Lucas A. Brown has authored 17 sequences. Here are the ten most recent ones:
A376552
Square root of the product of all sums and differences of the square roots of the first n primes.
Original entry on oeis.org
1, 24, 215, 44732416, 445902212680990209, 2470738560300573839567485058051752329216, 194775879942444285383551347529278187374780378665463617801353369255538909241232419740031
Offset: 2
The Swinnerton-Dyer polynomial for n=1 is x^2 - 2, which has negative constant term, so we skip n = 1.
For n = 2, the Swinnerton-Dyer polynomial is (x + sqrt(2) + sqrt(3)) * (x + sqrt(2) - sqrt(3)) * (x - sqrt(2) + sqrt(3)) * (x - sqrt(2) - sqrt(3)) = x^4 - 10*x^2 + 1, so a(2) = 1.
For n = 3, the Swinnerton-Dyer polynomial is x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, so a(3) = 24.
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p:= proc(n) option remember; expand(`if`(n=0, x, mul(
subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
end:
a:= n-> isqrt(coeff(p(n), x, 0)):
seq(a(n), n=2..8); # Alois P. Heinz, Nov 28 2024
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p[n_] := p[n] = Expand[If[n == 0, x, Product[p[n - 1] /. x -> x + i*Sqrt[Prime[n]], {i, {1, -1}}]]];
a[n_] := Coefficient[p[n], x, 1 - Sign[n]] // Sqrt // Floor;
Table[a[n], {n, 2, 10}] (* Jean-François Alcover, Jul 02 2025, after Alois P. Heinz *)
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# See LINKS.
A362026
Smallest unhappy number in base A161874(n).
Original entry on oeis.org
3, 7, 3, 5, 20, 3, 12, 3, 3, 14, 3, 3, 3, 3, 3, 3, 23, 3, 23, 3, 261, 6, 12
Offset: 1
The first term in this sequence corresponds to base 16. In base 16, 2 is happy because the sequence it generates is 2 -> 4 -> (1,0) -> 1, while 3 is unhappy because the sequence it generates is 3 -> 9 -> (5,1) -> (1,10) -> (6,5) -> (3,13) -> (11,2) -> (7,13) -> (13,10) -> (1,0,13) -> (10,10) -> (12,8) -> (13,0) -> (10,9) -> (11,5) -> (9,2) -> (5,5) -> (3,2) -> (0,13) -> (10,9) -> ..., which repeats with period 6.
A357808
Semiprimes k such that k is congruent to 4 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 6, 14, 115, 118, 178, 187, 214, 235, 3066899, 3067069, 3067079, 3067149, 3067429, 3067549, 3067594, 3067609, 3067669, 3067719, 3067999, 44690978147, 44690978217, 44690978245, 44690978623, 44690978903, 44690979022, 44690979442
Offset: 1
The 1st semiprime is 4, which is congruent to 4 (modulo 1), so 4 is in the sequence.
The 2nd semiprime is 6, which is congruent to 4 (modulo 2), so 6 is in the sequence.
The 3rd semiprime is 9, which is not congruent to 4 (modulo 3), so 9 is not in the sequence.
The 4th semiprime is 10, which is not congruent to 4 (modulo 4), so 10 is not in the sequence.
The 5th semiprime is 14, which is congruent to 4 (modulo 5), so 14 is in the sequence.
A357807
Semiprimes k such that k is congruent to 3 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 9, 15, 111, 141, 237, 27663, 27667, 3066878, 3066893, 3067023, 3067033, 3067073, 3067193, 3067243, 3067273, 3067283, 3067543, 3067598, 3067613, 3067663, 3067798, 3067843, 3067853, 3067913, 3067933, 3067993, 348933171, 348933219, 348933297
Offset: 1
The 1st semiprime is 4, which is congruent to 3 (mod 1), so 4 is in the sequence.
The 2nd semiprime is 6, which is not congruent to 3 (mod 2), so 6 is not in the sequence.
The 3rd semiprime is 9, which is congruent to 3 (mod 3), so 9 is in the sequence.
The 4th semiprime is 10, which is not congruent to 3 (mod 4), so 10 is not in the sequence.
A357023
Semiprimes k such that k is congruent to 5 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 185, 206, 209, 27681, 3066905, 3067135, 3067795, 3067985, 348933197, 348933239, 348933251, 348933257, 348933269, 44690978141, 44690978162, 44690978519, 44690978561, 44690978617, 44690978869, 44690978981, 44690979457, 44690979527, 6553736049293
Offset: 1
The 1st semiprime is 4, which is congruent to 5 (mod 1), so 4 is in the sequence.
