A363593 Numbers k such that both A359804(k) and A359804(k+1) are odd.

3, 8, 22, 29, 36, 42, 45, 53, 57, 64, 82, 85, 88, 94, 110, 119, 124, 132, 135, 141, 144, 152, 159, 165, 170, 177, 183, 190, 195, 201, 214, 220, 224, 231, 239, 246, 252, 264, 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

Offset: 1

Author

Michael De Vlieger and David James Sycamore, Jun 12 2023

Keywords

• nonn

Comments

Odd numbers may occur no more than twice in a row in A359804 as consequence of definition of that sequence.

Let b(n) = A359804(n). Let D(n) = b(a(n)..a(n)+1).

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.

b(a(n)+2) = 2k and b(a(n+j)+2) = 2m, j >= 1 imply m > k as consequence of definition of A359804.

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.

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.

Conjectured to be an infinite sequence, meaning that consecutive odd terms appear infinitely many times in A359804. - David James Sycamore, Jun 21 2023

Examples

			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):
   n  a(n)    i     j    k   p(i) p(j) m(i) m(j)
  ----------------------------------------------
   1    3     3     5    4     3    5    1    1
   2    8     7     9    8     7    3    1    3
   3   22    33    35   16    11    7    3    5
   4   29    45    49   26     5    7    9    7
   5   36    55    63   32     5    7   11    9
   6   42    13    65   34    13    5    1   13
   7   45    39    75   38     3    5   13   15
   8   53    85    51   46     5    3   17   17
   9   57    91    99   52     7   11   13    9
  10   64    57   105   58     3    7   19   15
  11   82   143    81   62    11    3   13   27
  12   85   135   147   64     5    7   27   21
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot A359804(n) mod 2 at (x,y) = (n mod 256, -floor(n/256)), 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.
• Michael De Vlieger, Plot A359804(n) mod 2 at (x,y) = (n mod 2^10, -floor(n/2^10)), 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.

Crossrefs

Cf. A000079, A053669, A100484, A359804, A361503, A361505, A363594.

Programs

Mathematica

nn = 432; c[] = False; q[] = 1;
Set[{i, j}, {1, 2}]; c[1] = c[2] = True; q[2] = 2; u = 3;
Reap[Do[
    (k = q[#]; While[c[k #], k++]; k *= #;
       While[c[# q[#]], q[#]++]) &[(p = 2;
      While[Divisible[i j, p], p = NextPrime[p]]; p)];
    If[OddQ[j k], Sow[n - 1]];
    Set[{c[k], i, j}, {True, j, k}];
    If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]]

Formula

A361503(a(n)+1) = 2, consequence of definition of A359804.