A002953 -id:A002953 - cp's OEIS frontend

A180277 a(n) = A002952(n) + A002953(n).

240, 2400, 40320, 72576, 94080, 120960, 141120, 311040, 403200, 483840, 483840, 725760, 673920, 1010880, 1209600, 1497600, 1411200, 1612800, 1747200, 2246400, 2177280, 2371200, 2449440, 2419200, 2620800, 2903040, 3144960, 3110400

Offset: 1

Jonathan Vos Post, Aug 23 2010

			a(1) = 114 + 126 = 240 = 2^4 * 3 * 5.
a(2) = 1140 + 1260 = 2400 = 2^5 * 3 * 5^2.
a(10) = a(11) - why?

Amiram Eldar, Table of n, a(n) for n = 1..7896

Cf. A002952, A002953, A161005, A180163.

a(n) = A002952(n) + A002953(n).

{(j+k) such that sigma*(j) = sigma*(k) = j+k, where sigma*(n) is the unitary divisor function A034448}.

Extended by R. J. Mathar, Aug 26 2010

Shorter name using given formula from Joerg Arndt, Jul 29 2024

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

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 *)

A324709 Larger of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

Original entry on oeis.org

126, 846, 1260, 8460, 11760, 10856, 14595, 17700, 49308, 83142, 62700, 71145, 73962, 83904, 107550, 88730, 131100, 168730, 149952, 196650, 203432, 306612, 365700, 399592, 419256, 548550, 721962, 669688, 831420, 686072, 691256, 712216, 652664, 661824, 827700

Offset: 1

Amiram Eldar, Mar 11 2019

nonn

The terms are ordered according to their lesser counterparts (A324708).

			126 is in the sequence since it is the larger of the amicable pair (114, 126): tsigma(114) = tsigma(126) = 114 + 126.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A002046, A002953, A126170, A292981, A322542, A324706, A324707, A324708.

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; tsigma[1]=1; tsigma[n_]:= Times @@ f @@@ FactorInteger[n]; s[n_] := tsigma[n] - n; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, m]] ,{n,1,700000}]; seq

A348344 Larger member of a noninfinitary amicable pair: numbers (k, m) such that nisigma(k) = m and nisigma(m) = k, where nisigma(k) is the sum of the noninfinitary divisors of k (A348271).

Original entry on oeis.org

448, 2032, 8128, 7168, 24384, 41984, 130048, 41940480, 102222432, 221316608, 34359738352

Offset: 1

Amiram Eldar, Oct 13 2021

The terms are ordered according to their smaller counterparts (A348343).

			448 is a term since A348271(448) = 336 and A348271(336) = 448.

Cf. A327633, A348271, A348343.

Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, m]], {n,1,10^4}]; seq

A357496 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

1136, 11696, 22256, 25472, 43424, 73664, 131355, 304336, 267968, 492608, 612704, 674920, 640305, 788697, 691292, 705344, 723392, 813728, 809776, 1117395, 1258335, 1559696, 1518570, 1598368, 1821376, 2218250, 2058944, 2678752, 2744288, 2765024, 2848864, 2610656, 3134224

Offset: 1

Amiram Eldar, Oct 01 2022

nonn

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The terms are ordered according to their lesser counterparts (A357495).
Both members of each pair are necessarily nonsquarefree numbers.

			1136 is a term since s(1136) = 880 and s(880) = 1136.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A162296, A325314, A322609, A357493, A357494, A357495.
Subsequence of A013929.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 2, 3*10^6}]; seq

A111904 Number of unitary amicable pairs in which the smallest member has n digits.

Original entry on oeis.org

0, 0, 1, 1, 5, 12, 59, 107, 269, 641, 1457, 3553

Offset: 1

N. J. A. Sloane, Nov 24 2005

J. O. 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]

Cf. A002952, A002953.

The next term, a(13), is at least 6664.

A306873 Larger of augmented unitary amicable pair.

Original entry on oeis.org

7336455, 41337555, 110691295, 108212055, 154646206, 313439511, 6400149855, 9971007915, 10049576691, 9849706755, 12125842995, 12180547995, 14064001666, 18225635506, 26623431835, 20500208806, 23746912995, 23952459706, 43137954706, 56039259255, 99517314526, 125782774755

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

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

The terms are ordered according to their lesser counterparts (A306872).

			7336455 is in the sequence since it is the larger of the amicable pair (6224890, 7336455): usigma(6224890) = usigma(7336455) = 13561344 = 6224890 + 7336455 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..27 (terms with lesser member below 10^12, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers
J. O. M. Pedersen, Tables of Aliquot Cycles

Cf. A002953, A015630, A034460, A126175, A306872.

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

A306876 Larger of reduced unitary amicable pair.

Original entry on oeis.org

175742294, 6263852385, 16082297385, 18625120185, 32553626105, 38947446285, 37626449685, 41194817265, 103052922665, 87988279533, 103552755405, 126755126589, 131943742994, 192245655405, 226960409585, 181521732405, 502566224565, 399451768365, 403080683565, 461943100905

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 terms are ordered according to their lesser counterparts (A306875).

			175742294 is in the sequence since it is the larger 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 with lesser member below 10^12, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers
J. O. M. Pedersen, Tables of Aliquot Cycles

Cf. A003503, A002953, A126173, A034460, A306875.

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

A333930 Larger of recursive amicable numbers pair: numbers m < k such that m = s(k) and k = s(m), where s(k) = A333926(k) - k is the sum of proper recursive divisors of k.

Original entry on oeis.org

284, 378, 2924, 4584, 5564, 16632, 16728, 28752, 30912, 53692, 76084, 69552, 87633, 124155, 139815, 179118, 168730, 225096, 202444, 256338, 245904, 266568, 365084, 389924, 320016, 430402, 391656, 353616, 387720, 393528, 486178, 525915, 555216, 642720, 814698, 682896

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

The terms are ordered according to their lesser counterparts (A333929).

			284 is a terms since A333926(284) - 284 = 220 and A333926(220) - 220 = 284.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A333926, A333927, A333929.

Analogous sequences: A002046, A002953 (unitary), A126166 (exponential), A126170 (infinitary), A292981 (bi-unitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); s[n_] := recDivSum[n] - n; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 1, 10^5}]; seq

cp's OEIS Frontend

A180277 a(n) = A002952(n) + A002953(n).

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A063991 Unitary amicable numbers.

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A002952 Smaller of unitary amicable pair.

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A324709 Larger of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

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A348344 Larger member of a noninfinitary amicable pair: numbers (k, m) such that nisigma(k) = m and nisigma(m) = k, where nisigma(k) is the sum of the noninfinitary divisors of k (A348271).

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A357496 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

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A111904 Number of unitary amicable pairs in which the smallest member has n digits.

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A306873 Larger of augmented unitary amicable pair.

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A306876 Larger of reduced unitary amicable pair.

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