Jinyuan Wang - cp's OEIS frontend

A386276 Numbers k such that the sequence defined by f(1) = k and f(x+1) = the sum of the three largest proper divisors of f(x), consists entirely of numbers having at least three proper divisors.

6, 18, 42, 54, 66, 72, 78, 102, 114, 126, 138, 162, 174, 186, 198, 216, 222, 234, 246, 258, 282, 294, 306, 318, 342, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 498, 504, 522, 534, 546, 558, 582, 594, 606, 618, 642, 648, 654, 666, 678, 702, 714, 726, 738

Offset: 1

Jinyuan Wang, Jul 17 2025

Numbers k of the form 6*i*12^j, where gcd(i, 10) = 1 and j >= 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..3709
Art of Problem Solving, 2025 IMO Problems/Problem 4

Cf. A080257.

is[k_]:=Module[{v},v=IntegerExponent[k,2]; OddQ[v]&&Mod[k,5]!=0&&2*IntegerExponent[k,3]>v] Select[Range[1,1000],is] (* Vincenzo Librandi, Jul 22 2025 *)

is(k) = my(v=valuation(k, 2)); v%2 && k%5 && 2*valuation(k, 3)>v;

a(n) = 55*n/4 + O(log n). - Charles R Greathouse IV, Aug 18 2025

A385290 Indices of records in A378865.

Original entry on oeis.org

1, 12, 24, 49, 101, 102, 497, 498, 499, 501, 1001, 1002, 2864, 4999, 5001, 10001, 10002, 10624, 12864, 18624, 27648, 123648, 249856, 442368, 786432, 1259874, 2159784, 8249175, 8759124, 10236587, 10236758, 10237649, 10239674, 10239786, 10268473, 10427539, 10476523

Offset: 1

Jinyuan Wang, Jun 24 2025

Cf. A276348, A378865, A379673.

A384851 Decimal expansion of minimal radius of a circle that contains 14 non-overlapping unit disks.

Original entry on oeis.org

4, 3, 2, 8, 4, 2, 8, 5, 5, 4, 8, 6, 0, 8, 3, 6, 6, 8, 1, 4, 0, 3, 9, 0, 9, 3, 6, 7, 4, 7, 8, 1, 8, 1, 0, 9, 1, 6, 0, 8, 4, 9, 5, 7, 2, 9, 6, 5, 8, 6, 7, 5, 7, 0, 1, 2, 4, 5, 7, 5, 4, 8, 5, 5, 2, 2, 1, 1, 3, 3, 7, 0, 4, 5, 4, 0, 2, 1, 3, 8, 6, 3, 1, 9, 7, 5, 7

Offset: 1

Jinyuan Wang, Jun 10 2025

			4.328428554860836681403909367478181...

Dinesh B. Ekanayake and Douglas J. LaFountain, Tight partitions for packing circles in a circle, Italian Journal of Pure and Applied Mathematics, 51 (2024), 115-136.
Michael Goldberg, Packing of 14, 16, 17 and 20 Circles in a Circle, Mathematics Magazine, Vol. 44, No. 3 (May, 1971), pp. 134-139 (provides D/d=4.3284 in Table 1, page 136).

Cf. A084618, A121570, A121598, A121601, A121602, A281115, A282279.

1+sqrt(1+polrootsreal(Pol([707281, -27270266, 472689975, -4930771548, 34449512067, -168736166642, 591369611109, -1498751280720, 2767422383674, -3746579404052, 3734397946902, -2743990597288, 1486108072662, -593729401364, 175537055738, -38557290064, 6295485573, -759438450, 66647843, -4134492, 172311, -4346, 49]))[10])

A383143 Number of positive integers with a shortest addition-subtraction chain of length n.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 16, 28, 49, 88, 156, 280, 499, 904, 1639, 2986, 5442, 9936, 18134

Offset: 0

Jinyuan Wang, Apr 17 2025

			a(6) = 16 because the number of occurrences of 6 in A128998 is 16. These numbers are 19, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 36, 40, 48, 64.

Neill Clift, Addition Subtraction Chains.
Index to sequences related to the complexity of n

Cf. A003065, A128998, A383002, A383142.

a(n) >= A003065(n).

