A328588 - OEIS

%I #17 Oct 27 2019 17:31:52

%S 2,4,8,10,14,16,20,22,26,28,32,34,38,40,44,46,50,52,56,58,62,64,68,70,

%T 74,76,80,82,86,88,92,94,98,100,104,106,110,112,116,118,122,124,128,

%U 130,134,136,140,142,146,148,152,154,158,160,164,166,170,172,176,178,182,184,188,190,194,196,200,202,206,208,212,214,218,220,224,226,230,232,236,238,240,242

%N Numbers n for which A257993(A276086(A276086(n))) is larger than A257993(n), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

%C Numbers n for which A328578(n) > A257993(n).

%C A047235 (numbers that are congruent to {2, 4} mod 6, thus even numbers that are not multiples of 3, with A257993(n) = 1) is a subsequence, because in primorial base (A049345) such numbers end with digits "10" or "20". A276086 will convert such a number to a number of the form p_k^e_k * ... * 7^b * 5^a * 3^{1,2} * 2^0 (an odd multiple of three, thus of the form 6k+3) which in primorial base will end with digits "11", thus on the second iteration A276086 will convert that to a number of the form p_k^e_k * ... * 7^b * 5^a * 3^1 * 2^1, with the least missing prime having an index (A257993) at least 3, which is larger than the original 1. Thus all terms of A047235 are included in this sequence.

%H Antti Karttunen, <a href="/A328588/b328588.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%o (PARI)

%o A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A328578(n) = A257993(A276086(A276086(n)));

%o isA328588(n) = (A328578(n) > A257993(n));

%Y Cf. A049345, A047235, A257993, A276086, A328578, A328585, A328586, A328587.

%Y Union of A047235 (terms of the form 6k+2 and 6k+4) and A328589 (gives the terms that are multiples of 6).

%Y Positions of positive terms in A328590.

%Y Differs from A047235 for the first time at n=81, with a(81) = 240, a term not present in A047235.

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 21 2019