A350877 -id:A350877 - cp's OEIS frontend

A350615 Indices of 1's in A350877.

1, 8, 12, 20, 742, 513152128

Offset: 1

Hans Havermann, Jan 22 2022

This sequence is probably infinite, but may grow extremely rapidly. Russ Cox has checked that there are no further terms below 4.9*10^12. - N. J. A. Sloane, Jan 25 2022

			The 1 at 513152128 is a down-drop from 2^32 at index 513152096 in A350877, 2^32 being 712032365 at index 513152095 plus the prime 3582934931.

Cf. A350877, A350616.

See A362107, A362108 for the chains of powers of 2 that lead to these 1's.

A350616 Indices of odd terms in A350877.

Original entry on oeis.org

1, 2, 4, 8, 12, 15, 20, 22, 25, 27, 30, 32, 34, 36, 38, 40, 46, 48, 51, 53, 56, 58, 60, 63, 65, 71, 73, 76, 78, 80, 83, 85, 88, 95, 97, 100, 102, 105, 108, 114, 117, 123, 126, 128, 132, 135, 137, 140, 143, 148, 152, 156, 161, 163, 166, 172, 174, 176, 178, 180, 182, 186, 188, 192, 195, 197, 201, 204, 207, 211, 213

Offset: 1

N. J. A. Sloane, Jan 23 2022

nonn

If we add 1 to these terms we get 2, 3, 5, 13, 16, 21, 23, 26, 28, 33, ..., which are the locations of record high points in A350877.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 338 terms from N. J. A. Sloane)

Cf. A350877, A350615, A350617, A350618, A350619, A350620, A350621, A350833.

j = 1; q = 2; {j}~Join~Reap[Do[If[EvenQ[j], Set[k, j/2], Set[k, j + q]; Set[q, NextPrime[q]]]; If[OddQ[k], Sow[i]]; j = k, {i, 2, 213}]][[-1, -1]] (* Michael De Vlieger, Jan 23 2022 *)

A350616_upto(N)=select(t->t%2, A350877_first(N), 1) \\ M. F. Hasler, Jan 23 2022

A350618 Terms in A350877 that immediately follow an odd term.

Original entry on oeis.org

3, 6, 8, 8, 12, 16, 18, 28, 30, 44, 42, 58, 70, 78, 86, 96, 62, 92, 90, 116, 102, 130, 148, 126, 160, 106, 156, 146, 182, 204, 178, 220, 192, 142, 220, 206, 260, 228, 224, 180, 224, 188, 238, 312, 236, 258, 340, 308, 304, 248, 264, 272, 258, 380, 352, 274, 406, 474, 514, 538, 552, 362, 488, 372, 406, 520, 396, 436

Offset: 1

N. J. A. Sloane, Jan 23 2022, revised Jan 28 2022

nonn

a(n) = A350617(n) + prime(n). Also a(n) = 2^A350833(n) * A350617(n+1).
This is a compressed version of A350877: when A350877 reaches an even number e, the following steps repeatedly divide e by 2 until an odd number is reached. In the present sequence the results of those divisions are suppressed.
For example, A350877 (n>=2) begins 1, 3, 6, [3,] 8, [4, 2, 1,] 8, [4, 2, 1,] 12, [6, 3,] 16, [8, 4, 2, 1,] 18, ..., where the suppressed terms are enclosed in square brackets.
The scatterplot of the present sequence is the same as the red-colored portion of Sigrist's colored scatterplot in A350877.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from Michael De Vlieger)

Cf. A350877, A350615, A350616, A350617, A350619, A350620, A350621, A350833.

j = 1; q = 2; Reap[Do[If[EvenQ[j], Set[k, j/2], Set[k, j + q]; Set[q, NextPrime[q]]]; If[OddQ[j], Sow[i + 1]]; j = k, {i, 2, 436}]][[-1, -1]] (* Michael De Vlieger, Jan 23 2022 *)

A350620 a(n) = smallest k such that A350877(k) = n, or -1 if n does not appear in A350877.

