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A329589
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Odd integers which are not a positive power of a single prime and have at least one prime divisor 1 (mod 4).
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3
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15, 35, 39, 45, 51, 55, 65, 75, 85, 87, 91, 95, 105, 111, 115, 117, 119, 123, 135, 143, 145, 153, 155, 159, 165, 175, 183, 185, 187, 195, 203, 205, 215, 219, 221, 225, 235, 245, 247, 255, 259, 261, 265, 267, 273, 275, 285, 287, 291, 295, 299, 303, 305, 315, 319, 323, 325, 327, 333, 335, 339, 345
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OFFSET
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1,1
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COMMENTS
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This sequence is a proper subsequence of A257591 where also odd numbers, not a prime power, and 1 (mod 4) divisors involving only primes congruent to 3 modulo 4 are included, like 63, 99, 147, 171, 189, ... .
This sequence gives all odd moduli m that have solutions of the complex congruence z^2 = +1 (mod m), with z = a + b*i, where a, b are positive integers (nonvanishing a*b case). For a proof one can use the formula for the number of solutions of this congruence for a*b vanishing, given in A329586 without powers of 2 (e2 = 0) and subtract it from the formula for the number of all representative solutions with modulus m >= 1 which is S(m) = 1 if m = 1, and S(m) = 2^(2*r1(m) + r3(m)), with r1(m) and r3(m) the number of distinct primes 1 (mod 4) (A002144) and 3 (mod 4) (A002145), respectively. This becomes the number of representative solutions 2^(r1(m) + r3(m))*(2^(r1(m)) - 1) - delta(r3(m), 0)*2^(r1(m))), with the Kronecker symbol. This shows that for odd modulus m >= 3 and nonvanishing a*b there is no solution if r1(m) = 0 and r3 >= 1. Moduli which are powers of a single prime have only solutions with a or b vanishing.
See A329587 for all moduli m with solutions of z^2 = +1 (mod m), with z = a + b*i and nonvanishing a*b, where all even numbers >= 4 appear.
For the representative solutions of this congruence with a*b = 0 see A329585 for all positive moduli m.
For the representative solutions of this congruence for all m >= 1 see A227091.
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LINKS
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FORMULA
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See the name.
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MATHEMATICA
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Select[Range[3, 235, 2], And[! PrimePowerQ@ #, AnyTrue[FactorInteger[#][[All, 1]], Mod[#, 4] == 1 &]] &] (* Michael De Vlieger, Dec 14 2019 *)
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PROG
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(PARI) isok(k) = if ((k%2) && !isprimepower(k), my(f=factor(k)); sum(i=1, #f~, (f[i, 1] % 4) == 1) >= 1); \\ Michel Marcus, Sep 18 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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