A063990 -id:A063990 - cp's OEIS frontend

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A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1

Offset: 1

Yasutoshi Kohmoto

The exponential divisors of a number x = Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.
Wu gives a complicated Dirichlet g.f.
a(1) = 1 by convention. This is also required for a function to be multiplicative. - N. J. A. Sloane, Mar 03 2009
The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - R. J. Mathar, Jul 14 2014

			a(8)=2 because 2 and 2^3 are e-divisors of 8.
The sets of e-divisors start as:
  1:{1}
  2:{2}
  3:{3}
  4:{2, 4}
  5:{5}
  6:{6}
  7:{7}
  8:{2, 8}
  9:{3, 9}
  10:{10}
  11:{11}
  12:{6, 12}
  13:{13}
  14:{14}
  15:{15}
  16:{2, 4, 16}
  17:{17}
  18:{6, 18}
  19:{19}
  20:{10, 20}
  21:{21}
  22:{22}
  23:{23}
  24:{6, 24}

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).
David Moews, A database of aliquot cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles.
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]
László Tóth and Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009.
Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Row lengths of A322791.
Cf. A049599, A061389, A051377 (sum of e-divisors).
Partial sums are in A099593.
Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

A049419:=n->Product(List(Collected(Factors(n)), p -> Tau(p[2]))); List([1..10^4], n -> A049419(n)); # Muniru A Asiru, Oct 29 2017

a049419 = product . map (a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A049419 := proc(n)
    local a;
    a := 1 ;
    for pf in ifactors(n)[2] do
        a := a*numtheory[tau](op(2,pf)) ;
    end do:
    a ;
end proc:
seq(A049419(n),n=1..20) ; # R. J. Mathar, Jul 14 2014

a[1] = 1; a[p_?PrimeQ] = 1; a[p_?PrimeQ, e_] := DivisorSigma[0, e]; a[n_] := Times @@ (a[#[[1]], #[[2]]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)

a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ Amiram Eldar, Mar 27 2023

Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic, Jul 23 2001

Sum_{k=1..n} a(k) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

More terms from Jud McCranie, May 29 2000

A063991 Unitary amicable numbers.

Original entry on oeis.org

114, 126, 1140, 1260, 18018, 22302, 32130, 40446, 44772, 49308, 56430, 64530, 67158, 73962, 142310, 168730, 180180, 197340, 223020, 241110, 242730, 286500, 296010, 308220, 365700, 429750, 462330, 548550, 591030, 618570, 669900, 671580, 739620, 785148, 815100, 827652, 827700, 932100

Offset: 1

N. J. A. Sloane, Sep 18 2001

nonn

From Amiram Eldar, Mar 09 2024: (Start)
The concept of unitary amicable numbers was introduced by Wall (1970), who proved that both members of a pair are either odd or even, and found 610 pairs (only 592 were correct, as found by te Riele, 1978).
Hagis (1971) calculated the first 19 pairs (the terms below 10^6).
Najar (1995) calculated the first 185 pairs (terms whose smaller member is below 10^8). (End)

Mariano Garcia, New unitary amicable couples, J. Recreational Math., Vol. 17, No. 1 (1984-5), pp. 32-35.
M. Lal, G. Tiller, and T. Summers, Unitary sociable numbers, Proceedings of the Second Manitoba Conference on Numerical Mathematics, Congressus Numerantium No. 7, 1972, pp. 211-216.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 148.

