Anton Mosunov - cp's OEIS frontend

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

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.

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.

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.

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.

A287238 Numbers whose sum of proper divisors is equal to 95186291194.

Original entry on oeis.org

118315669766, 130863307526, 181448867234, 184661404346, 184881064994, 185180602238, 186803095538, 187013065238, 187127594162, 187516992482, 187889460398, 188332874498, 188587837538, 188750086706, 189131019338, 189322730354, 189374121386, 189621107138

Offset: 1

Anton Mosunov, May 22 2017

The number 95186291194 is the 46th element of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function.
There are exactly 112 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 > 1 such that the number of preimages of an even number n is O((log n)^k).

			a(1) = 118315669766, because it is the smallest number whose sum of proper divisors is equal to 95186291194: 1 + 2 + 7 + 14 + 19 + 38 + 133 + 266 + 444795751 + 889591502 + 3113570257 + 6227140514 + 8451119269 + 16902238538 + 59157834883 = 95186291194.

Anton Mosunov, Table of n, a(n) for n = 1..112
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, A287247, A287251, A287262.

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.

A283183 Number of partitions of n into a prime and a square of an arbitrary integer.

Original entry on oeis.org

0, 1, 3, 2, 1, 4, 3, 2, 2, 0, 5, 4, 1, 4, 2, 2, 3, 4, 3, 4, 4, 2, 5, 2, 0, 2, 6, 4, 3, 4, 1, 6, 4, 0, 4, 2, 1, 8, 4, 2, 5, 4, 3, 4, 4, 2, 7, 4, 2, 2, 4, 4, 5, 6, 2, 6, 4, 0, 5, 4, 1, 8, 4, 0, 4, 6, 5, 8, 4, 2, 5, 6, 3, 2, 6, 2, 8, 4, 3, 6, 2, 2, 11, 6, 0, 6, 6

Offset: 1

Anton Mosunov, Mar 02 2017

a(n) is also the number of solutions to the equation n = p + m^2, where p is prime and m is an arbitrary integer. In comparison, the sequence A002471 counts representations with m being nonnegative.

a(n) is odd if and only if n is prime.

			a(11) = 5 because 11 = 11 + 0^2 = 7 + (-2)^2 = 7 + 2^2 = 2 + (-3)^2 = 2 + 3^2.

Anton Mosunov, Table of n, a(n) for n = 1..10000
H. Li, The exceptional set for the sum of a prime and a square, Acta Math. Hung., 99:123 (2003), 123-141.
R. J. Miech, On the equation n=p+x^2, Trans. of the AMS, 130:3 (1968), 494-512.
A. Nayebi, Upper bounds on the solutions to n=p+m^2, Bull of the Iran. Math. Soc., 37:4 (2011), 95-108.
W. Tianze, On the exceptional set for the equation n=p+k^2, Acta Math. Sinica, 11:2 (1995), 156-167.

Cf. A002471, A020495.

a[n_] := Boole@ PrimeQ[n] + 2 Length@ Select[n - Range[Sqrt@ n]^2, PrimeQ]; Array[a, 87] (* Giovanni Resta, Apr 09 2017 *)

local(i,j,k,total); for (i=1, 1000, j=1; k=1; total=isprime(i); while (j <= i, total += 2*isprime(i-j); j += (2*k+1); k++); print1(total, ", ")) \\ Anton Mosunov, Apr 09 2017

A284187 5-untouchable numbers.

Original entry on oeis.org

838, 904, 1970, 2066, 2176, 3134, 3562, 4226, 4756, 5038, 5312, 5580, 5692, 6612, 6706, 7096, 7210, 7384, 9266, 9530, 9704, 10316, 10742, 10828, 11482, 11578, 11724, 12384, 12592, 12682, 13098, 13236, 13772, 14582, 14846, 15184, 15284, 15338, 15484, 15520, 15578

Offset: 1

Anton Mosunov, Mar 21 2017

nonn

Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n, with the convention that s(0)=0. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.

Let sk(n) denote the k-th iterate of s(n). 5-untouchable numbers are the numbers that lie in the image of s4(n), but not in the image of s5(n). Question: does the set of 5-untouchable numbers have a natural asymptotic density?

