A336957 -id:A336957 - cp's OEIS frontend

A338062 Numbers k such that the Enots Wolley sequence A336957(k) is odd.

1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 128, 129, 132, 133, 136, 137

Offset: 1

N. J. A. Sloane, Oct 18 2020

nonn

a(n) = A042948(n) for n<=1065, but then the two sequences start to differ. - R. J. Mathar, Nov 06 2020

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957, A338063-A338069.

M = 1000;
A[1] = 1; A[2] = 2;
Clear[B]; B[_] = 0;
For[n = 3, True, n++, For[k = 3, k <= M, k++, If[B[k] == 0 && GCD[k, A[n-1]] > 1 && GCD[k, A[n-2]] == 1, If[Length[FactorInteger[k][[All, 1]] ~Complement~ FactorInteger[A[n-1]][[All, 1]]] > 0, A[n] = k; B[k] = 1; Break[]]]]; If[k > M, Break[]]];
Reap[For[k = 1, k <= M, k++, If[OddQ[A[k]], Sow[k]]]][[2, 1]] (* Jean-François Alcover, Oct 23 2020, after Maple code in A336957 *)

A338069 Even terms A336957(k) which are less than k, divided by 2, in order of appearance.

Original entry on oeis.org

15, 12, 18, 30, 24, 27, 33, 36, 39, 43, 44, 46, 47, 57, 60, 69, 72, 93, 90, 107, 108, 105, 111, 112, 114, 117, 127, 129, 141, 153, 154, 157, 159, 162, 165, 169, 168, 173, 171, 170, 174, 176, 177, 179, 180, 181, 183, 188, 184, 182, 187, 191, 193, 196, 189, 186, 192, 201, 198, 208

Offset: 1

N. J. A. Sloane, Oct 19 2020

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957, A338062-A338068.

A337644 Numbers k such that w(k), w(k+1), and w(k+2) are all odd, where w is A336957.

Original entry on oeis.org

2128, 4689, 7742, 11011, 11508, 12277, 16398, 20227, 22556, 23709, 26922, 31455, 36016, 36857, 39014, 39563, 45804, 47213, 47738, 48847, 48932, 50805, 53062, 57575, 58784, 60281, 63594, 66251, 68872, 74021, 79238, 84175, 89428, 91709, 92902, 92947, 94404, 98317

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Sep 24 2020

nonn

These terms are rare, since most of the time the parity of A336957 follows the pattern 1, 0,0, 1,1, 0,0, 1,1, 0,0, ... It would be useful to have a proof that the present sequence is (or is not) infinite. The graph strongly suggests it is an infinite sequence.
It is also possible that eventually there will be four or more odd terms in succession. However, this does not happen in the first eleven million terms, so probably it never happens.
If w(j) is even and w(j+1) is odd, then w(j+2) is forced to be also odd. In most cases w(j+3) is then even, but is occasionally odd (giving three odds in a row), and then the values of j+1 are given in the present sequence. For understanding the growth of A336957, the values of j+3 and w(j+3) are also important, and are given in A338070 and A338071, respectively.

N. J. A. Sloane, Table of n, a(n) for n = 1..5985 (computed from Frank Stevenson's file of the first 11333576 terms of A336957; terms 1..71 from N. J. A. Sloane, terms 72..575 from Scott R. Shannon)

Cf. A336957, A338070, A338071.

Comments revised by N. J. A. Sloane, Oct 12 2020

A337275 Index of appearance of 2prime(n) in A336957, or -1 if 2prime(n) never appears.

Original entry on oeis.org

-1, 3, 10, 6, 14, 18, 22, 23, 26, 51, 46, 54, 58, 87, 99, 106, 107, 110, 114, 134, 135, 142, 155, 171, 182, 195, 199, 215, 210, 214, 255, 259, 271, 274, 295, 299, 315, 323, 326, 347, 367, 371, 390, 391, 394, 398, 443, 451, 471, 475, 478, 491, 495, 511, 523, 531, 543, 547, 567, 575, 579, 599, 627

Offset: 1

N. J. A. Sloane, Sep 09 2020

sign

It is a strong conjecture that 2*prime(n) appears in A336957 for all n>1, and it is known that 4 does not appear.

See also the comment in A336957 discussing when primes first appear in A336957.

			A336957(14) = 22 = 2*prime(5), so a(5) = 14.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
N. J. A. Sloane, Table comparing the present sequence and A338074

Cf. A336957, A337276, A338074.

A337007 a(n) = gcd(b(n),b(n+1)), where b(n) = A336957(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 2, 3, 11, 5, 2, 3, 7, 11, 2, 5, 3, 13, 2, 7, 3, 17, 2, 19, 3, 23, 2, 5, 13, 7, 6, 5, 17, 7, 8, 3, 5, 19, 4, 3, 29, 5, 2, 11, 3, 31, 2, 13, 3, 35, 2, 29, 3, 37, 2, 17, 3, 41, 2, 5, 23, 7, 12, 5, 31, 7, 2, 3, 43, 5, 2, 3, 47, 5, 22, 3, 7, 19, 8, 9, 5, 7, 2, 13, 11

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Aug 11 2020

nonn

			A336957(3)=6, A336957(4) = 15, gcd(6,15) = 3 = a(3).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957, A337008.

