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%I M5291 #54 Mar 09 2024 06:00:11
%S 48,75,140,195,1050,1575,1648,1925,2024,2295,5775,6128,8892,9504,
%T 16587,20735,62744,75495,186615,196664,199760,206504,219975,266000,
%U 309135,312620,507759,526575,544784,549219,573560,587460,817479,1000824,1057595,1081184
%N Betrothed (or quasi-amicable) numbers.
%C Members of a pair (m,n) such that sigma(m) = sigma(n) = m+n+1, where sigma = A000203. - _M. F. Hasler_, Nov 04 2008
%C Also members of a pair (m,k) such that m = sum of nontrivial divisors of k and k = sum of nontrivial divisors of m. - _Juri-Stepan Gerasimov_, Sep 11 2009
%C Also numbers that are terms of cycles when iterating Chowla's function A048050. - _Reinhard Zumkeller_, Feb 09 2013
%C From _Amiram Eldar_, Mar 09 2024: (Start)
%C The first pair, (48, 75), was found by Nasir (1946).
%C Lehmer (1948) in a review of Nasir's paper, noted that "the pair (48, 75) behave like amicable numbers".
%C Makowski (1960) found the next 2 pairs, and called them "pairs of almost amicable numbers".
%C The next 6 pairs were found by independently by Garcia (1968), who named them "números casi amigos" and Lal and Forbes (1971), who named them "reduced amicable pairs".
%C Beck and Wajar (1971) found 6 more pairs, but missed the 15th and 16th pairs, (526575, 544784) and (573560, 817479).
%C Hagis and Lord (1977) found the first 46 pairs. They called them "quasi-amicable numbers", after Garcia (1968).
%C Beck and Wajar (1993) found the next 33 pairs.
%C According to Guy (2004; 1st ed., 1981), the name "betrothed numbers" was proposed by Rufus Isaacs. (End)
%D Mariano Garcia, Números Casi Amigos y Casi Sociables, Revista Annal, año 1, October 1968, Asociación Puertorriqueña de Maestros de Matemáticas.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B5, pp. 91-92.
%D D. H. Lehmer, Math. Rev., Vol. 8 (1948), p. 445.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Giovanni Resta, <a href="/A005276/b005276.txt">Table of n, a(n) for n = 1..7819</a> (terms < 10^13, terms 1..100 from T. D. Noe, 101..1000 from Donovan Johnson)
%H Walter E. Beck and Rudolph M. Wajar, <a href="https://www.fq.math.ca/Scanned/15-4/beck1.pdf">More reduced amicable pairs</a>, Fibonacci Quarterly, Vol. 15, No. 4 (1977), pp. 331-332.
%H Walter E. Beck and Rudolph M. Wajar, <a href="https://www.fq.math.ca/Scanned/31-4/beck.pdf">Reduced and Augmented Amicable Pairs to 10^8</a>, Fibonacci Quarterly, Vol. 31, No. 4 (1993), pp. 295-298.
%H Peter Hagis and Graham Lord, <a href="https://doi.org/10.1090/S0025-5718-1977-0434939-3">Quasi-amicable numbers</a>, Math. Comp. 31 (1977), 608-611.
%H M. Lal and A. Forbes, <a href="http://dx.doi.org/10.1090/S0025-5718-1971-0297685-X">A note on Chowla's function</a>, Math. Comp., Vol. 25, No. 116 (1971), pp. 923-925.
%H A. Makowski, <a href="https://www.jstor.org/stable/2310107">On Some Equations Involving Functions phi(n) and sigma(n)</a>, The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670.
%H Abdur Rahman Nasir, <a href="https://archive.org/details/dli.calcutta.11605/page/140/mode/2up">On a certain arithmetic function</a>, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
%H Paul Pollack, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pollack/pollack3.html">Quasi-Amicable Numbers are Rare</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuasiamicablePair.html">Quasiamicable Pair.</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Betrothed_numbers">Betrothed numbers</a>.
%F Equals A003502 union A003503. - _M. F. Hasler_, Nov 04 2008
%t bnoQ[n_]:=Module[{dsn=DivisorSigma[1,n],m,dsm},m=dsn-n-1; dsm= DivisorSigma[ 1,m];dsm==dsn==n+m+1]; Select[Range[2,1100000],bnoQ] (* _Harvey P. Dale_, May 12 2012 *)
%o (PARI) isA005276(n) = { local(s=sigma(n)); s>n+1 & sigma(s-n-1)==s }
%o for( n=1, 10^6, isA005276(n) & print1(n",")) \\ _M. F. Hasler_, Nov 04 2008
%o (Haskell)
%o a005276 n = a005276_list !! (n-1)
%o a005276_list = filter p [1..] where
%o p z = p' z [0, z] where
%o p' x ts = if y `notElem` ts then p' y (y:ts) else y == z
%o where y = a048050 x
%o -- _Reinhard Zumkeller_, Feb 09 2013
%Y Cf. A003502, A003503, A048050, A328370.
%Y Subsequence of A057533.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
%E Extended by _T. D. Noe_, Dec 29 2011
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