The 2nd semiprime is 6, which is not congruent to 5 (mod 6), so 6 is not in the sequence.
The 60th semiprime is 185, which is congruent to 5 (mod 60), so 185 is in the sequence.
A356764
Semiprimes divisible by their indices in the sequence of semiprimes, divided by those indices.
Original entry on oeis.org
4, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7
Offset: 1
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With[{sp=Select[Range[5*10^6],PrimeOmega[#]==2&]},Select[#[[1]]/#[[2]]&/@Thread[{sp,Range[Length[sp]]}],IntegerQ]] (* The program generates the first 13 terms of the sequence. *) (* Harvey P. Dale, Aug 07 2025 *)
A356357
Semiprimes k such that k is congruent to 7 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 21, 25, 205, 26707, 27679, 3066877, 3067067, 3067097, 3067117, 3067147, 3067177, 3067557, 3067567, 3067577, 3067607, 3067717, 348933193, 348933421, 348933439, 44690978633, 44690978899, 6553736049327, 6553736049407, 6553736049599, 6553736049631, 6553736049823, 6553736053327, 6553736054959
Offset: 1
The 1st semiprime is 4, which is congruent to 7 (mod 1), so 4 is in the sequence.
The 2nd semiprime is 6, which is not congruent to 7 (mod 2), so 6 is not in the sequence.
The 3rd semiprime is 9, which is not congruent to 7 (mod 3), so 9 is not in the sequence.
The 7th semiprime is 21, which is congruent to 7 (mod 7), so 21 is in the sequence.
A356755
Semiprimes k such that k is congruent to 2 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 6, 10, 119, 155, 158, 215, 27682, 3066887, 3066907, 3067027, 3067167, 3067187, 3067247, 3067277, 3067682, 3067687, 3067742, 3067787, 3067847, 3067907, 3067917, 3067937, 3067942, 3068042, 3068067, 348933302, 348933422, 44690978131, 44690978257, 44690978537, 44690978719, 44690978971
Offset: 1
The 1st semiprime is 4, which is congruent to 2 (mod 1), so 4 is in the sequence.
The 2nd semiprime is 6, which is congruent to 2 (mod 2), so 6 is in the sequence.
The 3rd semiprime is 9, which is congruent to 0 (mod 3), so 9 is not in the sequence.
The 4th semiprime is 10, which is congruent to 2 (mod 4), so 10 is in the sequence.
A356135
Semiprimes k such that k is congruent to 6 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 6, 9, 10, 22, 26, 177, 183, 213, 27662, 3066886, 3067021, 3067161, 3067166, 3067186, 3067241, 3067271, 3067421, 3067426, 3067541, 3067561, 3067571, 3067586, 3067661, 3067681, 3067711, 3067741, 3067901, 3067906, 3067991, 3068041, 44690978177, 44690978534, 44690978639, 44690978891
Offset: 1
The 1st semiprime is 4, which is congruent to 6 (mod 1), so 4 is in the sequence.
The 2nd semiprime is 6, which is congruent to 6 (mod 2), so 6 is in the sequence.
The 3rd semiprime is 9, which is congruent to 6 (mod 3), so 9 is in the sequence.
The 4th semiprime is 10, which is congruent to 6 (mod 4), so 10 is in the sequence.
The 5th semiprime is 14, which is not congruent to 6 (mod 5), so 14 is not in the sequence.
A357781
Semiprimes k such that k is congruent to 1 modulo k's index in the sequence of semiprimes.
Original entry on oeis.org
4, 82, 85, 106, 121, 133, 142, 166, 169, 217, 3067001, 3067006, 3067286, 3067411, 3067651, 3067691, 3067721, 3067751, 3067771, 3067781, 3067801, 3068071, 348933121, 348933127, 348933199, 348933223, 348933241, 348933259, 348933271, 348933427, 44690978221, 44690978543, 44690978669
Offset: 1
The 1st semiprime is 4, which is congruent to 1 (mod 1), so 4 is in the sequence.
The 4th semiprime is 10, which is congruent to 2 (mod 4), so 10 is not in the sequence.
The 27th semiprime is 82, which is congruent to 1 (mod 27), so 82 is in the sequence.
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