A063991 Unitary amicable numbers.

114, 126, 1140, 1260, 18018, 22302, 32130, 40446, 44772, 49308, 56430, 64530, 67158, 73962, 142310, 168730, 180180, 197340, 223020, 241110, 242730, 286500, 296010, 308220, 365700, 429750, 462330, 548550, 591030, 618570, 669900, 671580, 739620, 785148, 815100, 827652, 827700, 932100

Offset: 1

Author

N. J. A. Sloane, Sep 18 2001

Keywords

• nonn

Comments

From Amiram Eldar, Mar 09 2024: (Start)

The concept of unitary amicable numbers was introduced by Wall (1970), who proved that both members of a pair are either odd or even, and found 610 pairs (only 592 were correct, as found by te Riele, 1978).

Hagis (1971) calculated the first 19 pairs (the terms below 10^6).

Najar (1995) calculated the first 185 pairs (terms whose smaller member is below 10^8). (End)

References

• Mariano Garcia, New unitary amicable couples, J. Recreational Math., Vol. 17, No. 1 (1984-5), pp. 32-35.
• M. Lal, G. Tiller, and T. Summers, Unitary sociable numbers, Proceedings of the Second Manitoba Conference on Numerical Mathematics, Congressus Numerantium No. 7, 1972, pp. 211-216.
• Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 148.

Links

• Michel Marcus, Table of n, a(n) for n = 1..149
• Peter Hagis, Jr., Unitary amicable numbers, Math. Comp., Vol. 25, No. 116 (1971), pp. 915-918.
• O. William McClung, Generators of Unitary Amicable Numbers, Fibonacci Quarterly, Vol. 23, No. 2 (1985), pp. 158-164; Letter to the Editor, ibid., Vol. 24, No. 2 (1986), p. 106.
• Rudolph M. Najar, Operations on Generators of Unitary Amicable Pairs, Fibonacci Quarterly, Vol. 27, No. 2 (1989), pp. 144-152.
• Rudolph M. Najar, The Unitary Amicable Pairs To 10^8, International Journal of Mathematics and Mathematical Sciences, Volume 18, No. 2 (1995), pp. 405-410, Article ID 396507, 6 pages.
• J. O. M. Pedersen, Known Unitary Amicable Pairs.
• J. O. M. Pedersen, Known Unitary Amicable Pairs. [Via Wayback Machine]
• J. O. M. Pedersen, Tables of Aliquot Cycles.
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
• Ivars Peterson, Amicable Pairs, Divisors and a New Record, February 2, 2004.
• Herman J. J. te Riele, Further Results on Unitary Aliquot Sequences, Report NW 2/78, Math. Centre, Amsterdam, 2nd ed., Jan. 1978.
• Herman J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, Mathematisch Centrum, Amsterdam, 1976.
• Herman J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, Amsterdam, 1972.
• Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.
• Eric Weisstein's World of Mathematics, Unitary Amicable Pair.

Crossrefs

Union of A002952 and A002953.

Programs

PARI

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok1(n) = iferr(f(n) == n, E, 0);
isok2(n) = iferr(f(f(n)) == n, E, 0);
isok(n) = isok2(n) && !isok1(n); \\ Michel Marcus, Sep 29 2018

Extensions

More terms from Michel Marcus, Sep 29 2018