Original entry on oeis.org

1, 7, 2, 6, 71, 3, 25, 5, 22, 70, 30, 13, 345, 24, 27, 16, 161, 21, 148, 69, 32, 29, 51, 43, 1154, 344, 161336, 23, 34, 26, 48, 737, 156, 160, 36, 77534485877, 63, 147, 38, 68, 234, 31, 40, 28, 53, 50, 126, 42, 639, 1153, 58, 343, 73, 161335, 88, 111, 108, 33, 135, 614667, 192, 47, 65, 736, 60, 155, 454, 159, 186, 35, 97, 77534485876, 78, 62, 2340, 146, 143, 37, 24841, 67, 476, 233, 433, 10579, 140, 39, 359, 169, 85, 52, 80, 49, 195, 125, 166, 41, 17282073747557

Offset: 1

N. J. A. Sloane, Jan 23 2022

nonn

Martin Ehrenstein, Table of n, a(n) for n = 1..126

Cf. A350877. See A350621 for where the primes appear.

a(27) was found by Keith F. Lynch, and a(36) and a(72) by Russ Cox.
a(97) and a(115) were found by Martin Ehrenstein using Kim Walisch's primesieve package.
More than the usual number of terms are shown in order to give all the terms up to the extreme value a(97).

A350617 Odd-valued terms in A350877.

Original entry on oeis.org

1, 3, 3, 1, 1, 3, 1, 9, 7, 15, 11, 21, 29, 35, 39, 43, 3, 31, 23, 45, 29, 51, 65, 37, 63, 5, 53, 39, 73, 91, 51, 89, 55, 3, 71, 55, 103, 65, 57, 7, 45, 7, 47, 119, 39, 59, 129, 85, 77, 19, 31, 33, 17, 129, 95, 11, 137, 203, 237, 257, 269, 69, 181, 61, 93, 203, 65, 99, 109, 57, 203, 139, 249, 77, 225, 151, 267, 41, 219, 155

Offset: 1

N. J. A. Sloane, Jan 23 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 338 terms from N. J. A. Sloane)

Cf. A350877, A350615, A350616, A350618, A350619.

j = 1; q = 2; {j}~Join~Reap[Do[If[EvenQ[j], Set[k, j/2], Set[k, j + q]; Set[q, NextPrime[q]]]; If[OddQ[k], Sow[k]]; j = k, {i, 2, 240}]][[-1, -1]] (* Michael De Vlieger, Jan 23 2022 *)

A350619 Record high values in A350877.

Original entry on oeis.org

1, 3, 6, 8, 12, 16, 18, 28, 30, 44, 58, 70, 78, 86, 96, 116, 130, 148, 160, 182, 204, 220, 260, 312, 340, 380, 406, 474, 514, 538, 552, 556, 616, 656, 686, 786, 842, 878, 900, 918, 958, 982, 1000, 1092, 1190, 1272, 1290, 1378, 1428, 1432, 1540, 1612, 1776, 1880, 1946, 2126, 2226, 2284, 2290

Offset: 1

N. J. A. Sloane, Jan 23 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 408 terms from N. J. A. Sloane)

Cf. A350877, A350615, A350616, A350617, A350618.

r = 0; j = 1; q = 2; {j}~Join~Reap[Do[If[EvenQ[j], Set[k, j/2], Set[k, j + q]; Set[q, NextPrime[q]]]; If[k > r, Sow[k]; r = k]; j = k, {i, 2, 645}]][[-1, -1]] (* Michael De Vlieger, Jan 23 2022 *)

A350621 a(n) is the smallest k such that A350877(k) = prime(n), or -1 if prime(n) does not appear in A350877.

