A003502 -id:A003502 - cp's OEIS frontend

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

N. J. A. Sloane

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)

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).

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.

Cf. A003502, A003503, A048050, A328370.

Subsequence of A057533.

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

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 *)

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

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

Extended by T. D. Noe, Dec 29 2011

A003503 The larger of a betrothed pair.

Original entry on oeis.org

75, 195, 1925, 1648, 2295, 6128, 16587, 20735, 75495, 206504, 219975, 309135, 507759, 549219, 544784, 817479, 1057595, 1902215, 1331967, 1159095, 1763019, 1341495, 1348935, 1524831, 1459143, 2576945, 2226014, 2681019, 2142945, 2421704, 3220119, 3123735

Offset: 1

Robert G. Wilson v

It has been shown that (1) all known betrothed pairs are of opposite parity and (2) if a and b are a betrothed pair, and if a < b are of the same parity, then a > 10^10. See the reference for the Hagis & Lord paper. Can it be shown that all betrothed pairs are of opposite parity? - Harvey P. Dale, Apr 07 2013
From David A. Corneth, Jan 26 2019: (Start)
Let (k, m) be a betrothed pair. Then sigma(k) = sigma(m). Proof:
k = sigma(m) - m - 1 (1)
m = sigma(k) - k - 1 (2)
Partially substituting (1) in (2) gives
m = sigma(k) - (sigma(m) - m - 1) - 1 = sigma(k) - sigma(m) + m + 1 - 1 which simplifies to sigma(k) = sigma(m). QED.
If k and m are odd then they are both square. If k and m are even then they are square or twice a square (not necessarily both in the same family).
Proof: sigma(k) is odd iff k is a square or twice a square (cf. A028982). Hence if k isn't of that form (and sigma(k) is even) then the parity of sigma(k) - k - 1 is odd for odd k and even for even k.
If k is an odd square then sigma(k) - k - 1 is odd.
If k is twice a square or an even square then sigma(k) - k - 1 is even. QED.
Using inspection and the results above, if k and m are a betrothed pair of the same parity, the minimal term is > 2*10^14. (End)

			75 is a term because sigma(75) - 75 - 1 = 124 - 75 - 1 = 48 and 75 > 48 and sigma(48) - 48 - 1 = 124 - 48 - 1 = 75. - _David A. Corneth_, Jan 24 2019

R. K. Guy, Unsolved Problems in Number Theory, B5.

Giovanni Resta, Table of n, a(n) for n = 1..4122 (terms 1..1000 from Donovan Johnson, 1001..1126 from Amiram Eldar)
P. Hagis and G. Lord, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.
David Moews, Augmented amicable pairs
Jan Munch Pedersen, Tables of Aliquot Cycles.
Wikipedia, Betrothed numbers.

Cf. A000203, A003502, A005276, A028982.

aapQ[n_] := Module[{c=DivisorSigma[1, n]-1-n}, c!=n&&DivisorSigma[ 1, c]-1-c == n]; Transpose[Union[Sort[{#, DivisorSigma[1, #]-1-#}]&/@Select[Range[2, 10000], aapQ]]][[2]] (* Amiram Eldar, Jan 24 2019 after Harvey P. Dale at A015630 *)

is(n) = m = sigma(n) - n - 1; if(m < 1 || n <= m, return(0)); n == sigma(m) - m - 1 \\ David A. Corneth, Jan 24 2019

Computed by Fred W. Helenius (fredh(AT)ix.netcom.com)

Extended by T. D. Noe, Dec 29 2011

A306870 Lesser of reduced bi-unitary amicable pair.

Original entry on oeis.org

2024, 9504, 62744, 496320, 573560, 677144, 1000824, 1173704, 1208504, 1921185, 2140215, 2198504, 2312024, 2580864, 3847095, 4012184, 4682744, 5416280, 6618080, 9247095, 12500865, 12970880, 13496840, 14371104, 23939685, 25942784, 26409320, 28644704, 34093304

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

A pair m < n is a reduced bi-unitary amicable pair if bsigma(m) = bsigma(n) = m + n + 1, where bsigma(n) is the sum of bi-unitary divisors of n (A188999).

