A336957 -id:A336957 - cp's OEIS frontend

A338059 The Enots Wolley sequence A336957 with the missing prime powers interpolated.

1, 2, 4, 6, 3, 9, 15, 5, 25, 35, 7, 14, 8, 12, 27, 33, 11, 55, 10, 16, 18, 21, 49, 77, 22, 20, 45, 39, 13, 26, 28, 63, 51, 17, 34, 32, 38, 19, 57, 69, 23, 46, 40, 65, 91, 42, 30, 85, 119, 56, 24, 75, 95, 76, 36, 81, 87, 29, 145, 50, 44, 99, 93, 31, 62, 52, 117, 105, 70, 58, 261, 111, 37

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 17 2020

nonn

There is a strong conjecture that A336957 consists exactly of 1, 2, and all numbers with at least two different prime factors. [The only uncertainty is whether all numbers with at least two prime factors appear.]
The terms in A000961 greater than 2 are definitely missing from A336957, so A336957 is obviously not a permutation of the positive integers.
The present sequence is obtained by inserting the missing prime powers q = p^k, p >= 2, k >= 1, in their natural positions. More precisely, let the terms of A336957 be [W(i), i >= 1].
Between W(i) and W(i+1) we insert, in order, any prime powers q < W(i+1) which are not yet in the new sequence and satisfy gcd(q, W(i)) > 1 and gcd(q, W(i-1)) = 1.
It is conjectured that this is a permutation of the positive integers.

			Suppose n = 4.
The first 5 terms of A336957 are 1,2,6,15,35. The first 7 terms of the present sequence are 1, 2, 4, 6, 3, 9, 15. To see what comes after a(7) = W(4) = 15, we look at the missing prime powers less than W(5) = 35, which are 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31. Just two terms, 5 and 25, have a common factor with 15 and are relatively prime to W(3) = 6, so they are adjoined to the sequence.
In short, we adjoin any missing prime powers which are less than W(n+1), have a common factor with W(n), and are relatively prime to W(n-1). We insert them immediately after W(n).

N. J. A. Sloane, Table of n, a(n) for n = 1..5000

Cf. A000961, A336957, A338060 (inverse).

A338071 Values of w(k) when w(k-2), w(k-1), and w(k) are all odd, where w is A336957.

Original entry on oeis.org

3263, 7183, 11671, 16291, 16601, 20741, 23257, 28639, 37667, 33163, 38819, 43849, 51469, 52789, 48701, 50275, 63323, 65117, 67903, 67223, 79751, 72193, 71265, 79183, 80743, 74741, 106483, 90571, 94159, 104467, 108043, 135821, 109771, 112561, 119149, 149387, 116377, 137951

Offset: 1

N. J. A. Sloane, Oct 19 2020

nonn

See comments in A337644.

It would be nice to understand what is special about these numbers. The majority of them appear to products of two distinct primes. There seems to be very little overlap with either A337646 or A338057, although 1531513 appears both here and in A337646.

			The factorizations of the first 10 terms are:
1, (13)*(251)
2, (11)*(653)
3, (11)*(1061)
4, (11)*(1481)
5, (13)*(1277)
6, (7)*(2963)
7, (13)*(1789)
8, (13)*(2203)
9, (7)*(5381)
10, (13)*(2551)
The factorizations of terms 555 through 575 are:
555, (11)*(118681)
556, (7)*(213833)
557, (7)*(213887)
558, (11)*(118901)
559, (3)*(5)*(83059)
560, (11)*(120619)
561, (13)*(98867)
562, (11)*(121021)
563, (13)*(99391)
564, (7)*(218873)
565, (11)*(121621)
566, (13)*(99571)
567, (13)*(99989)
568, (11)*(122299)
569, (13)*(122503)
570, (11)*(122533)
571, (11)*(122579)
572, (13)*(100537)
573, (7)*(221537)
574, (11)*(123427)
575, (31)*(38393)

N. J. A. Sloane, Table of n, a(n) for n = 1..575

Cf. A336957, A337644, A337646, A338057, A338070.

A337647 Indices of record high points in A336957.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 13, 29, 33, 41, 52, 88, 100, 140, 148, 160, 168, 189, 193, 196, 200, 216, 296, 341, 368, 372, 444, 452, 741, 780, 841, 857, 869, 949, 1028, 1105, 1116, 1128, 1176, 1332, 1396, 1644, 1736, 1860, 2036, 2061, 2945, 3161, 3225, 3261, 3454, 4106, 5678, 5718, 5762, 5806, 6214, 6838, 7474

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

Cf. A336957, A337646.

A337648 Odd primes p such that the first term in A336957 that is divisible by p is 2*p.

Original entry on oeis.org

3, 19, 59, 73, 83, 89, 127, 131, 137, 149, 151, 157, 163, 193, 223, 227, 229, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491

Offset: 1

N. J. A. Sloane, Sep 26 2020

nonn

Conjecture 1: this sequence contains all primes > 367.
Conjecture 2: The set of odd primes is partitioned into A337648, A337649, and {7}.
(These conjectures have been checked for the first 161734 terms of A336957.)
When an odd prime p first divides a term of A336957 that term is equal to q*p where q < p is also a prime. It appears q is almost always 2 (the corresponding values of p form the present sequence), that there are 34 instances when q = 3 (see A337649), and q>3 happens just once, at A336957(5) = 35 when q=5 and p=7.
See also the comment in A336957 discussing when primes first appear in A336957.

