This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363593 #17 Jun 22 2023 06:01:26
%S A363593 3,8,22,29,36,42,45,53,57,64,82,85,88,94,110,119,124,132,135,141,144,
%T A363593 152,159,165,170,177,183,190,195,201,214,220,224,231,239,246,252,264,
%U A363593 270,281,287,292,299,302,306,309,323,328,334,341,347,350,356,361,372,378,381,386,397,402,411,418,424,431
%N A363593 Numbers k such that both A359804(k) and A359804(k+1) are odd.
%C A363593 Odd numbers may occur no more than twice in a row in A359804 as consequence of definition of that sequence.
%C A363593 Let b(n) = A359804(n). Let D(n) = b(a(n)..a(n)+1).
%C A363593 Since the product of 2 odd numbers b(n-2) and b(n-1) is odd, and since b(n) = mp, where p = A053669(b(n-2)*b(n-1)) = 2, D(n) implies b(a(n)+2) = 2m.
%C A363593 b(a(n)+2) = 2k and b(a(n+j)+2) = 2m, j >= 1 imply m > k as consequence of definition of A359804.
%C A363593 Perfect powers 2^k = b(j) occur such that j = a(n)+2 for some n. Therefore, A361505 is a subset of { a(n) + 2 }. Generally, perfect powers p^e in A246547 follow b(n-2) and b(n-1) such that b(n-2)*b(n-1) mod p != 0.
%C A363593 Conjecture: for prime q > 11, even squarefree semiprimes 2q follow D(n) for some n. Consider that primes in A359804 appear late for q > 11, yet pairs of successive odd numbers in that sequence occur rather often.
%C A363593 Conjectured to be an infinite sequence, meaning that consecutive odd terms appear infinitely many times in A359804. - _David James Sycamore_, Jun 21 2023
%H A363593 Michael De Vlieger, <a href="/A363593/b363593.txt">Table of n, a(n) for n = 1..10000</a>
%H A363593 Michael De Vlieger, <a href="/A363593/a363593.png">Plot A359804(n) mod 2 at (x,y) = (n mod 256, -floor(n/256))</a>, 8X magnification, where white represents even terms, and odd terms are shown in color. Singleton odd numbers are shown in dark blue, while red indicates two odd terms in a row. Shows A359804(n) mod 2 for n = 1..2^16.
%H A363593 Michael De Vlieger, <a href="/A363593/a363593_1.png">Plot A359804(n) mod 2 at (x,y) = (n mod 2^10, -floor(n/2^10))</a>, where white represents even terms, and odd terms are shown in color. Singleton odd numbers are shown in dark blue, while red indicates two odd terms in a row. Shows A359804(n) mod 2 for n = 1..2^20.
%F A363593 A361503(a(n)+1) = 2, consequence of definition of A359804.
%e A363593 Table of a(n) showing i = b(n) = p(i)*m(i), j = b(n+1) = p(j)*m(j), and k = b(n+2), where p(n) = A361503(n) and m(n) = A359804(n)/A361503(n):
%e A363593 n a(n) i j k p(i) p(j) m(i) m(j)
%e A363593 ----------------------------------------------
%e A363593 1 3 3 5 4 3 5 1 1
%e A363593 2 8 7 9 8 7 3 1 3
%e A363593 3 22 33 35 16 11 7 3 5
%e A363593 4 29 45 49 26 5 7 9 7
%e A363593 5 36 55 63 32 5 7 11 9
%e A363593 6 42 13 65 34 13 5 1 13
%e A363593 7 45 39 75 38 3 5 13 15
%e A363593 8 53 85 51 46 5 3 17 17
%e A363593 9 57 91 99 52 7 11 13 9
%e A363593 10 64 57 105 58 3 7 19 15
%e A363593 11 82 143 81 62 11 3 13 27
%e A363593 12 85 135 147 64 5 7 27 21
%e A363593 ...
%t A363593 nn = 432; c[_] = False; q[_] = 1;
%t A363593 Set[{i, j}, {1, 2}]; c[1] = c[2] = True; q[2] = 2; u = 3;
%t A363593 Reap[Do[
%t A363593 (k = q[#]; While[c[k #], k++]; k *= #;
%t A363593 While[c[# q[#]], q[#]++]) &[(p = 2;
%t A363593 While[Divisible[i j, p], p = NextPrime[p]]; p)];
%t A363593 If[OddQ[j k], Sow[n - 1]];
%t A363593 Set[{c[k], i, j}, {True, j, k}];
%t A363593 If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]]
%Y A363593 Cf. A000079, A053669, A100484, A359804, A361503, A361505, A363594.
%K A363593 nonn
%O A363593 1,1
%A A363593 _Michael De Vlieger_ and _David James Sycamore_, Jun 12 2023