, Structure issues for hyperperfect numbers, Fibonacci Quart. 19 (1981), no. 1, 6–14. MR 606102 D. M. YOUNG & R. T. GREGORY, A Survey of Numerical Analysis, AddisonWesley, Reading, Mass., 1973.
Additional Information
 © Copyright 1980 American Mathematical Society
 Journal: Math. Comp. 34 (1980), 639645
 MSC: Primary 10A20
 DOI: https://doi.org/10.1090/S00255718198005592069
 MathSciNet review: 559206