Michel Marcus, Table of n, a(n) for n = 1..149
Peter Hagis, Jr., Unitary amicable numbers, Math. Comp., Vol. 25, No. 116 (1971), pp. 915-918.
O. William McClung, Generators of Unitary Amicable Numbers, Fibonacci Quarterly, Vol. 23, No. 2 (1985), pp. 158-164; Letter to the Editor, ibid., Vol. 24, No. 2 (1986), p. 106.
Rudolph M. Najar, Operations on Generators of Unitary Amicable Pairs, Fibonacci Quarterly, Vol. 27, No. 2 (1989), pp. 144-152.
Rudolph M. Najar, The Unitary Amicable Pairs To 10^8, International Journal of Mathematics and Mathematical Sciences, Volume 18, No. 2 (1995), pp. 405-410, Article ID 396507, 6 pages.
J. O. M. Pedersen, Known Unitary Amicable Pairs.
J. O. M. Pedersen, Known Unitary Amicable Pairs. [Via Wayback Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Ivars Peterson, Amicable Pairs, Divisors and a New Record, February 2, 2004.
Herman J. J. te Riele, Further Results on Unitary Aliquot Sequences, Report NW 2/78, Math. Centre, Amsterdam, 2nd ed., Jan. 1978.
Herman J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, Mathematisch Centrum, Amsterdam, 1976.
Herman J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, Amsterdam, 1972.
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.
Eric Weisstein's World of Mathematics, Unitary Amicable Pair.

Union of A002952 and A002953.

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok1(n) = iferr(f(n) == n, E, 0);
isok2(n) = iferr(f(f(n)) == n, E, 0);
isok(n) = isok2(n) && !isok1(n); \\ Michel Marcus, Sep 29 2018

More terms from Michel Marcus, Sep 29 2018

A127666 Odd infinitary abundant numbers.

Original entry on oeis.org

945, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 37125, 38745, 39585, 40635, 41055, 42315

Offset: 1

Ant King, Jan 26 2007

nonn

This is also the sequence of odd integers whose infinitary aliquot sequences initially increase. Based on empirical evidence (up to 10 million), this applies to only about 0.1% of odd integers.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 77, 473, 5703, 53569, 561610, 5525461, 54979537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0005... . - Amiram Eldar, Sep 09 2022

			a(5)=16065 because 16065 is the fifth odd number that is exceeded by the sum of its proper infinitary divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., Vol. 54, No. 189, (1990), 395-411.
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]

Cf. A005231, A126168, A127661, A129656.

ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;Select[Range[1,50000,2],properinfinitarydivisorsum[ # ]># &] (* end of program *)
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[1, 50000, 2], isigma[#] > 2 # &] (* Amiram Eldar, Jun 09 2019 *)

A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
isok(k) = A049417(k)>2*k&&k%2==1; \\ Jinyuan Wang, Jun 09 2019

Odd values of n for which A126168(n)>n.

A002952 Smaller of unitary amicable pair.

Original entry on oeis.org

114, 1140, 18018, 32130, 44772, 56430, 67158, 142310, 180180, 197340, 241110, 296010, 308220, 462330, 591030, 669900, 671580, 785148, 815100, 1004850, 1077890, 1080150, 1156870, 1177722, 1222650, 1281540, 1475810, 1511930, 1571388

Offset: 1

N. J. A. Sloane; extended Nov 24 2005

I proved the following facts: (a) If (m,n) is a unitary amicable pair such that mod(m,4)= mod(n,4)=2 and 5 doesn't divide m*n then (10*m,10*n) is a unitary amicable pair. (b) If (m,n) is a unitary amicable pair such that m/12 and n/12 are natural numbers and gcd(m/12,12)=gcd(n/12,12)=1 then (3/2*m,3/2*n) is a unitary amicable pair. - Farideh Firoozbakht, Nov 27 2005

			(114,126) is a unitary amicable pair: 114 has unitary divisors 1, (2,57), (3,38) and (6,19), apart from 114 itself. Their sum is 126, whose unitary divisors < 126 are 1, (2,63), (7,18), (9,14) whose sum is 114.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..7896 (from Pedersen's website)
Peter Hagis, Jr., Unitary amicable numbers, Math. Comp., 25 (1971), 915-918.
J. M. Pedersen, Known Unitary Amicable Pairs
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]
Ivars Peterson, Amicable Pairs, Divisors, and a New Record, January 30 2004.
Eric Weisstein's World of Mathematics, Unitary Amicable Number.

