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nxt[{a_, b_, c_}] := {b, c, NextPrime[a + b + c] - (a+b + c)}; NestList[nxt, {0, 0, 1}, 100][[All, 1]] (* Harvey P. Dale, Sep 20 2022 *)

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Cf. A000040, A073737, A083242.

The sequence depends on initial three terms. Assuming no negative sequence we have ten distinct sets of first terms. We may denote them in short as s000,s001(=this sequence),s010,s011,s100,s101,s110,s002,s020 and s200. These sequences do not merge into each other, but maintain their individuality. E.g. terms nos. 97-100 are: {128,107,162,144},{127,107,162,145},{127,107,163,144},{126,107,163,145},{127,108,162,144},{126,108,162,145},{126,108,163,144},{126,107,162,146},{126,107,164,144},{126,109,162,144}, for above mentioned sequences, respectively. The same is true for the case of "sum of three successive terms" A073737, where we have six distinct sets of first terms s00,s01,s10,s11(=A073737),s02 and s20.

a(n) = A000040(n-3) - a(n-1) - a(n-2) - a(n-3). - Jason Yuen, Sep 01 2024

Cf. A073737, A083242.

nonn,easy,changed

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The sequence depends on initial three terms. Assuming no negative sequence we have ten distinct sets of first terms. We may denote them in short as s000,s001(=this sequence),s010,s011,s100,s101,s110,s002,s020 and s200. These sequences do not merge into each other, but maintain their individuality. E.g. terms nos. 97-100 are: {128,107,162,144},{127,107,162,145},{127,107,163,144}, {126,107,163,145},{127,108,162,144},{126,108,162,145},{126,108,163,144},{126,107,162,146},{126,107,164,144},{126,109,162,144}, for above mentioned sequences, respectively. The same is true for the case of "sum of three successive terms" A073737, where we have six distinct sets of first terms s00,s01,s10,s11(=A073737),s02 and s20.

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