A333417 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.

4, 9, 16, 27, 35, 49, 63, 65, 85, 95, 105, 121, 135, 145, 169, 175, 187, 203, 207, 221, 253, 265, 273, 289, 301, 305, 319, 351, 369, 387, 403, 407, 425, 445, 473, 485, 495, 517, 529, 545, 551, 567, 611, 615, 629, 637, 671, 679, 693, 697, 725, 747, 781, 793, 799

Offset: 1

Author

David James Sycamore, Mar 20 2020

Keywords

• nonn

Comments

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.

From Michael De Vlieger, Mar 20 2020: (Start)

a(n) = 0 for n = {90, 151, 349, 352, 444, ...}, cf. the comment from Alois P. Heinz at A330507.

Index m of last instance of A002110(n) in A333129 as m increases.

Last row n in A333238 that contains the consecutive primes (1...n).

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)

Examples

			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.
From _Michael De Vlieger_, Mar 20 2020: (Start)
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:
   n   A333259(n)            k
------------------------------
   2   1                     1
   3   . 1
   4   1                  -> 1
   5   1 . 1
   6   1 1                   2
   7   1 . . 1
   8   1 1                   2
   9   1 1                -> 2
  10   1 1 1                 3
  11   1 1 . . 1
  12   1 1 1                 3
  13   1 1 . . . 1
  14   1 1 . 1
  15   1 1 1                 3
  16   1 1 1              -> 3
  17   1 1 1 . . . 1
  18   1 1 1 1               4
  19   1 1 1 . . . . 1
  20   1 1 1 1               4
  ... (End)
		

Crossrefs

Cf. A000040, A002110, A051034, A331634, A332861, A330507, A333129, A333238, A333259, A333365.

Programs

Mathematica

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 *)

Extensions

More terms from Michael De Vlieger, Mar 20 2020