Original entry on oeis.org

7, 2, 71, 25, 30, 345, 161, 148, 51, 34, 48, 63, 234, 40, 126, 73, 135, 192, 454, 97, 78, 24841, 433, 85, 17282073747557, 322, 102, 106544217, 207, 556

Offset: 1

N. J. A. Sloane, Jan 23 2022

Based on the data in A350620.

Cf. A350877, A350618, A350620.

a(25)-a(30) from Martin Ehrenstein, Jan 26 2022

A350833 Run lengths of even terms in A350877 (half if even, add next prime if odd).

Original entry on oeis.org

0, 1, 3, 3, 2, 4, 1, 2, 1, 2, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 2, 1, 5, 1, 2, 1, 1, 2, 1, 2, 6, 1, 2, 1, 2, 2, 5, 2, 5, 2, 1, 3, 2, 1, 2, 2, 4, 3, 3, 4, 1, 2, 5, 1, 1, 1, 1, 1, 3, 1, 3, 2, 1, 3, 2, 2, 3, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 4, 3, 3, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2

Offset: 1

M. F. Hasler, Jan 23 2022

nonn

Equals first differences of indices of odd terms (A350616) minus one.
After the initial 0, also equals the 2-valuation (A007814) of A350618, the terms following odd terms in A350877.
Record values are a(1) = 0, a(2) = 1, a(3) = 3, a(6) = 4, a(16) = 5, a(33) = 6, a(146) = 7, a(243) = 11, a(1596) = 12, a(2092) = 13, ... The first occurrences of the other values are: a(5) = 2, a(8) = 510, a(9) = 667, a(10) = 1526. No others up to n = 33000. - M. F. Hasler, Jan 24 2022

			Between the first odd term, A350877(1) = 1, and second odd term, A350877(2) = 3, there are no even terms, therefore a(1) = 0.
Between the second and third odd term, A350877(4) = 3, there is one even term, A350877(3) = 6, therefore a(2) = 1.

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A007814, A350877, A350616, A350618.

A350833_upto(N)={N=A350616_upto(N);[N[k]-N[k-1]-1|k<-[2..#N]]}

a(n) = A350616(n+1) - A350616(n) - 1 = A007814(A350618(n-1)) for n > 1.

A351101 Variation of the Sisyphus sequence A350877: the same rules apply except that each time a(n) is divided by a prime the dividing prime is incremented to the next prime while the prime being added to each term is reset to 2.

Original entry on oeis.org

1, 3, 6, 3, 5, 8, 13, 20, 31, 44, 61, 80, 103, 132, 44, 46, 49, 54, 61, 72, 85, 17, 19, 22, 27, 34, 45, 58, 75, 94, 117, 146, 177, 214, 255, 298, 345, 398, 457, 518, 74, 76, 79, 84, 91, 102, 115, 132, 12, 14, 17, 22, 29, 40, 53, 70, 89, 112, 141, 172, 209, 250, 293, 340, 393, 452, 513, 580, 651

Offset: 1

Scott R. Shannon, Jan 31 2022

nonn

This sequence uses the same rules as the Sisyphus sequence, A350877, except that here, instead of always dividing by 2 whenever a(n) is divisible by 2, the prime that is acting as the divisor of a(n), initially 2, is incremented to the next prime once one or more divisions of a(n) by the current dividing prime occur. Additionally each time the dividing prime is incremented the prime being added to each term when a(n) is not divisible by the dividing prime is reset to 2. Once the dividing prime is incremented the terms are then checked for divisibility by this new prime. See the examples below.
In the first 100 million terms the values return to 1 twelve times, see A351278. It is likely this occurs infinitely often although this is unknown. In the same range the smallest number not seen is 5272. This suggests all numbers are eventually visited but this is also unknown. The first numbers to repeat are 3, 44, 61, 132, 17, 22, 75, ... .
In the first 100 million terms the longest gap between prime divisions is 299001 terms, ending at a(80477537) = 609823139121, which is divisible by 40471. Surprisingly the shortest possible gap between different dividing primes, two terms, does not occur until a(194517); the previous terms are a(194514) = 5421744, which is divisible by the dividing prime 1637, a(194515) = 5421744/1637 = 3312, a(194516) = a(194515) + 2 = 3314. The dividing prime is now 1657, and 3314/1657 = 2 = a(194517).

