A359804 -id:A359804 - cp's OEIS frontend

A361502 Index of n-th prime in A359804.

2, 3, 4, 8, 13, 42, 347, 3466, 49012, 528231, 717126, 63056215, 1375559400, 7038527851

Offset: 1

N. J. A. Sloane, Mar 18 2023, based on a comment made by Michael De Vlieger in A359804 in which he gave the values of a(1) to a(12)

Theorem: Every prime appears in A359804. For proof see A359804.

It appears that the primes in A359804 appear in order.

Rémy Sigrist, C++ program

Cf. A359804, A361504.

nn = 2^20; c[] = False; q[] = 1;
 i = 1; j = 2; c[1] = c[2] = True; u = 3;
 {2}~Join~Reap[Monitor[Do[
      (k = q[#]; While[c[k #], k++]; k *= #;
         While[c[# q[#]], q[#]++]) &[(p = 2;
        While[Divisible[i j, p], p = NextPrime[p]]; p)];
      If[PrimeQ[k], Sow[n]; Print[n]];
      Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}], n]][[-1, -1]] (* Michael De Vlieger, Mar 19 2023 *)

a(13)-a(14) from Rémy Sigrist, Mar 19 2023

A361503 a(1)=2; thereafter a(n) = smallest prime that does not divide b(n-1)*b(n), where b(k) = A359804(k).

Original entry on oeis.org

2, 3, 5, 2, 3, 5, 7, 3, 2, 5, 7, 11, 3, 5, 7, 11, 5, 3, 7, 5, 11, 7, 2, 3, 5, 7, 11, 5, 7, 2, 3, 5, 7, 3, 5, 7, 2, 5, 7, 11, 13, 5, 2, 3, 5, 2, 7, 3, 5, 7, 11, 5, 3, 2, 5, 7, 11, 2, 5, 3, 7, 5, 3, 7, 2, 11, 3, 7, 5, 3, 7, 5, 11, 7, 5, 11, 7, 5, 3, 7, 11, 3, 2, 5, 7, 2, 5, 3, 2, 5, 7, 13, 3, 5, 2, 3

Offset: 1

N. J. A. Sloane, Mar 18 2023

nonn

Understanding this sequence is the key to analyzing A359804.

			b(1)=1, b(2)=2, b(3)=3, b(4)=5, so a(2) = smallest prime not dividing 2, which is 3. a(3) = smallest prime not dividing 2*3, which is 5, and a(4) = smallest prime not dividing 3*5, which is 2.

Michael De Vlieger, Table of n, a(n) for n = 1..16383

Cf. A053669, A359804.

nn = 120; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
Set[{i, j}, {a[1], a[2]}]; u = 3;
{2}~Join~Reap[Do[
     (k = q[#]; While[c[k #], k++]; k *= #;
        While[c[# q[#]], q[#]++]) &[(p = 2;
       While[Divisible[i j, p], p = NextPrime[p]]; p)]; Sow[p];
     Set[{a[n], c[k], i, j}, {k, True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]] (* Michael De Vlieger, Mar 18 2023 *)

A361504 Index of n in A359804, or -1 if n never appears there.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 10, 9, 7, 13, 14, 42, 12, 11, 24, 347, 19, 3466, 15, 16, 17, 49012, 25, 18, 31, 32, 20, 528231, 21, 717126, 38, 22, 44, 23, 35, 63056215, 47, 45, 26, 1375559400, 27, 7038527851, 28, 29, 55

Offset: 1

N. J. A. Sloane, Mar 18 2023

Conjectured to be a permutation of the natural numbers.

Cf. A359804, A361502.

nn = 2^20; c[] = 0; q[] = 1;
Array[Set[{a[#], c[#]}, {#, #}] &, 2];
Set[{i, j}, {a[1], a[2]}]; u = 3;
Monitor[Do[
  (k = q[#]; While[c[k #] > 0, k++]; k *= #;
     While[c[# q[#]] > 0, q[#]++]) &[(p = 2;
    While[Divisible[i j, p], p = NextPrime[p]]; p)]; Sow[p];
  Set[{a[n], c[k], i, j}, {k, n, j, k}];
  If[k == u, While[c[u] > 0, u++]], {n, 3, nn}], n];
TakeWhile[Array[c, 120], # > 0 &] (* Michael De Vlieger, Mar 18 2023 *)

A361505 Index of 2^n in A359804.

