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%I #20 Apr 28 2020 14:59:26
%S 4,9,16,27,35,49,63,65,85,95,105,121,135,145,169,175,187,203,207,221,
%T 253,265,273,289,301,305,319,351,369,387,403,407,425,445,473,485,495,
%U 517,529,545,551,567,611,615,629,637,671,679,693,697,725,747,781,793,799
%N a(n) is the greatest number k having for every prime <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0.
%C Alternatively a(n) is the greatest number whose product of distinct least part primes from all prime partitions of n, is equal to primorial(n). Companion sequence to A330507.
%C From _Michael De Vlieger_, Mar 20 2020: (Start)
%C a(n) = 0 for n = {90, 151, 349, 352, 444, ...}, cf. the comment from _Alois P. Heinz_ at A330507.
%C Index m of last instance of A002110(n) in A333129 as m increases.
%C Last row n in A333238 that contains the consecutive primes (1...n).
%C Last index of the occurrence of 2^n - 1 in A333259, which is the decimal value of the characteristic function of primes in A333238 interpreted as a binary number. (End)
%e a(1) = 4 because [2,2] is the only prime partition of 4, and no greater number n has only 2 as least part in any partition of n into primes.
%e From _Michael De Vlieger_, Mar 20 2020: (Start)
%e Looking at this sequence as the first position of 2^n - 1 in A333259, which in binary is a k-bit repunit, we look for the last occasion of such in A333259, indicated by the arrows. a(k) = n for rows n that have an arrow. In the chart, we reverse the portrayal of the binary rendition of A333259(n), replacing zeros with "." for clarity:
%e n A333259(n) k
%e ------------------------------
%e 2 1 1
%e 3 . 1
%e 4 1 -> 1
%e 5 1 . 1
%e 6 1 1 2
%e 7 1 . . 1
%e 8 1 1 2
%e 9 1 1 -> 2
%e 10 1 1 1 3
%e 11 1 1 . . 1
%e 12 1 1 1 3
%e 13 1 1 . . . 1
%e 14 1 1 . 1
%e 15 1 1 1 3
%e 16 1 1 1 -> 3
%e 17 1 1 1 . . . 1
%e 18 1 1 1 1 4
%e 19 1 1 1 . . . . 1
%e 20 1 1 1 1 4
%e ... (End)
%t With[{s = TakeWhile[Import["https://oeis.org/A333259/b333259.txt", "Data"], Length@ # > 0 &][[All, -1]]}, Array[If[Length[#] == 0, 0, #[[-1, 1]] - 1] &@ Position[s, 2^# - 1] &, 55]] (* _Michael De Vlieger_, Mar 20 2020, using the b-file at A333259 *)
%Y Cf. A000040, A002110, A051034, A331634, A332861, A330507, A333129, A333238, A333259, A333365.
%K nonn
%O 1,1
%A _David James Sycamore_, Mar 20 2020
%E More terms from _Michael De Vlieger_, Mar 20 2020
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