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allocated for Manfred BoergensNumbers appearing exactly once in a Pythagorean triple.
3, 4, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 79, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 142, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 251, 254, 262, 263, 271, 278, 283, 302, 307, 311, 326, 331, 334, 347, 358, 359, 367, 379, 382, 383, 398, 419, 422, 431, 439, 443
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A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 8.
p or 2p with p prime and p = 3 mod 4, with 4 added to the sequence, in ascending order.
t={}; Do[If[(PrimeQ[n] && Mod[n, 4] == 3) || (PrimeQ[n/2] && Mod [n/2, 4] == 3), t = Join[t, {n}]], {n, 445}]; t = Insert[t, 4, 2]
(* Positions of the ones in A046081; based on program by Jean-François Alcover *)
a[1] = 0; a[n_] := Module[{f}, f = Select[FactorInteger[n], Mod[#[[1]], 4] == 1 &][[All, 2]]; (DivisorSigma[0, If[OddQ[n], n, n/2]^2] - 1)/2 + (Times @@ (2*f + 1) - 1)/2]; arr = Array[a, 445]; fl = Flatten[Position[arr, 1]]
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Manfred Boergens, Jul 22 2024
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allocated for Manfred Boergensp or 2p with p prime and p = 3 mod 4, in ascending order.
3, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 79, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 142, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 251, 254, 262, 263, 271, 278, 283, 302, 307, 311, 326, 331, 334, 347, 358, 359, 367, 379, 382, 383, 398, 419, 422, 431, 439, 443
1,1
Numbers appearing exactly once in a Pythagorean triple and as the smallest number in this triple.
Subsequence of A292762.
Inserting 4 as second term gives Axxxxxx.
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 8.
t={}; Do[If[(PrimeQ[n]&&Mod[n, 4] == 3)||(PrimeQ[n/2]&&Mod[n/2, 4] == 3), t=Join[t, {n}]], {n, 470}]; t
(* Positions of the ones in A046081, omitting position = 4; based on program by Jean-François Alcover *)
a[1] = 0; a[n_] := Module[{f}, f = Select[FactorInteger[n], Mod[#[[1]], 4] == 1 &][[All, 2]]; (DivisorSigma[0, If[OddQ[n], n, n/2]^2] - 1)/2 + (Times @@ (2*f + 1) - 1)/2]; arr = Array[a, nmax]; fl = Flatten[Position[arr, 1]]; Delete[fl, 2]
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Manfred Boergens, Jul 22 2024
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A(h,k) is the number of coverings of [h] by tuples (A_1,...,A_k) in P([h])^k with nonempty A_j, with P(.) denoting the power set. For the disjoint case see A019538. For tuples with "nonempty" dropped see A092477 and A329943 (amendment by _Manfred Boergens_, Jun 24 2024). - Manfred Boergens, May 26 2024
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