A383142 Smallest positive integer with shortest addition-subtraction chain of length n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 19, 29, 53, 87, 151, 267, 461, 811, 1383, 2357, 4277, 7499, 14003, 25931, 44269, 87773, 152947, 271563

Offset: 0

Jinyuan Wang, Apr 17 2025

			a(8) = 53 because 53 is the smallest positive integer k such that A128998(k) = 8. An example of a shortest addition-subtraction chain for 53 is (1 2 3 4 7 14 25 28 53). a(8) > A003064(8) because an optimal chain for A003064(8) = 47 has length 7: (1 2 3 6 12 24 23 47).

Neill Clift, Addition Subtraction Chains.
Index to sequences related to the complexity of n

Cf. A003064, A128998, A383001, A383143.

a(n) >= A003064(n).

A381229 a(n) is the number of distinct positive integers that can be obtained by starting with n, and optionally applying the operations square root, floor, and ceiling, in any order.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 7

Offset: 1

N. J. A. Sloane and Jinyuan Wang, Feb 25 2025

nonn

			For n = 15, sqrt(15) = 3.872..., floor and ceiling give 3 and 4. Sqrt(3) = 1.732..., and floor and ceiling give 1 and 2. 4 gives nothing new. In all, we get a(15) = 5 different numbers: 15, 3, 4, 1, 2.

Cf. A381226, A381227, A381228.

f(n) = my(t); if(n<4, [1..n], t=sqrtint(n); if(issquare(n), concat(f(t), n), Set(concat([f(t), f(t+1), [n]]))));
a(n) = #f(n);

A381026 Primitive solutions k to the Diophantine equation k^7 = Sum_{i=1..7} y_i^7 with y_i > 0.

Original entry on oeis.org

568, 626

Offset: 1

Jinyuan Wang, Feb 12 2025

			568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7.
626^7 = 91^7 + 148^7 + 158^7 + 255^7 + 258^7 + 309^7 + 625^7.

Jean-Charles Meyrignac, Computing Minimal Equal Sums Of Like Powers.
Eric Weisstein's World of Mathematics, Diophantine Equation - 7th powers.
Index to sequences related to Diophantine equations (7,1,7)

Cf. A380716.

A380716 Primitive solutions k to the Diophantine equation k^7 = Sum_{i=1..8} y_i^7 with y_i > 0.

Original entry on oeis.org

102, 377, 430, 454

Offset: 1

Jinyuan Wang, Jan 30 2025

			102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7.
377^7 = 5^7 + 23^7 + 47^7 + 97^7 + 108^7 + 179^7 + 315^7 + 359^7.
430^7 = 10^7 + 105^7 + 105^7 + 113^7 + 160^7 + 256^7 + 373^7 + 400^7.
454^7 = 50^7 + 52^7 + 65^7 + 252^7 + 266^7 + 312^7 + 319^7 + 440^7.

Eric Weisstein's World of Mathematics, Diophantine Equation - 7th powers.
Index to sequences related to Diophantine equations (7,1,8)

Cf. A381026.

a(4) from Jinyuan Wang, Feb 12 2025

A380150 a(n) is the least k such that there exists a number 1 <= m <= k-1 and exactly n different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2.

Original entry on oeis.org

2, 35, 385, 1872, 5670, 30030

Offset: 0

Jinyuan Wang and Jianing Song, Jan 13 2025

a(n) is also the least k such that there exists a number 1 <= m <= k-1 and at least n different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2: suppose on the contrary that the latter number is k' < a(n), then 1/x^2 - 1/y^2 = 1/m^2 - 1/k'^2 for some 1 <= m <= k'-1 and exactly n' > n pairs (x_1,y_1), ..., (x_{n'},y_{n'}) with 1 <= y_1 < ... < y_{n'} <= k'-1, so 1/x^2 - 1/y^2 = 1/(x_{n+1})^2 - 1/(y_{n+1})^2 has exactly n solutions (x_1,y_1), ..., (x_n,y_n) with 1 <= x < y <= y_{n+1}-1, which implies that a(n) <= y_{n+1} <= y_{n'} <= k'-1, a contradiction.
For a similar reason, this sequence is strictly increasing: if 1/x^2 - 1/y^2 = 1/m^2 - 1/a(n)^2 for some 1 <= m <= a(n)-1 and exactly n pairs (x_1,y_1), ... (x_n,y_n) with 1 <= y_1 < ... < y_n <= a(n)-1, then 1/x^2 - 1/y^2 = 1/(x_n)^2 - 1/(y_n)^2 has exactly n-1 solutions (x_1,y_1), ..., (x_{n-1},y_{n-1}) with 1 <= x < y <= y_n-1, so a(n-1) <= y_n.
a(6) <= 152152, a(7) <= 318240, a(8) <= 445536, a(9) <= 1191190 (see the Mathematics Stack Exchange link).