Original entry on oeis.org

1, 2, 5, 10, 24, 38, 87, 172, 349, 706, 1407, 2752, 5487, 11103, 22285, 44429, 88993, 177746, 356460, 712129, 1425163, 2849424, 5701776, 11401709, 22804522, 45608572, 91219022, 182438457, 364879209, 729757797, 1459514883, 2919031155, 5838065175, 11676129412, 23352260426

Offset: 0

N. J. A. Sloane, Mar 18 2023

nonn

Rémy Sigrist, C++ program

Cf. A359804, A361502, A361504.

nn = 2^20; c[] = False; q[] = 1;
i = 1; j = 2; c[1] = c[2] = True; u = 3;
{1, 2}~Join~Reap[Monitor[Do[
     (k = q[#]; While[c[k #], k++]; k *= #;
        While[c[# q[#]], q[#]++]) &[(p = 2;
       While[Divisible[i j, p], p = NextPrime[p]]; p)];
     If[IntegerQ@ Log2[k], Sow[n]];
     Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}], n]][[-1, -1]] (* Michael De Vlieger, Mar 18 2023 *)

a(26)-a(27) from Michael De Vlieger, Mar 19 2023

More terms from Rémy Sigrist, Mar 19 2023

A361639 For n > 1, A359804(n) is a multiple of A361503(n-1); a(n) = A359804(n) / A361503(n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 3, 4, 3, 2, 1, 4, 4, 3, 2, 5, 6, 4, 6, 3, 5, 8, 8, 8, 6, 4, 9, 7, 13, 9, 10, 8, 12, 11, 9, 16, 12, 10, 6, 1, 13, 17, 13, 15, 19, 11, 16, 16, 12, 8, 17, 17, 23, 18, 13, 9, 26, 19, 18, 14, 20, 19, 15, 29, 10, 23, 16, 23, 24, 17, 24, 11, 18

Offset: 2

Rémy Sigrist, Mar 19 2023

nonn

			a(10) = A359804(10) / A361503(9) = 8 / 2 = 4.

Michael De Vlieger, Table of n, a(n) for n = 2..16384
Rémy Sigrist, C program
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^20, showing 1s in red. Index on x-axis is n-1.

Cf. A359804, A361503.

C
```
See Links section.
```

nn = 120; c[] = False; q[] = 1;
i = 1; j = 2; c[1] = c[2] = True; u = 3;
{1}~Join~Reap[Do[
     (m = q[#]; While[c[m #], m++]; k = m #; q[#] = m) &[(p = 2;
       While[Divisible[i j, p], p = NextPrime[p]]; p)]; Sow[m];
     Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]] (* Michael De Vlieger, Mar 19 2023 *)

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

Original entry on oeis.org

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

Michael De Vlieger and David James Sycamore, Jun 12 2023

nonn

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

			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
  ...

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.

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

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]]

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

A363594 a(n) = the n-th instance of b(k)/2 such that b(k-1) and b(k-2) are both odd, where b(n) = A359804(n).

Original entry on oeis.org

2, 4, 8, 13, 16, 17, 19, 23, 26, 29, 31, 32, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 116, 118, 122, 124, 127, 128, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 152, 157, 158, 163, 164, 166, 167, 172

Offset: 1

Michael De Vlieger and David James Sycamore, Jun 12 2023

nonn

The sequence strictly increases, a consequence of definition of A359804.
Conjecture: { A000079 \ {1} } U { A000040 \ {3, 5, 7, 11} } is a subset. In other words, this sequence is the union of powers of 2 greater than 1, and primes greater than 11.
This sequence is conjectured to be infinite. It tracks all occurrences of consecutive odd terms in A359804, which are (by definition) always followed by an even term, from which a(n) is derived. - David James Sycamore, Jun 21 2023

			a(1) = 2 since b(3..5) = {3, 5, 4}; 4/2 = 2.
a(2) = 4 since b(8..10) = {7, 9, 8}; 8/2 = 4.
a(3) = 8 since b(22..24) = {33, 35, 16}; 16/2 = 8.
a(4) = 13 since b(29..31) = {45, 49, 26}; 26/2 = 13.
a(5) = 16 since b(36..38) = {55, 63, 32}; 32/2 = 16, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A000079, A100484, A359804, A361639, A363593.

nn = 500; 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[i j], Sow[k/2]];
    Set[{c[k], i, j}, {True, j, k}];
    If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, -1]]

a(n) = A359804(A363593(n)+2)/2 = A361639(A363593(n)+1).

A369685 Least common multiple of the first n terms of A359804.