A337008 Product of distinct primes dividing both b(n) and b(n+1), where b(n) = A336957(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 2, 3, 11, 5, 2, 3, 7, 11, 2, 5, 3, 13, 2, 7, 3, 17, 2, 19, 3, 23, 2, 5, 13, 7, 6, 5, 17, 7, 2, 3, 5, 19, 2, 3, 29, 5, 2, 11, 3, 31, 2, 13, 3, 35, 2, 29, 3, 37, 2, 17, 3, 41, 2, 5, 23, 7, 6, 5, 31, 7, 2, 3, 43, 5, 2, 3, 47, 5, 22, 3, 7, 19, 2, 3, 5, 7, 2, 13, 11, 17, 2, 43, 7

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Aug 11 2020

nonn

This is the radical (cf. A007947) of A337007. First differs from A337007 at n=34.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A007947, A336957, A337007.

A337645 a(1)=2; thereafter, a(n) = smallest number with at least two different prime factors that is missing from A336957 after A336957(n) has been found.

Original entry on oeis.org

2, 6, 10, 10, 10, 10, 10, 10, 10, 18, 20, 20, 20, 20, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 36, 36, 36, 36, 44, 44, 44, 44, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 54

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Sep 25 2020; corrected Oct 11 2020

nonn

The initial 2 is an exception. For n>2, A336957(n) is required to have at least two different prime factors.

It is conjectured that every number with at least two different prime factors will eventually appear in A336957.

			A336957 begins 1, 2, 6, 15, 35, 14, with A336957(6)=14. At that point 10 is the smallest legal candidate that has not yet appeared, so a(6) = 10.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957, A338052, A338058.

A337646 Record high points in A336957.

Original entry on oeis.org

1, 2, 6, 15, 35, 55, 77, 91, 119, 145, 261, 301, 329, 333, 451, 473, 493, 505, 515, 707, 721, 749, 1043, 1211, 1253, 1267, 1477, 2453, 2471, 2611, 2681, 2723, 2807, 3143, 3493, 3563, 3787, 3829, 4039, 4529, 4711, 5663, 5971, 6377, 6979, 9977, 10073, 10871, 11053, 11207, 16049, 19151

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Sep 25 2020

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..187
N. J. A. Sloane, Table showing, for i-th high point, the values of A336957(n), the index n, and the ratio A336957(n)/n

Cf. A336957, A337647.

A338050 Fixed points in A336957.

Original entry on oeis.org

1, 2, 10, 90, 106, 150, 162, 246, 394, 398, 406, 410, 442, 602, 1106, 1246, 1390, 1482, 1666, 1846, 5524, 6068, 6124, 6976, 35562, 42618, 42726, 45618, 1271596, 2634332, 2766702

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 10 2020

These are the values of n such that A336957(n) = n. Is this sequence finite or infinite?. We do not know. Also, why are the terms (except the first, which is special) even?
The known values of n such that A336957(n) = n+1 are 43, 103, 191, 203, 207, 211, 243, 247, 251, 287, 291, 303, 327, 399, 8133, 71071, 579451, 10955983. Are there more terms? We do not know. Also, why are all the terms odd?
The known values of n such that A336957(n) = n-1 are 31, 87, 187, 215, 255, 283, 307, 315, 347, 551, 10697, 30525, 46195, 478607, 3856311, 6405255.  Same questions!
All these values were computed from Frank Stevenson's file of the first 11333576 terms of A336957.

Cf A336957.

A338051 a(n) = A336957(n) - n.

Original entry on oeis.org

0, 0, 3, 11, 30, 8, 5, 25, 46, 0, 7, 9, 64, 8, 5, 29, 22, 8, 9, 43, 30, 12, 15, 33, 44, 20, 13, 37, 62, 12, -1, 53, 86, 22, -11, 39, 58, 38, -3, 47, 104, 8, 1, 55, 48, 16, 5, 69, 56, 20, 7, 209, 58, 20, 13, 97, 66, 24, 21, 55, 100, 22, -3, 91, 152, 32, -19, 61, 146, 30, -17, 69, 162, 36, -9, 71, 56, 74, -7, 55, 94

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 10 2020

sign

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957.

cp's OEIS Frontend

A338062 Numbers k such that the Enots Wolley sequence A336957(k) is odd.

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A338069 Even terms A336957(k) which are less than k, divided by 2, in order of appearance.

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A337644 Numbers k such that w(k), w(k+1), and w(k+2) are all odd, where w is A336957.

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A337275 Index of appearance of 2*prime(n) in A336957, or -1 if 2*prime(n) never appears.

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A337007 a(n) = gcd(b(n),b(n+1)), where b(n) = A336957(n).

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A337008 Product of distinct primes dividing both b(n) and b(n+1), where b(n) = A336957(n).

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A337645 a(1)=2; thereafter, a(n) = smallest number with at least two different prime factors that is missing from A336957 after A336957(n) has been found.

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A337646 Record high points in A336957.

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A338050 Fixed points in A336957.

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A338051 a(n) = A336957(n) - n.

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A337275 Index of appearance of 2prime(n) in A336957, or -1 if 2prime(n) never appears.