[seqfan] Pattern in A236019

[jnthn stdhr]

Hi all,

  Terms in A236019 (Smallest number having at least n partitions that
contain at least n primes.) come in "blocks" of integers of equal parity,
the blocks are always seperated by exactly two consecutive integers, and
the block-to-block parity always alternates:

{2},_,_,{5},_,_,{8,10},_,_,{13,15,17},_,_,{20,22,24,26,28},_,_,{31,33,35,37,39,41,43},...

  This behavior arises via a connection to https://oeis.org/A299731 (Number
of partitions of 3*n that have exactly n prime parts.).

  One way to demonstrate what's going on is to take the difference between
the actual number of prime parts in the partitions of n and the current
minimum for which we are testing.  E.g. n=0:  we are testing for at least
one partition with at least one prime, so we get 0 - 1 = -1;  n=1:  again,
0 - 1 = -1;  n=2:  1 - 1 = 0, so now the minimum is incremented and becomes
two, and we want at least two partitions with at least two primes each.;
n=3:  0 - 2 = -2;  and so on.  The resulting sequence starts:

{-1, -1, 0, -2, -1, 0, -2, -1, 1, -2, 0, -3, -1, 2, -2, 1, -3, 0, -4, -1,
4, -2, 3, -3, 2, -4, 1, -5, 0, -6, -1, 6, -2, 5, -3, 4, -4, 3, -5, 2, -6,
1, -7, 0, -8, -1, 10, -2,...}

where n is in A236019 iff a(n) >= 0.  The closest mathing signed sequences
are Jacobs's ladder and Ludic's ladder.

  The consecutive pairs not in A236019 occur iff a(n) and a(n-1) are both
less than zero.  The sequence is:

{0,1 ,3,4, 6,7, 11,12, 18,19,...}

  The size of the parity blocks in A236019 =
{1,1,2,3,5,7,11,14,21,29,40,...}

  If any of these are of interest, I would need help naming them.  If these
are of interest to anyone in particular, we can go over the specifics
off-list.

-Jonathan