Zhao Hui Du - cp's OEIS frontend

A386221 Numbers which can be expressed as the product of a number and its binary reversal in at least three different ways.

2371610879733375, 8379443074856875, 103889625367330285, 162508095102648823, 2143169709271976875, 2481725627762299375, 4055619414785589625, 8167773178498814075, 9027536760163222895, 133527604616779133915, 133893081609954481115, 137216105281788994475, 457495296809227508125

Offset: 1

Zhao Hui Du, Aug 12 2025

It appears that most numbers that can be expressed in three different ways can also be expressed in four different ways. For a(n) < 2^88, only 11 numbers can be expressed in exactly three ways while 691 numbers can be expressed in exactly four ways.

			2371610879733375 = 51606261*45955875 = 64244529*36915375 = 64338225*36861615 while 51606261 = 11000100110111001011110101_2, 45955875 = 10101111010011101100100011_2 and reverse(11000100110111001011110101) = 10101111010011101100100011.
8379443074856875 = 101377465*82655875 = 102886105*81443875 = 114021425*73490075 = 115718225*72412475.

Shou'en Wang, Binary reversal (Chinese)

Cf. A066531, A066598.

A382328 Maximum possible product of differences of every pair in a set of nonnegative integers with sum n.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 20, 48, 120, 240, 540, 1440, 4320, 11520, 30240, 64512, 207360, 725760, 2419200, 7257600, 17418240, 39191040, 174182400, 696729600, 2786918400, 9405849600, 25082265600, 65840947200, 182891520000, 1003290624000, 4514807808000, 21069103104000

Offset: 0

Zhao Hui Du, Mar 21 2025

nonn

It seems that for n>=45, if m(m+1)/2<=n<(m+1)(m+2)/2, the set to provide the maximum product has m+1 elements, such as for n=46, the maximum product is reached by set {0,1,2,3,4,5,6,7,8,10}.

			For n=7, the nonnegative integer set {0,1,2,4} has sum 7 and the product of number pairs is (1-0)*(2-0)*(4-0)*(2-1)*(4-1)*(4-2)=48 which is larger than any other sets with sum 7, so a(7)=48.

Chinese BBS, Find the heap array with the largest product of the differences between the two pairs in 2025

Cf. A002620.

A381760 Number of words of length 2n+1 with one 0 entry and two entries of each of 1..n so that there are exactly k numbers between two equal k's and so that the first element is not 0 and also does not exceed the last.

Original entry on oeis.org

1, 1, 1, 3, 11, 38, 130, 638, 4158, 23384, 124520, 847484, 6987380, 53746000, 400346544, 3529108816, 35963592624, 351432650816, 3346590201888, 36341624453568

Offset: 1

Zhao Hui Du, Mar 06 2025

			For n=4, there are 3 different ways {131423024, 141302432, 240231413} so that a(4)=3 and both 023421314 and 041312432 are invalid since 0 in boundary.

Chinese BBS for the problem

Cf. A014552, A176127, A381759.

a(n) = A381759(n) - A176127(n).

A381759 Number of words of length 2n+1 with one 0 entry and two entries of each of 1..n so that there are exactly k numbers between two equal k's and so that the first element does not exceed the last.

Original entry on oeis.org

1, 1, 3, 5, 11, 38, 182, 938, 4158, 23384, 160104, 1063772, 6987380, 53746000, 479965824, 4182552416, 35963592624, 351432650816, 3860219984448, 41614300175968

Offset: 1

Zhao Hui Du, Mar 06 2025

			a(4) = 5 since there are 5 different solutions: 131423024, 141302432, 240231413, 023421314, 041312432.

Chinese BBS for the problem

Cf. A014552, A176127, A381760.

a(n) = A381760(n) + A176127(n).

A372849 Number of pairs of two disjoint sets of n positive integers more than 1 with product A354697.

Original entry on oeis.org

1, 4, 2, 25, 7, 31, 55, 114, 237, 695, 1666, 2646, 6928, 42986, 79098, 126721, 375348, 667321, 1831927, 7130833, 12067929, 42973699, 105786888, 218943019, 646950177, 1476274502, 3846678717, 14320262729, 46445678648, 91771247330, 182567269925

Offset: 2

Zhao Hui Du, May 14 2024

Zhao Hui Du, The result on a Chinese BBS.

Cf. A354697.

A372794 Number of pairs of two disjoint sets of n positive integers with product A354457(n).

Original entry on oeis.org

1, 1, 3, 4, 31, 15, 55, 110, 280, 797, 1419, 3557, 5647, 19559, 59708, 360726, 346487, 2018032, 2106172, 13228494, 13994982, 65426469, 110980446, 257148638, 660593345, 1966842579, 3909078573, 20820932559, 26864715089, 144689720443, 307476230099, 571509614773

Offset: 2

Zhao Hui Du, May 13 2024

nonn

			For n=2, there exists only one pair of sets <{1,6},{2,3}> with product 1*6=2*3, so a(2)=1.

Zhao Hui Du, C++ Source Code.

Cf. A354457.

A368676 Number of 4 X 4 prime magic squares with magic sum 2n.

