For 1 ≤ n ≤ 27, i.e. for 29 ≤ k ≤ 107, the terms and their prime factorizations are:
  2*29^2 + 29 = 1711 = 29*59,
  2*30^2 + 29 = 1829 = 31*59,
  2*32^2 + 29 = 2077 = 31*67,
  2*35^2 + 29 = 2479 = 37*67,
  2*39^2 + 29 = 3071 = 37*83,
  2*44^2 + 29 = 3901 = 47*83,
  2*50^2 + 29 = 5029 = 47*107,
  2*57^2 + 29 = 6527 = 61*107,
  2*58^2 + 29 = 6757 = 29*233 (note: 58 = 2*29),
  2*61^2 + 29 = 7471 = 31*241,
  2*63^2 + 29 = 7967 = 31*257,
  2*65^2 + 29 = 8479 = 61*139,
  2*72^2 + 29 = 10397 = 37*281,
  2*74^2 + 29 = 10981 = 79*139,
  2*76^2 + 29 = 11581 = 37*313,
  2*84^2 + 29 = 14141 = 79*179,
  2*87^2 + 29 = 15167 = 29*523 (note: 87 = 3*29),
  2*88^2 + 29 = 15517 = 59*263,
  2*89^2 + 29 = 15871 = 59*269,
  2*91^2 + 29 = 16591 = 47*353,
  2*92^2 + 29 = 16957 = 31*547,
  2*94^2 + 29 = 17701 = 31*571,
  2*95^2 + 29 = 18079 = 101*179,
  2*97^2 + 29 = 18847 = 47*401,
  2*99^2 + 29 = 19631 = 67*293,
  2*102^2 + 29 = 20837 = 67*311,
  2*107^2 + 29 = 22927 = 101*227.
For 56 ≤ n ≤ 67, i.e. for 174 ≤ k ≤ 200, the terms and their prime factorizations are:
  2*174^2 + 29 = 60581 = 29*2089 (note: 174 = 6*29),
  2*175^2 + 29 = 61279 = 233*263,
  2*179^2 + 29 = 64111 = 61*1051,
  2*180^2 + 29 = 64829 = 241*269,
  2*182^2 + 29 = 66277 = 191*347,
  2*183^2 + 29 = 67007 = 37*1811,
  2*185^2 + 29 = 68479 = 31*47*47 (note: first term that is not semiprime),
  2*187^2 + 29 = 69967 = 31*37*61 (note: second term that is not semiprime),
  2*191^2 + 29 = 72991 = 47*1553,
  2*194^2 + 29 = 75301 = 257*293,
  2*196^2 + 29 = 76861 = 101*761,
  2*200^2 + 29 = 80029 = 191*419.
The third, fourth, fifth terms that are not semiprime, and their prime factorizations, are:
  2*232^2 + 29 = 107677 = 29*47*79 (note: 232 = 8*29),
  2*247^2 + 29 = 122047 = 31*31*127,
  2*261^2 + 29 = 136271 = 29*37*127 (note: 261 = 9*29).
List « 68479, 69967, 107677, 122047, 136271, ... » by Amiram Eldar.
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None of the primes 2, 3, 5, 7, 11, 13, 17, 19, 23 (and 41, 43, 53, 71, 73, 89, 97, 103, 109, ...) divides any number of the form 2*k^2 + 29.
First example proofs:
o 2*k^2 + 29 = 2h iff 2(k^2 - h) = -29. Impossible, because -29 is odd.
o 2*k^2 + 29 = 3h iff 2*k^2 + 29 = 3(2m + 1), iff 2*k^2 = 2*3m - 26, iff k^2 = 3m - 13, i.e. k^2 ≡ -13 [3] ≡ 2 [3]. Impossible, because:
  if k ≡ 0 [3] then k^2 ≡ 0 [3];
  if k ≡ 1 [3] or if k ≡ 2 [3] ≡ -1 [3], then k^2 ≡ 1 [3].
o 2*k^2 + 29 = 5h iff 2*k^2 + 29 = 5(2m + 1), iff 2*k^2 = 2*5m - 24, iff k^2 = 5m - 12, i.e. k^2 ≡ -12 [5] ≡ 3 [5]. Impossible, because:
  if k ≡ 0 [5] then k^2 ≡ 0 [5];
  if k ≡ 1 [5] or if k ≡ 4 [5] ≡ -1 [5], then k^2 ≡ 1 [5];
  if k ≡ 2 [5] or if k ≡ 3 [5] ≡ -2 [5], then k^2 ≡ 4 [5].
