A287238 -id:A287238 - cp's OEIS frontend

A283157 Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.

2, 4, 6, 40, 106, 314, 1954, 2234, 2794, 11194, 22394, 58234, 111994, 160154, 291194, 425594, 560554, 1022554, 1455994, 1601594, 3203194, 11703994, 16743994, 21781754, 24751994, 53253194, 60860794, 79587194, 95295194, 181060874, 287123194, 435635194, 973772794

Offset: 1

Anton Mosunov, Mar 01 2017

nonn

Let sigma(n) denote the sum of divisors function, and s(n):=sigma(n)-n. Let r(n) denote the number of solutions to n=s(m) and put a(1):=2. a(2) is equal to the smallest number such that r(a(2)) > r(a(1)). a(3) is equal to the smallest number such that r(a(3)) > r(a(2)), and so on.
Pomerance proved that, for every e > 0, the number of solutions to n = s(m) when n is even is O_e(n^{2/3+e}).
There are 49 elements in this sequence which do not exceed 2^40. The largest element, 690100611194, has 139 preimages.

			a(1)=2, because 2=s(m) has 0 solutions;
a(2)=4, because 4=s(9);
a(3)=6, because 6=s(6)=s(25);
a(4)=40, because 40=s(44)=s(74)=s(81);
a(5)=106, because 106=s(80)=s(104)=s(110)=s(206);
a(6)=314, because 314=s(370)=s(406)=s(442)=s(622)=s(313^2);
a(7)=1954, because 1954=s(1856)=s(1952)=s(2216)=s(2702)=s(3014)=s(3902);
a(8)=2234, because 2234=s(2536)=s(2770)=s(3454)=s(3562)=s(3706)=s(3886)=s(3922);
a(9)=2794, because 2794=s(3176)=s(3716)=s(3470)=s(3878)=s(4334)=s(4658)=s(4958)=s(4982)=s(5582).

Anton Mosunov, Table of n, a(n) for n = 1..49
Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, Numerical and Statistical Analysis of Aliquot Sequences, arXiv:2110.14136 [math.NT], 2021.
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A005114, A283156, A287233, A287238, A287247, A287251, A287262.

v=vectorsmall(10^8);
for(n=2,#v,t=(sigma(n)-n)/2;if(denominator(t)==1 && t<=#v, v[t]++))
r=0;for(n=1,#v, if(v[n]>r,r=v[n];print1(2*n", "))) \\ Charles R Greathouse IV, Mar 02 2017

a(20)-a(25) from Charles R Greathouse IV, Mar 02 2017
a(26)-a(31) from Anton Mosunov, Mar 03 2017
a(32)-a(49) from Anton Mosunov, Apr 20 2017

A287233 Numbers whose sum of proper divisors is equal to 88978489594.

Original entry on oeis.org

111223111970, 119597953286, 153690118286, 162254892614, 165823548554, 170330251118, 172618269242, 173103606398, 174143614538, 174490283894, 174816560918, 174923620562, 175023621326, 175949944022, 176622299474, 176749123766, 176986301486, 177090301922

Offset: 1

Anton Mosunov, May 22 2017

The number 88978489594 is the 45th element of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function.
There are exactly 95 elements in the sequence.
In 2016, C. Pomerance proved that, for every e>0, the number of preimages is O_e(n^{2/3+e}).
Conjecture: there exists a positive real number k such that the number of preimages of an even number n is O((log n)^k).

			a(1) = 111223111970, because it is the smallest number whose sum of proper divisors is equal to 88978489594: 1 + 2 + 5 + 10 + 11122311197 + 22244622394 + 55611555985 = 88978489594.

Anton Mosunov, Table of n, a(n) for n = 1..95
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A001065, A283156, A283157, A287238, A287251, A287262.

A287251 Numbers whose sum of proper divisors is equal to 666304038394.

Original entry on oeis.org

1082744062322, 1178845606262, 1207676069426, 1215025011014, 1279464378926, 1309091462678, 1309893165362, 1310880770114, 1312211013242, 1315226230958, 1317231828218, 1318629668702, 1324707235382, 1325469101618, 1326419490542, 1328065089458, 1328085645914

Offset: 1

Anton Mosunov, May 22 2017

The number 666304038394 is the 48th element of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function.
There are exactly 130 elements in the sequence.
In 2016, C. Pomerance proved that, for every e>0, the number of preimages is O_e(n^{2/3+e}).
Conjecture: there exists a positive real number k such that the number of preimages of an even number n is O((log n)^k).

			a(1) = 1082744062322  because it is the smallest number whose sum of proper divisors is equal to 666304038394: 1 + 2 + 13 + 26 + 41644002397 + 83288004794 + 541372031161 = 666304038394.