Original entry on oeis.org

128, 0, 0, 160, 0, 0, 224, 0, 64, 384, 64, 192, 2112, 96, 224, 4768, 0, 480, 5472, 160, 1088, 6688, 160, 1632, 13600, 416, 1728, 24640, 544, 3008, 40736, 512, 6720, 45504, 672, 11776, 41984, 2752, 17888, 65760, 4416, 18688, 128544, 4544, 21888, 162240, 3712

Offset: 60

Zhao Hui Du, Jan 02 2024

nonn

			[17, 11, 31, 61]
[43, 67,  3,  7]
[41, 37, 13, 29]
[19,  5, 73, 23]
 is a 4 X 4 prime magic square in which the elements in each row and column and both diagonals sum to 120 and all elements are prime numbers. There are a total of 128 such prime magic squares so a(60)=128.

Eric Weisstein's World of Mathematics, Prime Magic Square.

Cf. A006052, A073520, A073521, A271578.

A363799 Numbers whose binary representation has more 1-bits than its cube.

Original entry on oeis.org

407182835067, 445317119867, 478351981947, 814365670134, 873268508637, 890634239734, 956703963894, 956703964539, 1628731340268, 1746537017274, 1781268479468, 1913407927788, 1913407929078, 2774213097787, 3257462680536, 3493074034548, 3562536958936, 3573277243773

Offset: 1

Zhao Hui Du, Jun 23 2023

a(n) must have more 1-bits than a(n)^3 when they are written in binary.

			407182835067 is a term because A000120(407182835067) = 29, while A192085(407182835067) = A000120(407182835067^3) = 28.

Chris K. Caldwell and G. L. Honaker, Jr., 445317119867, Prime Curios!
K. G. Hare, S. Laishram, and T. Stoll, Stolarsky's conjecture and the sum of digits of polynomial values, arXiv:1001.4169 [math.NT], 2010. See p. 3.
Thomas Stoll, On Stolarsky's conjecture: The sum of digits of n and n^h, Slides, 2010.

Cf. A000120, A192085, A138597 (equality).

Cf. A094694 (for squares).

isok(k) = hammingweight(k) > hammingweight(k^3); \\ Michel Marcus, Aug 07 2023

a(9)-a(18) from Martin Ehrenstein, Jul 31 2023

A363287 Numbers which cannot be written as the sum of 4 distinct proper prime powers (A246547).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 50, 51, 52, 57, 59, 62, 63, 66, 67, 68, 73, 75, 80, 90, 95, 107, 134, 135, 136, 140, 145, 151, 152, 256, 2040, 340473

Offset: 1

Zhao Hui Du, May 25 2023

nonn

A proper prime power is an integer which is at least the 2nd power of a prime, such as 4, 8, 9, 16, 25, 27, as in A246547.
It is likely that all numbers above 162 can be written as the sum of 5 distinct proper prime powers.
a(72)=340473, a(73)=3881313, a(74)=4657401 and a(75) >= 10^9, if it exists.

			The smallest integer which can be written as the sum of 4 proper prime powers is 37 = 4+8+9+16 so a(n)=n for n <= 36 and a(37) = 38.

Cf. A246547.

A363240 Number of distinct resistances that can be produced from a circuit that is a 2-connected loopless multigraph with n edges and each edge having a unit resistor.

Original entry on oeis.org

1, 2, 5, 12, 32, 88, 260, 819, 2680, 8642, 27976, 88946, 281541, 893028, 2841344, 9092174, 29176634, 93854841, 302611365

Offset: 2

Zhao Hui Du, May 23 2023

The resistances between any two nodes of the graph are counted.

All resistances in A337517 can be obtained by serial combinations of resistances of one or more 2-connected loopless multigraphs.

			a(2)=1 since the only multigraph with 2 edges is a double edge graph which forms resistance 1/2.
For n=4, there are a quadruple edge graph (resistance 1/4), a triangle graph with one double edge (2/5 between double edge and 3/5 between single edge) and square graph (3/4 between neighbor nodes and 1 between opposite nodes) so a(4)=5.

Cf. A010357, A337517.

cp's OEIS Frontend

User: Zhao Hui Du

A386221 Numbers which can be expressed as the product of a number and its binary reversal in at least three different ways.

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A382328 Maximum possible product of differences of every pair in a set of nonnegative integers with sum n.

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A381760 Number of words of length 2n+1 with one 0 entry and two entries of each of 1..n so that there are exactly k numbers between two equal k's and so that the first element is not 0 and also does not exceed the last.

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A381759 Number of words of length 2n+1 with one 0 entry and two entries of each of 1..n so that there are exactly k numbers between two equal k's and so that the first element does not exceed the last.

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A372849 Number of pairs of two disjoint sets of n positive integers more than 1 with product A354697.

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A372794 Number of pairs of two disjoint sets of n positive integers with product A354457(n).

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A368676 Number of 4 X 4 prime magic squares with magic sum 2n.

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A363799 Numbers whose binary representation has more 1-bits than its cube.

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A363287 Numbers which cannot be written as the sum of 4 distinct proper prime powers (A246547).

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A363240 Number of distinct resistances that can be produced from a circuit that is a 2-connected loopless multigraph with n edges and each edge having a unit resistor.

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