o 2*k^2 + 29 = 7h iff 2*k^2 + 29 = 7(2m + 1), iff 2*k^2 = 2*7m - 22, iff k^2 = 7m - 11, i.e. k^2 ≡ -11 [7] ≡ 3 [7]. Impossible, because:
  if k ≡ 0 [7] then k^2 ≡ 0 [7];
  if k ≡ 1 [7] or if k ≡ 6 [7] ≡ -1 [7], then k^2 ≡ 1 [7];
  if k ≡ 2 [7] or if k ≡ 5 [7] ≡ -2 [7], then k^2 ≡ 4 [7];
  if k ≡ 3 [7] or if k ≡ 4 [7] ≡ -3 [7], then k^2 ≡ 9 [7] ≡ 2 [7].
o 2*k^2 + 29 = 11h iff 2*k^2 + 29 = 11(2m + 1), iff 2*k^2 = 2*11m - 18, iff k^2 = 11m - 9, i.e. k^2 ≡ -9 [11] ≡ 2 [11]. Impossible, because:
  if k ≡ 0 [11] then k^2 ≡ 0 [11];
  if k ≡ 1 [11] or if k ≡ 10 [11] ≡ -1 [11], then k^2 ≡ 1 [11];
  if k ≡ 2 [11] or if k ≡ 9 [11] ≡ -2 [11], then k^2 ≡ 4 [11];
  if k ≡ 3 [11] or if k ≡ 8 [11] ≡ -3 [11], then k^2 ≡ 9 [11];
  if k ≡ 4 [11] or if k ≡ 7 [11] ≡ -4 [11], then k^2 ≡ 16 [11] ≡ 5 [11];
  if k ≡ 5 [11] or if k ≡ 6 [11] ≡ -5 [11], then k^2 ≡ 25 [11] ≡ 3 [11].
o 2*k^2 + 29 = 13h iff 2*k^2 + 29 = 13(2m + 1), iff 2*k^2 = 2*13m - 16, iff k^2 = 13m - 8, i.e. k^2 ≡ -8 [13] ≡ 5 [13]. Impossible, because:
  if k ≡ 0 [13] then k^2 ≡ 0 [13];
  if k ≡ 1 [13] or if k ≡ 12 [13] ≡ -1 [13], then k^2 ≡ 1 [13];
  if k ≡ 2 [13] or if k ≡ 11 [13] ≡ -2 [13], then k^2 ≡ 4 [13];
  if k ≡ 3 [13] or if k ≡ 10 [13] ≡ -3 [13], then k^2 ≡ 9 [13];
  if k ≡ 4 [13] or if k ≡ 9 [13] ≡ -4 [13], then k^2 ≡ 16 [13] ≡ 3 [13];
  if k ≡ 5 [13] or if k ≡ 8 [13] ≡ -5 [13], then k^2 ≡ 25 [13] ≡ 12 [13];
  if k ≡ 6 [13] or if k ≡ 7 [13] ≡ -6 [13], then k^2 ≡ 36 [13] ≡ 10 [13].
o 2*k^2 + 29 = 17h iff 2*k^2 + 29 = 17(2m + 1), iff 2*k^2 = 2*17m - 12, iff k^2 = 17m - 6, i.e. k^2 ≡ -6 [17] ≡ 11 [17]. Impossible, because:
  if k ≡ 0 [17] then k^2 ≡ 0 [17];
  if k ≡ 1 [17] or if k ≡ 16 [17] ≡ -1 [17], then k^2 ≡ 1 [17];
  if k ≡ 2 [17] or if k ≡ 15 [17] ≡ -2 [17], then k^2 ≡ 4 [17];
  if k ≡ 3 [17] or if k ≡ 14 [17] ≡ -3 [17], then k^2 ≡ 9 [17];
  if k ≡ 4 [17] or if k ≡ 13 [17] ≡ -4 [17], then k^2 ≡ 16 [17];
  if k ≡ 5 [17] or if k ≡ 12 [17] ≡ -5 [17], then k^2 ≡ 25 [17] ≡ 8 [17];
  if k ≡ 6 [17] or if k ≡ 11 [17] ≡ -6 [17], then k^2 ≡ 36 [17] ≡ 2 [17];
  if k ≡ 7 [17] or if k ≡ 10 [17] ≡ -7 [17], then k^2 ≡ 49 [17] ≡ 15 [17];
  if k ≡ 8 [17] or if k ≡ 9 [17] ≡ -8 [17], then k^2 ≡ 64 [17] ≡ 13 [17].
