A005276 -id:A005276 - cp's OEIS frontend

A048050 Chowla's function: sum of divisors of n except for 1 and n.

0, 0, 0, 2, 0, 5, 0, 6, 3, 7, 0, 15, 0, 9, 8, 14, 0, 20, 0, 21, 10, 13, 0, 35, 5, 15, 12, 27, 0, 41, 0, 30, 14, 19, 12, 54, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 75, 7, 42, 20, 45, 0, 65, 16, 63, 22, 31, 0, 107, 0, 33, 40, 62, 18, 77, 0, 57, 26, 73, 0, 122, 0, 39, 48, 63, 18, 89

Offset: 1

N. J. A. Sloane

a(n) = 0 if and only if n is a noncomposite number (cf. A008578). - Omar E. Pol, Jul 31 2012
If n is semiprime, a(n) = A008472(n). - Wesley Ivan Hurt, Aug 22 2013
If n = p*q where p and q are distinct primes then a(n) = p+q.
If k,m > 1 are coprime, then a(k*m) = a(k)*a(m) + (m+1)*a(k) + (k+1)*a(m) + k + m. - Robert Israel, Apr 28 2015
a(n) is also the total number of parts in the partitions of n into equal parts that contain neither 1 nor n as a part (see example). More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that contain neither k nor k*n as a part. - Omar E. Pol, Nov 24 2019
Named after the Indian-American mathematician Sarvadaman D. S. Chowla (1907-1995). - Amiram Eldar, Mar 09 2024

			For n = 20 the divisors of 20 are 1,2,4,5,10,20, so a(20) = 2+4+5+10 = 21.
On the other hand, the partitions of 20 into equal parts that contain neither 1 nor 20 as a part are [10,10], [5,5,5,5], [4,4,4,4,4], [2,2,2,2,2,2,2,2,2,2]. There are 21 parts, so a(20) = 21. - _Omar E. Pol_, Nov 24 2019

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., 25 (1971), 923-925.
Abdur Rahman Nasir, On a certain arithmetic function, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
Omar E. Pol, Illustration of initial terms obtained geometrically.

Cf. A000203, A001065, A000593, A002954, A048995, A007956, A048671, A182936.

Cf. A057533, A005276, A027751, A237593, A245092, A244049 (partial sums).

a048050 1 = 0
a048050 n = (subtract 1) $ sum $ a027751_row n
-- Reinhard Zumkeller, Feb 09 2013

A048050:=func< n | n eq 1 or IsPrime(n) select 0 else &+[ a: a in Divisors(n) | a ne 1 and a ne n ] >; [ A048050(n): n in [1..100] ]; // Klaus Brockhaus, Mar 04 2011

A048050 := proc(n) if n > 1 then numtheory[sigma](n)-1-n ; else 0; end if; end proc:

f[n_]:=Plus@@Divisors[n]-n-1; Table[f[n],{n,100}] (*Vladimir Joseph Stephan Orlovsky, Sep 13 2009*)
Join[{0},DivisorSigma[1,#]-#-1&/@Range[2,80]] (* Harvey P. Dale, Feb 25 2015 *)

a(n)=if(n>1,sigma(n)-n-1,0) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import divisors
def a(n): return sum(divisors(n)[1:-1]) # Indranil Ghosh, Apr 26 2017

from sympy import divisor_sigma
def A048050(n): return 0 if n == 1 else divisor_sigma(n)-n-1 # Chai Wah Wu, Apr 18 2021

a(n) = A000203(n) - A065475(n).
a(n) = A001065(n) - 1, n > 1.
For n > 1: a(n) = Sum_{k=2..A000005(n)-1} A027750(n,k). - Reinhard Zumkeller, Feb 09 2013
a(n) = A000203(n) - n - 1, n > 1. - Wesley Ivan Hurt, Aug 22 2013
G.f.: Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017

A003502 The smaller of a betrothed pair.

Original entry on oeis.org

48, 140, 1050, 1575, 2024, 5775, 8892, 9504, 62744, 186615, 196664, 199760, 266000, 312620, 526575, 573560, 587460, 1000824, 1081184, 1139144, 1140020, 1173704, 1208504, 1233056, 1236536, 1279950, 1921185, 2036420, 2102750, 2140215, 2171240, 2198504, 2312024

Offset: 1

Robert G. Wilson v

			48 is a term because sigma(48) - 48 - 1 = 124 - 48 - 1 = 75 and 48 < 75 and sigma(75) - 75 - 1 = 124 - 75 - 1 = 48. - _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 < 10^13, terms 1..1000 from Donovan Johnson, 1001..1126 from Amiram Eldar)
Shyam Sunder Gupta, Amicable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 5, 159-183.
Peter Hagis and Graham 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, A003503, A005276.