Cf. A002953, A063991, A111904.

uDivisors[n_] := Select[Divisors[n], # < n && GCD[#, n/#] == 1 & ]; mate[n_] := If[m = Total[uDivisors[n]]; n == Total[uDivisors[m]], m, 0]; Reap[Do[If[n < mate[n], Print[n]; Sow[n]], {n, 2, 2000000}]][[2, 1]] (* Jean-François Alcover, Jun 12 2012 *)

A002953 Larger of unitary amicable pair.

Original entry on oeis.org

126, 1260, 22302, 40446, 49308, 64530, 73962, 168730, 223020, 286500, 242730, 429750, 365700, 548550, 618570, 827700, 739620, 827652, 932100, 1241550, 1099390, 1291050, 1292570, 1241478, 1398150, 1621500, 1669150, 1598470, 1654212

Offset: 1

N. J. A. Sloane; extended Nov 24 2005

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..7896 (from Pedersen's website)
Peter Hagis, Jr., Unitary amicable numbers, Math. Comp., 25 (1971), 915-918.
J. M. Pedersen, Known Unitary Amicable Pairs
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]
Ivars Peterson, Amicable Pairs, Divisors, and a New Record, January 30 2004.
Eric Weisstein's World of Mathematics, Unitary Amicable Pair

Cf. A002952, A063991, A111904.

uDivisors[n_] := Select[Divisors[n], # < n && GCD[#, n/#] == 1 & ]; mate[n_] := If[m = Total[uDivisors[n]]; n == Total[uDivisors[m]], m, 0]; Reap[Do[If[n < (m = mate[n]), Print[m]; Sow[m]], {n, 2, 2000000}]][[2, 1]] (* Jean-François Alcover, Jun 12 2012 *)

A259933 Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.

Original entry on oeis.org

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 66928, 66992, 67095, 71145, 63020, 76084, 69615, 87633, 79750, 88730, 100485, 124155, 122368, 123152, 122265, 139815, 141664, 153176, 142310, 168730, 171856, 176336, 176272, 180848, 185368, 203432, 196724, 202444, 280540, 365084, 308620, 389924

Offset: 1

Omar E. Pol, Jul 09 2015

nonn

A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.
By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
The amicable pairs (x < y) are adjacent to each other in the list.
Also A260086 and A260087 interleaved.
Another version (A259180) lists the amicable pairs (x < y) ordered by increasing x.
Amicable numbers A063990 are the terms of this sequence in increasing order.
First differs from both A063990 and A259180 at a(17).

			-----------------------------------
       Amicable pair         Sum
          x      y          x + y
-----------------------------------
n     A260086 A260087      A259953
-----------------------------------
1        220     284          504
2       1184    1210         2394
3       2620    2924         5544
4       5020    5564        10584
5       6232    6368        12600
6      10744   10856        21600
7      12285   14595        26880
8      17296   18416        35712
9      66928   66992       133920
10     67095   71145       138240
11     63020   76084       139104
12     69615   87633       157248
...      ...     ...          ...
32    609928  686072      1296000
33    643336  652664      1296000
...
The sum of the proper divisors (or aliquot parts) of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. On the other hand the sum of the proper divisors (or aliquot parts) of 284 is 1 + 2 + 4 + 71 + 142 = 220. Note that 220 + 284 = sigma(220) = sigma(284) = 504. The sum 220 + 284 = 504 is the smallest sum of an amicable pair, so a(1) = 220 and a(2) = 284.
Note that some pairs (x, y) share the same sum (x + y), for example: (609928 + 686072) = (643336 + 652664) = sigma(609928) = sigma(686072) = sigma(643336) = sigma(652664) = 1296000, thus in the list first appears the pair (609928, 686072) and then (643336, 652664) because 609928 < 643336.