			All even numbers less than 838 have a preimage under s5(n), so they are not 5-untouchable.
a(1) = 838, because 838 = s4(2588) but 2588 is untouchable. Therefore 838 is not in the image of s5(n). Note that 2588 is the only preimage of 838 under s4(n).
a(2) = 904, because 904 = s4(4402) = s4(5378) but both 4402 and 5378 are untouchable.
a(3) = 1970, because 1970 = s4(4312) but 4312 is untouchable.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
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.
R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
Carl 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.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.

Cf. A005114, A152454, A283152, A284147, A284156, A363461.

Several missing terms inserted by Jinyuan Wang, Jan 07 2025

A284156 4-untouchable numbers.

Original entry on oeis.org

298, 1006, 1016, 1108, 1204, 1492, 1502, 1940, 2164, 2344, 2370, 2770, 3116, 3358, 3410, 3482, 3596, 3676, 3688, 3976, 4076, 4164, 4354, 4870, 5206, 5634, 5770, 6104, 6206, 6332, 6488, 6696, 6850, 7008, 7118, 7290, 7496, 7586, 7654, 7812, 7922, 8164, 8396, 8434

Offset: 1

Anton Mosunov, Mar 21 2017

nonn

Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n, with the convention that s(0)=0. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.

Let sk(n) denote the k-th iterate of s(n). 4-untouchable numbers are the numbers that lie in the image of s3(n), but not in the image of s4(n). Question: does the set of 4-untouchable numbers have a natural asymptotic density?

			All even numbers less than 298 have a preimage under s4(n), so they are not 4-untouchable.
a(1) = 298, because 298 = s3(668) but 668 is untouchable. Therefore 298 is not in the image of s4(n). Note that 668 is the only preimage of 298 under s3(n).
a(2) = 1006, because 1006 = s3(5366) but 5366 is untouchable.
a(3) = 1016, because 1016 = s3(4402) = s3(5378) but both 4402 and 5378 are untouchable.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
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.
R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
Carl 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.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.

Cf. A005114, A152454, A283152, A284147, A284187.

Several missing terms inserted by Jinyuan Wang, Jan 07 2025

A284147 3-untouchable numbers.

Original entry on oeis.org

388, 606, 696, 790, 918, 1264, 1330, 1344, 1350, 1468, 1480, 1496, 1634, 1688, 1800, 1938, 1966, 1990, 2006, 2026, 2102, 2122, 2202, 2220, 2318, 2402, 2456, 2538, 2780, 2830, 2916, 2962, 2966, 2998, 3224, 3544, 3806, 3926, 4208, 4292, 4330, 4404, 4446, 4466

Offset: 1

Anton Mosunov, Mar 20 2017

nonn

Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n, with the convention that s(0)=0. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.

Let sk(n) denote the k-th iterate of s(n). 3-untouchable numbers are the numbers that lie in the image of s2(n), but not in the image of s3(n). Question: does the set of 3-untouchable numbers have a natural asymptotic density?

			All even numbers less than 388 have a preimage under s3(n), so they are not 2-untouchable.
a(1) = 388, because 388 = s2(668) but 668 is untouchable. Therefore 388 is not in the image of s3(n). Note that 668 is the only preimage of 388 under s2(n).
a(2) = 606, because 606 = s2(474) but 474 is untouchable.
a(3) = 696, because 696 = s2(276) = s2(306) but both 276 and 306 are untouchable.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
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. and Exp. Math. 29 (2020) 414-425
R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
Carl 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.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.

Cf. A005114, A152454, A283152, A284156, A284187.

Several missing terms inserted by Jinyuan Wang, Jan 05 2025

cp's OEIS Frontend

User: Anton Mosunov

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

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

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

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

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A287238 Numbers whose sum of proper divisors is equal to 95186291194.

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

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A283183 Number of partitions of n into a prime and a square of an arbitrary integer.

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A284187 5-untouchable numbers.

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A284156 4-untouchable numbers.

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A284147 3-untouchable numbers.