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Multiply perfect number
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===Sources=== *{{cite journal |first1=Kevin A. |last1=Broughan |first2=Qizhi |last2=Zhou |title=Odd multiperfect numbers of abundancy 4 |journal= Journal of Number Theory |doi=10.1016/j.jnt.2007.02.001 |year=2008 |mr=2419178 |volume=126 |number=6 |pages=1566–1575 |url=https://researchcommons.waikato.ac.nz/bitstream/10289/1796/1/Odd%20multiperfect%20numbers%20of%20abundancy%204.pdf |hdl=10289/1796 |hdl-access=free }} * {{cite book |last=Guy | first=Richard K. | author-link=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=B2 }} *{{cite journal |first1=Pentti |last1=Haukkanen |first2=V. |last2=Sitaramaiah |title=Bi-unitary multiperfect numbers, I |journal= Notes on Number Theory and Discrete Mathematics |doi=10.7546/nntdm.2020.26.1.93-171 |year=2020a |volume=26 |number=1 |pages=93–171 |url=http://nntdm.net/papers/nntdm-26/NNTDM-26-1-093-171.pdf |doi-access=free }} *{{cite journal |first1=Pentti |last1=Haukkanen |first2=V. |last2=Sitaramaiah |title=Bi-unitary multiperfect numbers, II |journal= Notes on Number Theory and Discrete Mathematics |doi=10.7546/nntdm.2020.26.2.1-26 |year=2020b |volume=26 |number=2 |pages=1–26 |url=http://nntdm.net/papers/nntdm-26/NNTDM-26-2-001-026.pdf |doi-access=free }} *{{cite journal |first1=Pentti |last1=Haukkanen |first2=V. |last2=Sitaramaiah |title=Bi-unitary multiperfect numbers, III |journal= Notes on Number Theory and Discrete Mathematics |doi=10.7546/nntdm.2020.26.3.33-67 |year=2020c |volume=26 |number=3 |pages=33–67 |url=http://nntdm.net/papers/nntdm-26/NNTDM-26-3-033-067.pdf |doi-access=free }} *{{cite journal |first1=Pentti |last1=Haukkanen |first2=V. |last2=Sitaramaiah |title=Bi-unitary multiperfect numbers, IV(a) |journal= Notes on Number Theory and Discrete Mathematics |doi=10.7546/nntdm.2020.26.4.2-32 |year=2020d |volume=26 |number=4 |pages=2–32 |url=https://nntdm.net/papers/nntdm-26/NNTDM-26-4-002-032.pdf |doi-access=free }} *{{cite journal |first1=Pentti |last1=Haukkanen |first2=V. |last2=Sitaramaiah |title=Bi-unitary multiperfect numbers, IV(b) |journal= Notes on Number Theory and Discrete Mathematics |doi=10.7546/nntdm.2021.27.1.45-69 |year=2021a |volume=27 |number=1 |pages=45–69 |url=https://nntdm.net/papers/nntdm-27/NNTDM-27-1-045-069.pdf |doi-access=free }} *{{cite journal |first1=Pentti |last1=Haukkanen |first2=V. |last2=Sitaramaiah |title=Bi-unitary multiperfect numbers, V |journal= Notes on Number Theory and Discrete Mathematics |doi=10.7546/nntdm.2021.27.2.20-40 |year=2021b |volume=27 |number=2 |pages=20–40 |url=https://nntdm.net/papers/nntdm-27/NNTDM-27-2-020-040.pdf |doi-access=free }} * {{cite journal | zbl=0612.10006 | last=Kishore | first=Masao | title=Odd triperfect numbers are divisible by twelve distinct prime factors | journal= Journal of the Australian Mathematical Society, Series A | volume=42 | issue=2 | pages=173–182 | year=1987 | issn=0263-6115 | doi=10.1017/s1446788700028184| doi-access=free }} * {{cite journal |first1=Richard |last1=Laatsch |title=Measuring the abundancy of integers |journal=[[Mathematics Magazine]] |jstor=2690424 |year=1986 |volume=59 |number=2 |pages=84–92 |mr=0835144| issn=0025-570X | zbl=0601.10003 |doi=10.2307/2690424}} * {{cite journal |first1=James G. |last1=Merickel |title= Divisors of Sums of Divisors: 10617 |journal= The American Mathematical Monthly |year=1999 |jstor=2589515 |volume=106 |number=7 |page=693 |mr=1543520 |doi=10.2307/2589515 }} * {{cite journal |first1=Richard F. |last1=Ryan |title=A simpler dense proof regarding the abundancy index |journal= Mathematics Magazine |year=2003 |volume=76 |number=4 |pages=299–301 |doi=10.1080/0025570X.2003.11953197 |jstor=3219086 |mr=1573698 |s2cid=120960379 }} * {{cite book | editor1-last=Sándor | editor1-first=Jozsef | editor2-last=Crstici | editor2-first=Borislav | title=Handbook of number theory II | url=https://archive.org/details/handbooknumberth00sand_741 | url-access=limited | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=[https://archive.org/details/handbooknumberth00sand_741/page/n33 32]–36 | zbl=1079.11001 }} * {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }} * {{Cite thesis|degree=PhD |first1= Ronald M. |last1=Sorli |title=Algorithms in the study of multiperfect and odd perfect numbers |year=2003 |hdl=10453/20034 |publisher=University of Technology|location= Sydney }} * {{cite journal|last=Tóth|first=László|title=Odd Spoof Multiperfect Numbers|journal=Integers|volume=25|year=2025|issue=A19 |arxiv=2502.16954|url=https://math.colgate.edu/~integers/z19/z19.pdf}} * {{cite journal |first1= Paul A. |last1=Weiner |title=The abundancy ratio, a measure of perfection |journal= Mathematics Magazine |year=2000 |jstor=2690980 |volume=73 |number=4 |pages=307–310 |mr=1573474 |doi=10.1080/0025570x.2000.11996860 |s2cid=119773896 }}
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