Original entry on oeis.org

1, 1, 2, 6, 30, 60, 60, 60, 420, 1260, 2520, 2520, 2520, 27720, 27720, 27720, 27720, 27720, 138600, 138600, 138600, 138600, 138600, 138600, 277200, 277200, 277200, 277200, 277200, 277200, 1940400, 25225200, 75675600, 75675600, 75675600, 75675600, 75675600, 75675600

Offset: 0

David James Sycamore, Jan 28 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..6187

Cf. A003418, A359804, A368901, A369686.

nn = 40; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {1, True}] &, 2];
Set[{i, j}, {1, 2}]; m = 2; u = 3;
Do[
  (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[
  (p = 2; While[Divisible[i j, p], p = NextPrime[p]]; p)];
  Set[{a[n], c[k], i, j, m}, {m, True, j, k, #}] &[LCM[m, k]];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jan 29 2024 *)

a(0) = 1, and for n >0, a(n) = lcm(A359804(n), a(n-1)).

A369686 LCM-transform of A359804 (see Comment and links).

Original entry on oeis.org

1, 2, 3, 5, 2, 1, 1, 7, 3, 2, 1, 1, 11, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 13, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 17, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 11, 1, 5, 1, 1, 1, 1, 1, 1, 3, 31, 1

Offset: 1

David James Sycamore, Jan 28 2024

nonn

Let b(k) be the Least Common Multiple (LCM) of the first k terms of A359804, then a(n) = b(n)/b(n-1), where sequence b(n) is A369685.

The property S (as defined in A368900) refers to what is observed in the positive integers (A000027), and also in the Doudna sequence (A005940), whereby each prime power appears prior to any of its multiples. The present sequence does not have this property since, for example, 26 = a(31) precedes 13 = a(42). Thus A369804 represents a significant disturbance of A000027 in that whereas it is conjectured to be a permutation of the positive integers, it does not preserve one of the basic properties of that sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Andrzej Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
Andrzej Nowicki, Strong divisibility and LCM-sequences, Am. Math. Monthly (2015) Vol. 122, 958-966.

Cf. A014963, A359804, A368900, A369685.

nn = 120; c[] = False; q[] = 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
Set[{i, j}, {1, 2}]; m = 2; u = 3;
Do[
  (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[
  (p = 2; While[Divisible[i j, p], p = NextPrime[p]]; p)];
  Set[{a[n], c[k], i, j, m}, {#/m, True, j, k, #}] &[LCM[m, k]];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jan 29 2024 *)

a(n) = A369685(n)/A369685(n-1).

More terms from Michael De Vlieger, Jan 29 2024

A361643 The binary expansion of a(n) specifies which primes divide A359804(n).

Original entry on oeis.org

0, 1, 2, 4, 1, 3, 5, 8, 2, 1, 6, 9, 16, 3, 5, 10, 17, 4, 3, 9, 7, 18, 12, 1, 3, 5, 11, 17, 6, 8, 33, 2, 5, 9, 3, 20, 10, 1, 7, 13, 19, 32, 36, 65, 34, 6, 129, 24, 3, 5, 11, 17, 68, 66, 257, 7, 40, 18, 33, 132, 3, 9, 5, 130, 14, 513, 21, 258, 9, 260, 3, 72, 7

Offset: 1

Rémy Sigrist, Mar 19 2023

			The first terms, in decimal and in binary, alongside A359804(n) and its divisibility by small prime numbers, are:
  n   a(n)  bin(a(n))  A359804(n)  Divisibility by:
                                      7  5  3  2
  --  ----  ---------  ----------     -  -  -  -
   1     0          0           1
   2     1          1           2              X
   3     2         10           3           X
   4     4        100           5        X
   5     1          1           4              X
   6     3         11           6           X  X
   7     5        101          10        X     X
   8     8       1000           7     X
   9     2         10           9           X
  10     1          1           8              X

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A087207, A359804.

PARI
```
See Links section.
```

a(n) = A087207(A359804(n)).

cp's OEIS Frontend

A361502 Index of n-th prime in A359804.

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A361503 a(1)=2; thereafter a(n) = smallest prime that does not divide b(n-1)*b(n), where b(k) = A359804(k).

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A361504 Index of n in A359804, or -1 if n never appears there.

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A361505 Index of 2^n in A359804.

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A361639 For n > 1, A359804(n) is a multiple of A361503(n-1); a(n) = A359804(n) / A361503(n-1).

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

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A363594 a(n) = the n-th instance of b(k)/2 such that b(k-1) and b(k-2) are both odd, where b(n) = A359804(n).

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A369685 Least common multiple of the first n terms of A359804.

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A369686 LCM-transform of A359804 (see Comment and links).