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A232527
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Numbers n such that there are no positive integers m such that m^3 is of the form x^2 + n*y^2 and m is not of the form x^2 + n*y^2.
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1
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1, 2, 3, 5, 6, 9, 10, 12, 13, 14, 17, 21, 22, 30, 33, 34, 37, 42, 46, 49, 57, 58, 65, 66, 69, 70, 73, 77, 78, 82, 85, 90, 93, 97, 102, 105, 114, 117, 130, 133, 138, 141, 142, 145, 154, 165, 177, 190, 193, 198, 205, 210, 213, 217, 238, 253, 258, 265, 273, 282, 285, 301, 310, 322, 330, 333, 345, 357, 385, 390
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OFFSET
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1,2
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COMMENTS
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Equivalently, numbers n such that there are no positive integers m such that m^r is of the form x^2 + n*y^2 and m is not of the form x^2 + n*y^2, for any odd integer r >= 3.
Because if m is not of the form x^2 + n*y^2 and m^(r-2) is not of the form x^2 + n*y^2, but m^r = a^2 + n*b^2, then m^(r+2) = (m*a)^2+n*(m*b)^2, m^(r+4) = (m^2*a)^2+n*(m^2*b)^2, etc. are also of the form x^2 + n*y^2. Then m^(3*r-6) = (m^(r-2))^3 = (m^(r-3)*a)^2 + n*(m^(r-3)*b)^2 is of the form x^2 + n*y^2 and m^(r-2) is not of the form x^2 + n*y^2.
All squarefree convenient numbers (A000926) congruent to {1, 2, 3, 5, 6} mod 8 are members of this sequence.
It appears that for a given value of n, if there is an m such that m^3 is of the form x^2 + n*y^2 and m is not of the form x^2 + n*y^2, then there exists some m < n that satisfies this condition.
Is this sequence finite?
Is 66045 the largest term of this sequence?
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LINKS
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EXAMPLE
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n = 11 is not a member of this sequence because for m = 23, 23 is not of form x^2 + 11*y^2, but 23^3 = 12167 = 54^2 + 11*29^2.
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PROG
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(PARI) for(n=1, 100000, flag=0; for(m=1, n, a=0; b=0; for(x=0, ceil(sqrt(m/n)), if(issquare(m-n*x^2), a=1; break)); if(a==0, for(y=0, ceil(sqrt(m^3/n)), if(issquare(m^3-n*y^2), b=1; break))); if(a==0&&b==1, flag=1)); if(flag==0, print1(n", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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