Laszlo Hars, Performance compared to mathematica Julia-users (2014)
Khelleos, Amicable numbers, CyberForum.ru (2011)
OEIS Wiki, Amicable numbers (This page needs work)
Wikipédia, Barátságos számok (contains a mistake: A063990 should be replaced with A259933)

Cf. A000203, A063990, A259180, A259953, A260086, A260087.

a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A259953(n).

A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 139104, 133920, 138240, 157248, 168480, 224640, 262080, 245520, 294840, 311040, 348192, 357120, 388800, 399168, 645624, 698544, 749952, 756000, 892800, 955206, 1017792, 1048320

Offset: 1

T. D. Noe, Aug 14 2010

nonn

This sequence initially shares many terms with A161005 because small amicable pairs are sometimes consecutive terms in the sorted list of amicable numbers, A063990.
This sequence is sorted by the smaller (abundant) member from A002025, so a(n) is not increasing. - Jeppe Stig Nielsen, Jan 27 2015
Duplicates occur, e.g., a(32)=a(35)=1296000. - Jeppe Stig Nielsen, Jan 27 2015
Comment originally by M. F. Hasler, Dec 14 2013, in A161005: "Also: The common value of sigma(a) = sigma(b) of the amicable pairs (a,b). See A137231 for the analog for amicable triples, and A116148 for quadruples." - Jeppe Stig Nielsen, Jan 27 2015
It is not known if a(n) is always even (see Hagis links). - Jeppe Stig Nielsen, Jan 31 2015
Are all terms abundant (A005101)? The first 10000 terms are. - Ivan N. Ianakiev, Apr 15 2021

			a(9) = A002025(9) + A002046(9) = 63020 + 76084 = 139104.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Peter Hagis, Lower bounds for relatively prime amicable numbers of opposite parity, Math. Comp. 24 (1970), 963-968.
Peter Hagis, Relatively Prime Amicable Numbers of Opposite Parity, Mathematics Magazine, Vol. 43, No. 1 (Jan., 1970), pp. 14-20.
Eric W. Weisstein's World of Mathematics, Pair Sum.

Cf. A002025, A002046, A066539, A259180 (amicable pairs).

s[n_] := DivisorSigma[1,n]-n; smallAmicableQ[n_] := Module[{b=s[n]}, n

Formula

a(n) = A259180(2n-1) + A259180(2n). - Omar E. Pol, Oct 22 2017

A074902 Known friendly numbers.

Original entry on oeis.org

6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84, 96, 102, 108, 114, 120, 132, 135, 138, 140, 150, 168, 174, 186, 200, 204, 210, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 294, 300, 308, 312, 318, 330, 348, 354, 360, 364, 366, 372

Offset: 1

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Author

N. J. A. Sloane, Sep 15 2002

Keywords

nonn

Comments

The sequence is not known to be complete up to 372, since there are many small numbers, including 10, 14, 15 and 20, which have not been proved to be solitary. If any other numbers up to 372 are friendly, the smallest corresponding values of m are > 10^30.

A positive integer n is 'friendly' if abundancy(n) = abundancy(m) for some positive integer m not equal to n, where abundancy(n) = sigma(n)/n (cf. A000203); otherwise n is 'solitary'. (The name "friendly" is also sometimes mistakenly used with other meanings; cf. A063990 and A007770.)

All perfect numbers are friendly numbers, but they are only friendly with each other (a perfect number being defined as having abundancy index of 2.) - Daniel Forgues, Jun 23 2009

Triangle A211679 has rows that list the first numbers that have n-1 smaller friends. Sequence A211677 lists just the last number in each row. - T. D. Noe, May 10 2012

Examples

24 is in the sequence since abundancy(24) = abundancy(91963648) = 5/2.

Links

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.

Sagar Mandal, Prime Divisors of 10's Friends: A Generalization of Prior Bounds, arXiv:2412.02701 [math.GM], 2024. See p. 10.