The larger members are in A306871.

			2024 is in the sequence since it is the lesser of the amicable pair (2024, 2295): bsigma(2024) = bsigma(2295) = 4320 = 2024 + 2295 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..528 (terms below 2*10^11, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers.

Cf. A003502, A126172, A188999, A292980, A306871.

fun[p_, e_]:=If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); f[n_] := bsigma[n] - n - 1; s={}; Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, n]], {n, 1, 10^7}]; s

A306875 Lesser of reduced unitary amicable pair.

Original entry on oeis.org

172622505, 6217560734, 16017860054, 18102483014, 20021589510, 31285993970, 32576024810, 39270110990, 68700877014, 80170395410, 81142298930, 99542647490, 125182657005, 144194617490, 153113328654, 181335043274, 318710758730, 374642686418, 378482712530, 455440763414

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

A pair m < n is a reduced unitary amicable pair if usigma(m) = usigma(n) = m + n + 1, where usigma(n) is the sum of unitary divisors of n (A034460).

The larger members are in A306876.

			172622505 is in the sequence since it is the lesser of the amicable pair (172622505, 175742294): usigma(172622505) = usigma(175742294) = 348364800 = 172622505 + 175742294 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..28 (terms below 10^12, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers
J. O. M. Pedersen, Tables of Aliquot Cycles

Cf. A003502, A002952, A126172, A034460, A306876.

us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  s={}; Do[m = us[n] - 1; If[m > n && us[m] == n + 1, AppendTo[s, n]], {n, 1, 10^9}]; s

A126160 Number of betrothed pairs (m,n) with m <=10^k (and k=1,2,3,...), where a betrothed pair satisfies sigma(m)=sigma(n)=m+n+1 and m

Original entry on oeis.org

0, 1, 2, 8, 9, 17, 46, 79, 180, 404, 882, 1946, 4122

Offset: 1

Ant King, Dec 19 2006

Also called quasi-amicable pairs, or reduced amicable pairs.

			a(7)=46 because there are 46 betrothed pairs (m,n) with m<=10^7

J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
P. Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq. 14 (2011) # 11.5.2.

Cf. A003502, A003503, A005276.

s[n_]:=DivisorSigma[1,n]-n;BetrothedNumberQ[n_]:=If[s[s[n]-1]==n+1 && n>1,True,False];BetrothedPairList[k_]:=(anlist=Select[Range[k],BetrothedNumberQ[ # ] &]; prlist=Table[Sort[{anlist[[n]],s[anlist[[n]]]-1}],{n,1,Length[anlist]}]; Union[prlist,prlist]);data=BetrothedPairList[10^6];Table[Length[Select[data,First[ # ]<10^k &]],{k,1,6}]

a(13) from Giovanni Resta, Jul 24 2019

A179612 Sums of pairs of betrothed (or quasi-amicable) numbers.

Original entry on oeis.org

123, 335, 2975, 3223, 4319, 11903, 25479, 30239, 138239, 393119, 416639, 508895, 773759, 861839, 1071359, 1391039, 1645055, 2903039, 2413151, 2298239, 2903039, 2515199, 2557439, 2757887, 2695679, 3856895, 4147199, 4717439, 4245695, 4561919, 5391359, 5322239

Offset: 1

Jonathan Vos Post, Jan 08 2011

nonn

This is to A161005 sums of pairs of amicable numbers as betrothed (or quasi-amicable) numbers A005276 are to A063990 amicable numbers. The subsequence of primes begins: 11903, 138239, 1071359.

			a(1) = 48 + 75 = 123 = 3 * 41.
a(2) = 140 + 195 = 335 = 5 * 67.
a(6) = 5775 + 6128 = 11903 is the smallest prime in these pair-sums.

