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A005276 Betrothed (or quasi-amicable) numbers.
(Formerly M5291)
15
48, 75, 140, 195, 1050, 1575, 1648, 1925, 2024, 2295, 5775, 6128, 8892, 9504, 16587, 20735, 62744, 75495, 186615, 196664, 199760, 206504, 219975, 266000, 309135, 312620, 507759, 526575, 544784, 549219, 573560, 587460, 817479, 1000824, 1057595, 1081184 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Members of a pair (m,n) such that sigma(m) = sigma(n) = m+n+1, where sigma = A000203. - M. F. Hasler, Nov 04 2008
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
Also numbers that are terms of cycles when iterating Chowla's function A048050. - Reinhard Zumkeller, Feb 09 2013
From Amiram Eldar, Mar 09 2024: (Start)
The first pair, (48, 75), was found by Nasir (1946).
Lehmer (1948) in a review of Nasir's paper, noted that "the pair (48, 75) behave like amicable numbers".
Makowski (1960) found the next 2 pairs, and called them "pairs of almost amicable numbers".
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".
Beck and Wajar (1971) found 6 more pairs, but missed the 15th and 16th pairs, (526575, 544784) and (573560, 817479).
Hagis and Lord (1977) found the first 46 pairs. They called them "quasi-amicable numbers", after Garcia (1968).
Beck and Wajar (1993) found the next 33 pairs.
According to Guy (2004; 1st ed., 1981), the name "betrothed numbers" was proposed by Rufus Isaacs. (End)
REFERENCES
Mariano Garcia, Números Casi Amigos y Casi Sociables, Revista Annal, año 1, October 1968, Asociación Puertorriqueña de Maestros de Matemáticas.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B5, pp. 91-92.
D. H. Lehmer, Math. Rev., Vol. 8 (1948), p. 445.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..7819 (terms < 10^13, terms 1..100 from T. D. Noe, 101..1000 from Donovan Johnson)
Walter E. Beck and Rudolph M. Wajar, More reduced amicable pairs, Fibonacci Quarterly, Vol. 15, No. 4 (1977), pp. 331-332.
Walter E. Beck and Rudolph M. Wajar, Reduced and Augmented Amicable Pairs to 10^8, Fibonacci Quarterly, Vol. 31, No. 4 (1993), pp. 295-298.
Peter Hagis and Graham Lord, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.
M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., Vol. 25, No. 116 (1971), pp. 923-925.
A. Makowski, On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670.
Abdur Rahman Nasir, On a certain arithmetic function, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
Paul Pollack, Quasi-Amicable Numbers are Rare, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.2.
Eric Weisstein's World of Mathematics, Quasiamicable Pair..
Wikipedia, Betrothed numbers.
FORMULA
Equals A003502 union A003503. - M. F. Hasler, Nov 04 2008
MATHEMATICA
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 *)
PROG
(PARI) isA005276(n) = { local(s=sigma(n)); s>n+1 & sigma(s-n-1)==s }
for( n=1, 10^6, isA005276(n) & print1(n", ")) \\ M. F. Hasler, Nov 04 2008
(Haskell)
a005276 n = a005276_list !! (n-1)
a005276_list = filter p [1..] where
p z = p' z [0, z] where
p' x ts = if y `notElem` ts then p' y (y:ts) else y == z
where y = a048050 x
-- Reinhard Zumkeller, Feb 09 2013
CROSSREFS
Subsequence of A057533.
Sequence in context: A165039 A211721 A057533 * A328370 A143722 A261709
KEYWORD
nonn,nice,changed
AUTHOR
EXTENSIONS
Extended by T. D. Noe, Dec 29 2011
STATUS
approved

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Last modified March 18 21:02 EDT 2024. Contains 370951 sequences. (Running on oeis4.)