o 2*k^2 + 29 = 19h iff 2*k^2 + 29 = 19(2m + 1), iff 2*k^2 = 2*19m - 10, iff k^2 = 19m - 5, i.e. k^2 ≡ -5 [19] ≡ 14 [19]. Impossible, because:
  if k ≡ 0 [19] then k^2 ≡ 0 [19];
  if k ≡ 1 [19] or if k ≡ 18 [19] ≡ -1 [19], then k^2 ≡ 1 [19];
  if k ≡ 2 [19] or if k ≡ 17 [19] ≡ -2 [19], then k^2 ≡ 4 [19];
  if k ≡ 3 [19] or if k ≡ 16 [19] ≡ -3 [19], then k^2 ≡ 9 [19];
  if k ≡ 4 [19] or if k ≡ 15 [19] ≡ -4 [19], then k^2 ≡ 16 [19];
  if k ≡ 5 [19] or if k ≡ 14 [19] ≡ -5 [19], then k^2 ≡ 25 [19] ≡ 6 [19];
  if k ≡ 6 [19] or if k ≡ 13 [19] ≡ -6 [19], then k^2 ≡ 36 [19] ≡ 17 [19];
  if k ≡ 7 [19] or if k ≡ 12 [19] ≡ -7 [19], then k^2 ≡ 49 [19] ≡ 11 [19];
  if k ≡ 8 [19] or if k ≡ 11 [19] ≡ -8 [19], then k^2 ≡ 64 [19] ≡ 7 [19];
  if k ≡ 9 [19] or if k ≡ 10 [19] ≡ -9 [19], then k^2 ≡ 81 [19] ≡ 5 [19].
o 2*k^2 + 29 = 23h iff 2*k^2 + 29 = 23(2m + 1), iff 2*k^2 = 2*23m - 6, iff k^2 = 23m - 3, i.e. k^2 ≡ -3 [23] ≡ 20 [23]. Impossible, because:
  if k ≡ 0 [23] then k^2 ≡ 0 [23];
  if k ≡ 1 [23] or if k ≡ 22 [23] ≡ -1 [23], then k^2 ≡ 1 [23];
  if k ≡ 2 [23] or if k ≡ 21 [23] ≡ -2 [23], then k^2 ≡ 4 [23];
  if k ≡ 3 [23] or if k ≡ 20 [23] ≡ -3 [23], then k^2 ≡ 9 [23];
  if k ≡ 4 [23] or if k ≡ 19 [23] ≡ -4 [23], then k^2 ≡ 16 [23];
  if k ≡ 5 [23] or if k ≡ 18 [23] ≡ -5 [23], then k^2 ≡ 25 [23] ≡ 2 [23];
  if k ≡ 6 [23] or if k ≡ 17 [23] ≡ -6 [23], then k^2 ≡ 36 [23] ≡ 13 [23];
  if k ≡ 7 [23] or if k ≡ 16 [23] ≡ -7 [23], then k^2 ≡ 49 [23] ≡ 3 [23];
  if k ≡ 8 [23] or if k ≡ 15 [23] ≡ -8 [23], then k^2 ≡ 64 [23] ≡ 18 [23];
  if k ≡ 9 [23] or if k ≡ 14 [23] ≡ -9 [23], then k^2 ≡ 81 [23] ≡ 12 [23];
  if k ≡ 10 [23] or if k ≡ 13 [23] ≡ -10 [23], then k^2 ≡ 100 [23] ≡ 8 [23];
  if k ≡ 11 [23] or if k ≡ 12 [23] ≡ -11 [23], then k^2 ≡ 121 [23] ≡ 6 [23].
o 2*k^2 + 29 = 41h iff 2*k^2 + 29 = 41(2m + 1), iff 2*k^2 = 2*41m + 12, iff k^2 = 41m + 6, i.e. k^2 ≡ 6 [41]. Impossible, because: ...