Cf. A336957, A337275, A337649.

A338053 "Early" terms in A336957, in order of appearance.

Original entry on oeis.org

6, 15, 35, 14, 12, 33, 55, 18, 21, 77, 22, 20, 45, 39, 26, 28, 63, 51, 34, 38, 57, 69, 46, 40, 65, 91, 42, 85, 119, 56, 75, 95, 76, 87, 145, 50, 44, 99, 93, 62, 52, 117, 105, 70, 58, 261, 111, 74, 68, 153, 123, 82, 80, 115, 161, 84, 155, 217, 98, 129, 215, 100, 141, 235, 110, 147, 133

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 11 2020, following a suggestion from Christopher Landauer

nonn

A term A336957(k) is early if A336957(k) > k; punctual if A336957(k) = k (see A338050); and late if A336957(k) < k.

It appears that the majority of terms are late.

Cf. A336957, A338050, A338051, A338054.

A338054 "Early" terms in A336957, arranged in increasing order.

Original entry on oeis.org

6, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 68, 69, 70, 74, 75, 76, 77, 80, 82, 84, 85, 87, 91, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 11 2020, following a suggestion from Christopher Landauer

nonn

A term A336957(k) is early if A336957(k) > k; punctual if A336957(k) = k (see A338050); and late if A336957(k) < k.

It appears that the majority of terms are late.

Cf. A336957, A338050, A338051, A338053.

A338057 Values of A336957(k) where A336957(k)/k sets a new record.

Original entry on oeis.org

1, 6, 15, 35, 1531513, 2042057, 3828689, 4849879, 5615321, 8933993, 9189469, 14038957, 16591609, 17867867, 20165077, 20420417, 28843849, 31396331, 34204153, 37522621, 43648643, 82447517

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 12 2020

See A338056 for further information.

Cf. A336957, A338056.

A338070 Numbers k such that w(k-2), w(k-1), and w(k) are all odd, where w is A336957.

Original entry on oeis.org

2130, 4691, 7744, 11013, 11510, 12279, 16400, 20229, 22558, 23711, 26924, 31457, 36018, 36859, 39016, 39565, 45806, 47215, 47740, 48849, 48934, 50807, 53064, 57577, 58786, 60283, 63596, 66253, 68874, 74023, 79240, 84177, 89430, 91711, 92904, 92949, 94406, 98319, 103480

Offset: 1

N. J. A. Sloane, Oct 19 2020

nonn

See comments in A337644.

N. J. A. Sloane, Table of n, a(n) for n = 1..5985

Cf. A336957, A337644, A338071.

A338074 Numbers k such that A336957(k) is twice a prime.

Original entry on oeis.org

3, 6, 10, 14, 18, 22, 23, 26, 46, 51, 54, 58, 87, 99, 106, 107, 110, 114, 134, 135, 142, 155, 171, 182, 195, 199, 210, 214, 215, 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

Offset: 1

N. J. A. Sloane, Oct 21 2020

nonn

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

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
N. J. A. Sloane, Table comparing the present sequence and A337275

Cf. A336957, A337275.

A337066 a(n) is the smallest k such that A336957(k) = n, or -1 if n does not appear in A336957.

Original entry on oeis.org

1, 2, -1, -1, -1, 3, -1, -1, -1, 10, -1, 7, -1, 6, 4, -1, -1, 11, -1, 15, 12, 14, -1, 35, -1, 18, -1, 19, -1, 31, -1, -1, 8, 22, 5, 39, -1, 23, 17, 27, -1, 30, -1, 43, 16, 26, -1, 67, -1, 42, 21, 47, -1, 71, 9, 34, 24, 51, -1, 63, -1, 46, 20, -1, 28, 75, -1, 55, 25, 50, -1, 79, -1, 54, 36, 38

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Aug 13 2020

sign

There is a strong conjecture that numbers n >= 3 appear in A336957 if and only if they are divisible by at least two different primes.

Scott R. Shannon, Table of n, a(n) for n = 1..20000

Cf. A098550, A336957, A337007, A337008.

cp's OEIS Frontend

A338059 The Enots Wolley sequence A336957 with the missing prime powers interpolated.

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A338071 Values of w(k) when w(k-2), w(k-1), and w(k) are all odd, where w is A336957.

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Examples

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A337647 Indices of record high points in A336957.

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A337648 Odd primes p such that the first term in A336957 that is divisible by p is 2*p.

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A338053 "Early" terms in A336957, in order of appearance.

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A338054 "Early" terms in A336957, arranged in increasing order.

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A338057 Values of A336957(k) where A336957(k)/k sets a new record.

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A338070 Numbers k such that w(k-2), w(k-1), and w(k) are all odd, where w is A336957.

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A338074 Numbers k such that A336957(k) is twice a prime.

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A337066 a(n) is the smallest k such that A336957(k) = n, or -1 if n does not appear in A336957.

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