Cf. A000203, A003502, A003503, A005276, A063990, A161005.

a(n) = A003502(n) + A003503(n). {(j + k) such that sigma(j)=sigma(k)=j+k+1, where sigma=A000203}.

More terms from Amiram Eldar, Jan 27 2019

A328370 Quasi-amicable pairs.

Original entry on oeis.org

48, 75, 140, 195, 1050, 1925, 1575, 1648, 2024, 2295, 5775, 6128, 8892, 16587, 9504, 20735, 62744, 75495, 186615, 206504, 196664, 219975, 199760, 309135, 266000, 507759, 312620, 549219, 526575, 544784, 573560, 817479, 587460, 1057595, 1000824, 1902215, 1081184, 1331967, 1139144, 1159095, 1140020, 1763019

Offset: 1

Omar E. Pol, Oct 14 2019

nonn

Also called betrothed pairs, or quasiamicable pairs, or reduced amicable pairs.
A pair of numbers x and y is called quasi-amicable if sigma(x) = sigma(y) = x + y + 1, where sigma(n) is the sum of the divisors of n.
All known quasi-amicable pairs have opposite parity.
First differs from A005276 at a(6).
According to Hisanori Mishima (see link) there are 404 quasi-amicable pairs where the smaller part is less than 10^10. See A126160 for more values. - Peter Luschny, Nov 18 2019

			Initial quasi-amicable pairs:
    48,   75;
   140,  195;
  1050, 1925;
  1575, 1648;
  2024, 2295;
...
The sum of the divisors of 48 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 48 = 124. On the other hand the sum of the divisors of 75 is 1 + 3 + 5 + 15 + 25 + 75 = 124. Note that 48 + 75 + 1 = sigma(48) = sigma(75) = 124. The smallest quasi-amicable pair is (48, 75), so a(1) = 48 and a(2) = 75.

R. K. Guy, Unsolved Problems in Number Theory, B5.
P. Hagis and G. Lord, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.
Hisanori Mishima, Table of quasi-amicable pairs under 10^10.
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.

A003502 and A003503 interleaved.

Cf. A000203, A005276, A126160, A179612, A259180.

with(numtheory): aList := proc(searchbound)
local r, n, m, L: L := []:
for m from 1 to searchbound do
   n := sigma(m) - m - 1:
   if n <= m then next fi;
   r := sigma(n) - n - 1:
   if r = m then L := [op(L), m, n] fi;
od; L end:
aList(10000); # Peter Luschny, Nov 18 2019

a(2*n-1) = A003502(n); a(2*n) = A003503(n).

A166385 Primes between the lower and upper member of the n-th pair of betrothed numbers.

Original entry on oeis.org

53, 59, 61, 67, 71, 73, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289

Offset: 1

Juri-Stepan Gerasimov, Oct 13 2009

Primes in one of the intervals [A003502(k),A003503(k)], k>=1.

			The group of primes from 53 to 73 is in the sequence, because they are between 48 and 75.
The group of primes from 149 to 193 is in the sequence, because they are between 140 and 195.

Cf. A000040, A005276.

1063 inserted by R. J. Mathar, Oct 21 2009

cp's OEIS Frontend

A005276 Betrothed (or quasi-amicable) numbers.

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A003503 The larger of a betrothed pair.

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A306870 Lesser of reduced bi-unitary amicable pair.

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A306875 Lesser of reduced unitary amicable pair.

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A126160 Number of betrothed pairs (m,n) with m <=10^k (and k=1,2,3,...), where a betrothed pair satisfies sigma(m)=sigma(n)=m+n+1 and m

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A179612 Sums of pairs of betrothed (or quasi-amicable) numbers.

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A328370 Quasi-amicable pairs.

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A166385 Primes between the lower and upper member of the n-th pair of betrothed numbers.