cp's OEIS Frontend

A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, 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.

%I A328633 #16 Oct 27 2019 17:32:14
%S A328633 2,6,18,34,36,48,66,78,96,108,122,126,138,154,156,168,186,198,212,222,
%T A328633 234,244,252,264,282,294,312,324,332,342,354,364,372,384,402,414,422,
%U A328633 426,438,454,456,468,486,498,516,528,542,546,558,574,576,588,606,618,632,642,654,664,672,684,702,714,732,744,752,762,774,784,792,804
%N A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, 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 A328633 Numbers n for which A276087(n) is a multiple of 6, but not of 5.
%C A328633 Question: Is the even bisection of A328316, starting from A328316(4) as: 6, 18, 43218, ..., a subsequence of this sequence? See also A328317.
%C A328633 Subsequence such that both k and A276087(k) are in this sequence starts as: 2, 6, 18, 34, 36, 48, 66, 154, 156, 186, 234, 244, 294, 312, 324, 354, 364, 384, 426, 438, 454, 456, 542, 546, 558, 588, 606, ...
%C A328633 When A276086 is applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, implying that the number's primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), the original number will be converted to a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401", with the least significant zero in position A328578(n), which is seen to be always either 3 or 2.
%H A328633 Antti Karttunen, <a href="/A328633/b328633.txt">Table of n, a(n) for n = 1..10000</a>
%H A328633 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A328633 294 = 7^2 * 3 * 2 has primorial base expansion (A049345) "12400", which, when converted to a prime product form (A276086) yields 11^1 * 7^2 * 5^4 * 3^0 * 2^0 = 336875. This in turn has primorial base representation [11,2,9,1,0,2,1], which when converted to prime product form gives 17^11 * 13^2 * 11^9 * 7^1 * 5^0 * 3^2 * 2^1 = 1720796647657111567992931482, which has the required property of being a multiple of 6 but not of 5, thus 294 is included in this sequence.
%o A328633 (PARI)
%o A328633 A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
%o A328633 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A328633 A328578(n) = A257993(A276086(A276086(n)));
%o A328633 isA328633(n) = (3==A328578(n));
%o A328633 (PARI) isA328633(n) = { my(u=A276086(A276086(n))); ((u%5)&&!(u%6)); };
%Y A328633 Row 3 of A328631.
%Y A328633 Cf. A049345, A257993, A276086, A276087, A328578.
%Y A328633 Cf. also A328316, A328317, A328589, A328762.
%K A328633 nonn
%O A328633 1,1
%A A328633 _Antti Karttunen_, Oct 27 2019