Sagar Mandal, Exploring the relationships between the divisors of friends of 10, Bull. Calcutta Math. Soc. (2025) Vol. 48, No. 1-3, 21-32.

Sagar Mandal and Sourav Mandal, Upper bounds for the prime divisors of friends of 10-II, arXiv:2412.02701 [math.GM], 2024. See p. 8.

Eric Weisstein's World of Mathematics, Friendly Pair

Eric Weisstein's World of Mathematics, Friendly Number

Crossrefs

Union of A050972 and A050973. Cf. A014567.

Extensions

Edited by Dean Hickerson, Sep 19 2002

A003416 Sociable numbers: smallest member of each cycle (conjectured).

Original entry on oeis.org

12496, 14316, 1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 805984760, 1095447416, 1236402232, 1276254780, 1799281330

Offset: 1

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Author

N. J. A. Sloane

Keywords

nonn

Comments

Numbers belonging to aliquot cycles of length greater than 2.

There is no proof that 564 (for example) is missing from this sequence (cf. A122726). - N. J. A. Sloane, Sep 17 2021

The lengths of the corresponding aliquot cycles are given by A052470. - Michel Marcus, Nov 15 2013

References

R. K. Guy, Unsolved Problems Number Theory, B7.

Paul Poulet, Parfaits, amiables et extensions, Editions Stevens, Bruxelles, 1918.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 91-92.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 174 Penguin Books 1987.

Links

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..53

Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.

Verner E. Hoggart, Jr., Letters to N. J. A. Sloane, 1974-1975

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 112

David Moews, Perfect, amicable and sociable numbers

David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]

Jan O. M. Pedersen, Tables of Aliquot Cycles [Broken link]

Jan O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]

Jan O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Eric Weisstein's World of Mathematics, Sociable Numbers.

Crossrefs

Cf. A052470, A072890, A072891, A122726.

Extensions

Incorrect g.f. deleted by N. J. A. Sloane, Sep 20 2008

Added "conjectured" to definition. - N. J. A. Sloane, Sep 17 2021

A066539 Difference between larger and smaller terms of n-th amicable pair.

Original entry on oeis.org

64, 26, 304, 544, 136, 112, 2310, 1120, 13064, 64, 4050, 18018, 8980, 23670, 17550, 784, 11512, 26420, 4480, 4576, 18064, 5720, 84544, 81304, 110852, 43184, 17888, 17150, 11680, 3510, 69296, 76144, 67072, 76592, 9328, 115592, 70592, 61110, 21712

Offset: 1

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Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 06 2002

Keywords

nonn

Comments

Values are sorted along increasing A002025. - R. J. Mathar, Jul 19 2009, Jul 23 2009

Examples

a(7) = 2310 because the 7th pair of amicable numbers is 12285 and 14595; and 14595 - 12285 = 2310.

References

For references see A002025.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..415523

Crossrefs

Cf. A002025, A002046, A063990, A259180 (amicable pairs).

Programs

Mathematica

With[{s = PositionIndex@Array[DivisorSigma[1, #] &, 10^6]}, Flatten@ Map[Differences, Apply[Join, Map[Function[n, Select[Subsets[Lookup[s, n], {2}], Total@ # == n &]], Sort@ Select[Keys@ s, Length@ Lookup[s, #] > 1 &]]]]] (* Michael De Vlieger, Oct 22 2017 *)

Formula

a(n) = A002046(n) - A002025(n).

a(n) = A259180(2n) - A259180(2n-1). - Omar E. Pol, Oct 22 2017

Extensions

Sorted on smaller term for alignment with A002046 and A002025. - R. J. Mathar, Jul 19 2009

A-number in comment corrected by R. J. Mathar, Jul 23 2009

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A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

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A063991 Unitary amicable numbers.

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A127666 Odd infinitary abundant numbers.

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A002952 Smaller of unitary amicable pair.

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A002953 Larger of unitary amicable pair.

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A259933 Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.

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A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

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