			The smallest k such that there exists a number 1 <= m <= k-1 and exactly three different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2 is k = 1872: we have 1/300^2 - 1/325^2 = 1/468^2 - 1/585^2 = 1/624^2 - 1/1040^2 = 1/720^2 - 1/1872^2. See the Mathematics Stack Exchange link for more examples.

Mathematics Stack Exchange, Finding multiple ways of representing a number by a difference of inverse squares

Cf. A094191, A355812, A379983.

f(k) = my(v=List([]), m2); for(y=1, k-1, for(x=1, y-1, m2=1/(1/x^2-1/y^2+1/k^2); if(m2==m2\1 && issquare(m2), listput(v, m2)))); if(#v, vecmax(vector(#v, i, sum(j=1, #v, v[i]==v[j]))), 0); \\ Gives the maximum number of pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2, where m runs through 1..k-1
lista(nn) = my(k=1); for(n=0, nn, until(f(k)==n, k++); print1(k, ", "));

A363875 Numbers k such that there is no odd number whose aliquot sequence contains k.

Original entry on oeis.org

2, 28, 52, 88, 96, 120, 124, 146, 162, 188, 206, 208, 210, 216, 238, 246, 248, 250, 262, 268, 276, 288, 290, 292, 298, 304, 306, 322, 324, 326, 336, 342, 362, 372, 388, 396, 406, 408, 412, 426, 428, 430, 438, 448, 452, 472, 474, 478, 486, 494, 498, 508, 516

Offset: 1

Jinyuan Wang, Jun 25 2023

nonn

k is in sequence iff k can never be reached when iterating the map x -> A001065(x) starting with any odd number m.
Assuming the stronger version of Goldbach conjecture, iff k is in the sequence, there are infinitely many odd numbers whose aliquot sequence contain k.
Supersequence of A005114 (except 5), A283152, A284147, A284156, A284187, ..., and untouchable perfect numbers (28, 137438691328, ...), untouchable amicable numbers (A238382), untouchable sociable numbers.

Andrew R. Booker, Finite connected components of the aliquot graph, arXiv:1610.07471 [math.NT], 2017.
Jean-Luc Garambois, Le graphe infini des suites aliquotes (in French).
Wikipedia, Aliquot sequence

Cf. A001065, A005114, A238382, A283152, A284147, A284156, A284187.

cp's OEIS Frontend

User: Jinyuan Wang

A386276 Numbers k such that the sequence defined by f(1) = k and f(x+1) = the sum of the three largest proper divisors of f(x), consists entirely of numbers having at least three proper divisors.

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A385290 Indices of records in A378865.

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A384851 Decimal expansion of minimal radius of a circle that contains 14 non-overlapping unit disks.

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A383143 Number of positive integers with a shortest addition-subtraction chain of length n.

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A383142 Smallest positive integer with shortest addition-subtraction chain of length n.

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A381229 a(n) is the number of distinct positive integers that can be obtained by starting with n, and optionally applying the operations square root, floor, and ceiling, in any order.

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A381026 Primitive solutions k to the Diophantine equation k^7 = Sum_{i=1..7} y_i^7 with y_i > 0.

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A380716 Primitive solutions k to the Diophantine equation k^7 = Sum_{i=1..8} y_i^7 with y_i > 0.

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A380150 a(n) is the least k such that there exists a number 1 <= m <= k-1 and exactly n different pairs (x,y), 1 <= x < y <= k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2.

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A363875 Numbers k such that there is no odd number whose aliquot sequence contains k.

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