A005276 Betrothed (or quasi-amicable) numbers.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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.

Crossrefs

Cf. A003502, A003503, A048050, A328370.

Subsequence of A057533.

Programs

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

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 *)

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

Formula

Equals A003502 union A003503. - M. F. Hasler, Nov 04 2008

Extensions

Extended by T. D. Noe, Dec 29 2011