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19, 35, 38, 41, 45, 50, 53, 56, 57, 58, 59, 63, 76, 77, 78, 79, 80, 81, 83, 84, 85, 92, 93, 95, 96, 108, 109, 112, 113, 116, 117, 124, 125, 126, 142, 143, 146, 154, 157, 173, 184, 185, 186, 193, 194, 195, 196, 197, 203, 215, 217, 224, 227, 232, 233, 237, 241
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OFFSET
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1,1
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COMMENTS
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The sequence is infinite, in view of a strong closeness between counting functions of numbers N_1 for which lpf(N_1-3) > lpf(N_1-1) >= prime(n) and numbers N_2 for which lpf(N_2-1) > lpf(N_2-3) >= prime(n), if {N_2-3, N_2-1} is not a pair of twin primes, where p_n=prime(n) and lpf=least prime factor (A020639). (Cf., for example, A243803-A243804). This closeness is explained by a somewhat symmetry (for details, see Shevelev's link).
However, it is very interesting to find an analytical proof of infinity of this and complementory sequences.
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LINKS
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MATHEMATICA
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lpf[k_] := FactorInteger[k][[1, 1]];
a19[n_ /; n>1] := a19[n] = For[k = If[n == 2, 10, a19[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
a20[n_ /; n>1] := a20[n] = For[k = If[n <= 2, 2, a20[n-1]], True, k = k+2, If[Not[PrimeQ[k-3] && PrimeQ[k-1]] && lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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