A274226 -id:A274226 - cp's OEIS frontend

A274227 Primes in A274226.

29, 53, 61, 109, 157, 277, 397

Offset: 1

Andreas Boe, Jun 14 2016

If a(8) exists it must be larger than 10^8.

From a proof outline of Wagner, the discriminants of Q(sqrt(-p)) with class number 6 end at -1588, ending this sequence at 397. - Travis Scott, Feb 09 2023

			29 is a term because 2^2 + 3^2 + 4^2 = 29 is the only representation of 29 as a sum of 3 positive squares, and those squares are distinct.
41 is not a term because, even though it can be represented in just one way as a sum of 3 distinct squares (1^2 + 2^2 + 6^2) it can also be represented as 3^2 + 4^2 + 4^2.

Andreas Boe, List of values with values of x, y and z attached
C. Wagner, Class number 5, 6 and 7, Math. Comput. 64 (1996), pp. 785-800.

Cf. A274226, A025339.

rp[n_] := Flatten@ IntegerPartitions[n, {3}, Range[Sqrt@n]^2]; Select[
Range[265] // Prime, Length[u = rp[#]] == 3 && Union[u] == Sort[u] &] (* Giovanni Resta, Jun 16 2016 *)
Select[Prime@Range@78,Sum[(-1)^Boole@Xor[Mod[t,4]==1,PowerMod[t,(#-1)/2,#]==1],{t,1,#-1,2}]==6&] (* Travis Scott, Feb 09 2023 *)

A025339 Numbers that are the sum of 3 distinct nonzero squares in exactly one way.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 56, 59, 61, 65, 66, 70, 75, 78, 81, 83, 84, 91, 93, 104, 106, 107, 109, 113, 114, 115, 116, 118, 120, 121, 133, 137, 139, 140, 142, 145, 147, 152, 153, 157, 162, 164, 168, 169, 171, 178, 180, 184, 190, 196, 198, 200

Offset: 1

David W. Wilson

nonn

Numbers n such that there is a unique triple (i,j,k) with 0 < i < j < k and n = i^2 + j^2 + k^2.

By removing the terms that have a factor of 4 we obtain A096017. - T. D. Noe, Jun 15 2004

			14 is a term since 14 = 1^2+2^2+3^2.
38 is a term since 38 = 2^2+3^2+5^2 (note that 38 is also 1^2+1^2+6^2, but that is not a contradiction since here i=j).

Robert Israel, Table of n, a(n) for n = 1..1019
Index entries for sequences related to sums of squares

A subsequence of A004432.
Cf. A001974, A024803, A096017.
A274226 has a somewhat similar definition but is actually a different sequence.

N:= 10^4; # to get all terms <= N
S:= Vector(N):
for a from 1 to floor(sqrt(N/3)) do
  for b from a+1 to floor(sqrt((N-a^2)/2)) do
    c:= [$(b+1) .. floor(sqrt(N-a^2-b^2))]:
    v:= map(t -> a^2 + b^2 + t^2, c):
    S[v]:= map(`+`,S[v],1)
od od:
select(t -> S[t]=1, [$1..N]); # Robert Israel, Jan 03 2016

Select[Range[200], (pr = PowersRepresentations[#, 3, 2]; Length[Select[pr, Union[#] == # && #[[1]] > 0&]] == 1)&] (* Jean-François Alcover, Feb 27 2019 *)

cp's OEIS Frontend

A274227 Primes in A274226.

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