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]]] [[1]] (* Amiram Eldar, Jan 24 2019 after Harvey P. Dale at A007992 *)

is(n) = m = sigma(n) - n - 1; if(m == 0 || 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

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

A057533 Values of n for which iteration of Chowla's function loops.

Original entry on oeis.org

48, 75, 92, 140, 146, 176, 195, 215, 255, 267, 287, 312, 332, 369, 386, 407, 411, 519, 527, 551, 627, 734, 744, 818, 972, 973, 984, 1027, 1050, 1078, 1096, 1149, 1175, 1185, 1387, 1408, 1472, 1474, 1535, 1575, 1648, 1651, 1784, 1792, 1880, 1888, 1891

Offset: 1

Asher Auel, Sep 03 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A048050.

Cf. A005276 (subsequence). - Reinhard Zumkeller, Feb 09 2013

a057533 n = a057533_list !! (n-1)
a057533_list = filter (\z -> p z [z]) [1..] where
   p x ts = y > 0 && (y `elem` ts || p y (y:ts)) where y = a048050 x
-- Reinhard Zumkeller, Feb 09 2013

A192292 Pairs of numbers a, b for which sigma(a)=b and sigma(b)-b-1=a, where sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

7, 10, 14, 16, 45, 86, 2379, 2324, 4213, 5866, 27323, 33604, 1303227, 1737628, 3722831, 4208308, 15752651, 18706108, 6094085371, 8114352508, 30090695519, 40119052564

Offset: 1

Paolo P. Lava, Jun 29 2011

Betrothed numbers mixed with anti-divisors.

a(23) > 10^11. - Hiroaki Yamanouchi, Sep 28 2015

			sigma*(45)= 2+6+7+10+13+18+30 = 86.
sigma(86)-86-1 = 2+43 = 45.
sigma*(2379) = 2+6+26+67+71+78+122+366+1586 = 2374.
sigma(2324)-2324-1 = 2+4+7+14+28+83+166+332+581+1162 = 2379.

Cf. A005276, A066272, A192290, A192291, A192293.

with(numtheory); P:= proc(n) local b,c,i,j,k;
for i from 3 to n do k:=0; j:=i;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
b:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
if sigma(b)-b-1=i then print(i); print(b); fi;
od; end: P(10^9);

a(13)-a(14) from Paolo P. Lava, Dec 03 2014

a(7)-a(8) swapped and a(15)-a(22) added by Hiroaki Yamanouchi, Sep 28 2015

A285889 The smaller of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = n + x + y.

Original entry on oeis.org

220, 48, 174, 390, 102, 280, 160, 500, 66, 132, 54, 24280, 992, 560, 140, 168, 60, 10360, 1120, 1232, 198, 210, 132, 2170, 520, 1520, 96, 168, 330, 732, 60, 4424, 270, 540, 144, 1000, 1484, 4080, 220, 840, 1144, 16500, 1988, 5456, 210, 528, 150, 4158, 1180, 12236

Offset: 0

Paolo P. Lava, Apr 28 2017

In the first 1000 terms the most repeated number is 840 with 15 occurrences.

			a(0) = 220: sigma(220) = sigma(284) = 220 + 284 = 504;
a(1) = 48: sigma(48) = sigma(75) = 48 + 75 + 1 = 124;
a(2) = 174: sigma(174) = sigma(184) = 174 + 184 + 2 = 360.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A000203, A285890, A285892.

See first terms of A002025 and A005276.

with(numtheory): P:=proc(q) local a,b,k,n; for n from 0 to q do for k from 1 to q do
a:=sigma(k)-k-n; b:=sigma(a)-a-n; if a>0 and b=k and a<>b then print(k); break;
fi; od; od; end: P(10^9);