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Each of the primes 29, 31, 37, 47, 59, 61, 67, 79, 83, 101, 107, ... divides certain numbers of the form 2*k^2 + 29.
First example proofs:
o 2*k^2 + 29 = 29h iff 2*k^2 + 29 = 29(2m + 1), iff 2*k^2 = 2*29m, iff k^2 = 29m, i.e. k^2 ≡ 0 [29], iff k ≡ 0 [29], because 29 is prime.
(So, all the prime factors of the numbers of the form 2*k^2 + 29 are ≥ 29.)
o 2*k^2 + 29 = 31h iff 2*k^2 + 29 = 31(2m + 1), iff 2*k^2 = 2*31m + 2, iff k^2 = 31m + 1, i.e. k^2 ≡ 1 [31], iff k ≡ 1 [31] or k ≡ -1 [31] ≡ 30 [31], because 31 is prime.
o 2*k^2 + 29 = 37h iff 2*k^2 + 29 = 37(2m + 1), iff 2*k^2 = 2*37m + 8, iff k^2 = 37m + 4, i.e. k^2 ≡ 4 [37], iff k ≡ 2 [37] or k ≡ -2 [37] ≡ 35 [37], because 37 is prime.
o 2*k^2 + 29 = 47h iff 2*k^2 + 29 = 47(2m + 1), iff 2*k^2 = 2*47m + 18, iff k^2 = 47m + 9, i.e. k^2 ≡ 9 [47], iff k ≡ 3 [47] or k ≡ -3 [47] ≡ 44 [47], because 47 is prime.
o 2*k^2 + 29 = 59h iff 2*k^2 + 29 = 59(2m + 1), iff 2*k^2 = 2*59m + 30, iff k^2 = 59m + 15, i.e. k^2 ≡ 15 [59], iff k ≡ 29 [59] or k ≡ -29 [59] ≡ 30 [59], because 59 is prime.
o 2*k^2 + 29 = 61h iff 2*k^2 + 29 = 61(2m + 1), iff 2*k^2 = 2*61m + 32, iff k^2 = 61m + 16, i.e. k^2 ≡ 16 [61], iff k ≡ 4 [61] or k ≡ -4 [61] ≡ 57 [61], because 61 is prime.
o 2*k^2 + 29 = 67h iff 2*k^2 + 29 = 67(2m + 1), iff 2*k^2 = 2*67m + 38, iff k^2 = 67m + 19, i.e. k^2 ≡ 19 [67], iff k ≡ 32 [67] or k ≡ -32 [67] ≡ 35 [67], because 67 is prime.
o 2*k^2 + 29 = 79h iff 2*k^2 + 29 = 79(2m + 1), iff 2*k^2 = 2*79m + 50, iff k^2 = 79m + 25, i.e. k^2 ≡ 25 [79], iff k ≡ 5 [79] or k ≡ -5 [79] ≡ 74 [79], because 79 is prime.
o 2*k^2 + 29 = 83h iff 2*k^2 + 29 = 83(2m + 1), iff 2*k^2 = 2*83m + 54, iff k^2 = 83m + 27, i.e. k^2 ≡ 27 [83], iff k ≡ 39 [83] or k ≡ -39 [83] ≡ 44 [83], because 83 is prime.
o 2*k^2 + 29 = 101h iff 2*k^2 + 29 = 101(2m + 1), iff 2*k^2 = 2*101m + 72, iff k^2 = 101m + 36, i.e. k^2 ≡ 36 [101], iff k ≡ 6 [101] or k ≡ -6 [101] ≡ 95 [101], because 101 is prime.
o 2*k^2 + 29 = 107h iff 2*k^2 + 29 = 107(2m + 1), iff 2*k^2 = 2*107m + 78, iff k^2 = 107m + 39, i.e. k^2 ≡ 39 [107], iff k ≡ 50 [107] or k ≡ -50 [107] ≡ 57 [107], because 107 is prime.
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Lists « 2, 3, 5, 7, 11, 13, 17, 19, 23 (and 41, 43, 53, 71, 73, 89, 97, 103, 109, ...) » and « 29, 31, 37, 47, 59, 61, 67, 79, 83, 101, 107, ... » by Jon E. Schoenfield, details by Rémi Guillaume.