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|*b*^*n*+2|*x*{0}*y*|1|2|3|1|https://oeis.org/A138066<br>https://oeis.org/A084713 (corresponding primes)<br>https://oeis.org/A138067 (*n* = 1 not allowed)|https://oeis.org/A087576<br>https://oeis.org/A095302 (corresponding primes)<br>https://oeis.org/A087575 (only allow prime *b*)|(none)||167|100000|*b* == 0 mod 2: always divisible by 2<br>*b* == 1 mod 3: always divisible by 3<br>*b* = 2^*r* such that the equation 2^*x* == −1 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((2^*r*)^*n*+2 = ×ばつ(2^{*n×ばつ*r*−1}+1), and if 2^{*n×ばつ*r*−1}+1 has no algebraic factorization, then *n×ばつ*r*−1 must be a power of 2 (otherwise, if *n×ばつ*r*−1 has an odd prime factor *p*, then 2^{*n×ばつ*r*−1}+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −1 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord*_*r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord*_*r*(*p*) is even, e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 32768 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 2097152 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 8388608 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 549755813888 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 35184372088832 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 140737488355328 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|47 (114)<br>89 (256)<br>159 (137)|