Table[m = 1; While[MissingQ@ Set[k, SelectFirst[Range[m - 1], DivisorSigma[1, m] == DivisorSigma[1, #] == m + # + n &]], m++]; {k, m}, {n, 0, 10}][[All, 1]] (* Version 10.2, or *)
Do[m = 1; While[Set[k, Module[{k = 1}, While[! Xor[DivisorSigma[1, m] == DivisorSigma[1, k] == m + k + n, k >= m], k++]; k]] >= m, m++]; Print@ k, {n, 0, 10}] (* Michael De Vlieger, Apr 28 2017 *)

getfirstterms(n)={my(L:list,S:list,k:small,t);L=List();S=List([1,3]);k=0;forstep(i=3,+oo,1,listput(S,sigma(i));forvec(j=[[2,i],[2,i]],t=vecsum(j)+k;if((S[j[1]]==t)&&(t==S[j[2]]),listput(L,j[1]);if(k==n,break(2),k++)),2));return(Vec(L))} \\ R. J. Cano, May 03 2017

A239436 Members of a pair (m,k) such that sigma(m) = sigma(k) = sigma(m+k), m < k where sigma = A000203.

Original entry on oeis.org

1288, 1485, 5775, 6128, 8008, 11685, 16744, 19305, 21896, 25245, 24472, 28215, 26488, 35505, 32620, 45441, 37352, 43065, 39928, 46035, 47656, 54945, 50260, 65637, 52808, 60885, 55384, 63855, 62744, 75495, 72772, 79365, 68264, 78705, 75075, 79664, 80584, 90915

Offset: 1

Michel Lagneau, Mar 18 2014

nonn

The numbers such that sigma(k) = sigma(m) = m+k+1 and m+k is prime are in the sequence since sigma(k+m) = m+k+1 (see A005276). - Giovanni Resta, Mar 20 2014

			The pair (1288, 1485) is in the sequence because sigma(1288) = sigma(1485) = 2880 and sigma(1288+1485) = sigma(2773) = 2880.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000203, A005276.

a[n1_, n2_] := (t = Table[{DivisorSigma[1, n], n}, {n, n1, n2}] // Sort; s = Select[Split[t, #1[[1]] == #2[[1]] &], Length[#] >= 2 &]; f[lst_] := Select[Table[{lst[[i]], lst[[j]]}, {i, 1, Length[lst] - 1}, {j, i + 1, Length[lst]}] // Flatten[#, 1] &, #[[1, 1]] == DivisorSigma[1, #[[1, 2]] + #[[2, 2]]] &]; Select[f /@ s, # != {} &]); Flatten[a[1, 10^5], 2][[All, 2]] (* Jean-François Alcover, Mar 20 2014 *)

More terms from Jean-François Alcover, Mar 20 2014

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

A309227 Quasi-sociable numbers.

Original entry on oeis.org

1215571544, 1270824975, 1467511664, 1512587175, 1530808335, 1579407344, 1638031815, 1727239544

Offset: 1

Donghwi Park, Jul 16 2019

The smallest quasi-sociable number cycle is {1215571544, 1270824975, 1467511664, 1530808335, 1579407344, 1638031815, 1727239544, 151258717, 1215571544}. They are the only known quasi-sociable numbers in 2019.

a(9) > 10^12, if it exists. - Giovanni Resta, Jul 25 2019

			1215571544 = 2^3 * 11 * 13813313
1270824975 = 3^2 * 5^2 * 7 * 19 * 42467
1467511664 = 2^4 * 19 * 599 * 8059
1530808335 = 3^3 * 5 * 7 * 1619903
1579407344 = 2^4 * 31^2 * 59 * 1741
1638031815 = 3^4 * 5 * 7 * 521 * 1109
1727239544 = 2^3 * 2671 * 80833
1512587175 = 3 * 5^2 * 11 * 1833439

David Moews, Reduced sociable numbers (e-mailed by Mitchell Dickerman on Sep. 10, 1997).

Cf. A122726 (sociable numbers), A005276 (betrothed (or quasi-amicable) numbers).

cp's OEIS Frontend

A048050 Chowla's function: sum of divisors of n except for 1 and n.

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A003502 The smaller of a betrothed pair.

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A003503 The larger of a betrothed pair.

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Mathematica

PARI

Extensions

A057533 Values of n for which iteration of Chowla's function loops.

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A192292 Pairs of numbers a, b for which sigma*(a)=b and sigma(b)-b-1=a, where sigma*(n) is the sum of the anti-divisors of n.

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A285889 The smaller of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = n + x + y.

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A239436 Members of a pair (m,k) such that sigma(m) = sigma(k) = sigma(m+k), m < k where sigma = A000203.

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A192292 Pairs of numbers a, b for which sigma(a)=b and sigma(b)-b-1=a, where sigma(n) is the sum of the anti-divisors of n.