Anton Mosunov, Table of n, a(n) for n = 1..130
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A001065, A283156, A283157, A287233, A287238, A287247, A287262.

A287262 Numbers whose sum of proper divisors is equal to 690100611194.

Original entry on oeis.org

1258418761414, 1276686130498, 1286096593354, 1290188098942, 1306261870882, 1321049741038, 1338795185146, 1350625481098, 1359498202882, 1365723585502, 1367261834038, 1371277504834, 1372962401386, 1373062247098, 1373771709754, 1374112095298, 1374709701094

Offset: 1

Anton Mosunov, May 22 2017

The number 690100611194 is the 49th term of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function. Up to 2^40, this is the even number with the greatest number of preimages. As of May 22 2017, this is the largest known even number with the greatest number of preimages.
There are exactly 139 terms in the sequence.
In 2016, C. Pomerance proved that, for every e > 0, the number of preimages is O_e(n^{2/3+e}).
Conjecture: there exists a positive real number k such that the number of preimages of an even number n is O((log n)^k).

			a(1) = 1258418761414, because it is the smallest number whose sum of proper divisors is equal to 690100611194: 1 + 2 + 31 + 62 + 20297076797 + 40594153594 + 629209380707 = 690100611194.

Anton Mosunov, Table of n, a(n) for n = 1..139
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A001065, A283156, A283157, A287233, A287238, A287247, A287251.

A287247 Numbers whose sum of proper divisors is equal to 289697407994.

Original entry on oeis.org

300629385430, 331082751976, 348602203870, 539890623754, 552235683634, 556381352806, 556523967562, 557844696646, 562012970938, 569170200598, 569518766962, 573004430386, 574282506778, 575462269366, 576199620754, 577107726658, 577305647026, 577419601138

Offset: 1

Anton Mosunov, May 22 2017

The number 289697407994 is the 47th element of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function.
There are exactly 123 elements in the sequence.
In 2016, C. Pomerance proved that, for every e>0, the number of preimages is O_e(n^{2/3+e}).
Conjecture: there exists a positive real number k such that the number of preimages of an even number n is O((log n)^k).

			a(1) = 300629385430, because it is the smallest number whose sum of proper divisors is equal to 289697407994: 1 + 2 + 5 + 10 + 11 + 22 + 55 + 110 + 2732994413 + 5465988826 + 13664972065 + 27329944130 + 30062938543 + 60125877086 + 150314692715 = 289697407994.

Anton Mosunov, Table of n, a(n) for n = 1..123
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A001065, A283156, A283157, A287233, A287238, A287251, A287262.

A287246 Numbers whose sum of proper divisors is equal to 57939481594.

Original entry on oeis.org

77753398058, 94151657522, 98497118618, 105654348614, 107396027126, 107978124554, 112402593722, 112536300194, 113395841738, 113591877506, 113834039318, 113903752562, 114541896698, 114637401218, 114663447902, 114856499738, 115175241854, 115246892846, 115271202986

Offset: 1

Anton Mosunov, May 22 2017

The number 57939481594 is the 44th element of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function.
There are exactly 94 elements in the sequence.
In 2016, C. Pomerance proved that, for every e>0, the number of preimages is O_e(n^{2/3+e}).
Conjecture: there exists a positive real number k such that the number of preimages of an even number n is O((log n)^k).

			a(1) = 77753398058, because it is the smallest number whose sum of proper divisors is equal to 57939481594: 1 + 2 + 7 + 14 + 49 + 98 + 6793 + 13586 47551 + 95102 + 116797 + 233594 + 332857 + 665714 + 817579 + 1635158 + 5723053 + 11446106 + 793402021 + 1586804042 + 5553814147 + 11107628294 + 38876699029 = 57939481594.

Anton Mosunov, Table of n, a(n) for n = 1..94
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A001065, A283156, A283157, A287233, A287238, A287247, A287251, A287262.

cp's OEIS Frontend

A283157 Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.

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A287233 Numbers whose sum of proper divisors is equal to 88978489594.

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A287251 Numbers whose sum of proper divisors is equal to 666304038394.

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A287262 Numbers whose sum of proper divisors is equal to 690100611194.

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A287247 Numbers whose sum of proper divisors is equal to 289697407994.

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A287246 Numbers whose sum of proper divisors is equal to 57939481594.

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