			a(3) = 6 as a(2) = 3, which is not divisible by the current dividing prime 2, and the next additive prime is 3, so a(3) = 3 + 3 = 6.
a(4) = 3 as a(3) = 6, the current dividing prime is 2, and 6/2 = 3. As 3 is not divisible by 2, the divisions by 2 stop, and the dividing prime becomes 3 while the additive prime resets to 2.
a(5) = 5 as a(4) = 3 and the additive prime is 2, so a(5) = 3 + 2 = 5.
a(6) = 8 as a(5) = 5, which is not divisible by 3, and the next additive prime is 3, so a(6) = 5 + 3 = 8.
a(15) = 44 as a(43) = 132, the current dividing prime is 3, and 132/3 = 44. As 44 is not divisible by 3, the divisions by 3 stop, the dividing prime becomes 5 and the additive prime resets to 2.

Scott R. Shannon, Image of the first 10 million terms.

Cf. A350877, A351278, A351102, A000040.

A351102 Variation of the Sisyphus sequence A350877: the same rules apply except that each time a(n) is divided by a prime the dividing prime is incremented to the next prime.

Original entry on oeis.org

1, 3, 6, 3, 8, 15, 5, 16, 29, 46, 65, 13, 36, 65, 96, 133, 19, 60, 103, 150, 203, 262, 323, 390, 461, 534, 613, 696, 785, 882, 983, 1086, 1193, 1302, 1415, 1542, 1673, 1810, 1949, 2098, 2249, 2406, 2569, 2736, 2909, 3088, 3269, 3460, 3653, 3850, 350, 549, 760, 983, 1210, 1439, 1672

Offset: 1

Scott R. Shannon, Jan 31 2022

nonn

This sequence uses the same rules as the Sisyphus sequence, A350877, except that here, instead of always dividing by 2 whenever a(n) is divisible by 2, the prime that is acting as the divisor of a(n), initially 2, is incremented to the next prime once one or more divisions of a(n) by the current dividing prime occur. Once the dividing prime is incremented the terms are then checked for divisibility by this new prime. See the examples below.

In the first 25 million terms the only term where a(n) = 1 is the initial term. In the same range many small values do not appear, e.g., 2, 4, 7, 9, 11, 12, ... . It is unknown if these, and eventually all, numbers are visited. The first numbers to be repeated are 3, 65, 983, 60, 78228, 46254, 540140, ... . In the first 25 million terms the longest gap between prime divisions is 124970 terms, ending at a(20217061) = 47062257110333, which is divisible by 19483.

			a(3) = 6 as a(2) = 3, which is not divisible by the current dividing prime 2, and the next additive prime is 3, so a(3) = 3 + 3 = 6.
a(4) = 3 as a(3) = 6, the current dividing prime is 2, and 6/2 = 3. As 3 is not divisible by 2, the divisions by 2 stop, and the dividing prime now becomes 3.
a(5) = 8 as a(4) = 3 and the next additive prime is 5, so a(5) = 3 + 5 = 8.
a(6) = 15 as a(5) = 8, which is not divisible by 3, and the next additive prime is 7, so a(6) = 8 + 7 = 15.
a(7) = 5 as a(6) = 15, the current dividing prime is 3, and 15/3 = 5. As 5 is not divisible by 3, the divisions by 3 stop, and the dividing prime now becomes 5.
a(8) = 16 as a(7) = 5 and the next additive prime is 11, so a(8) = 5 + 11 = 16.
a(446) = 22090, a(447) = 470, a(448) = 10. This is the first time that the current term and the resulting quotient are both divisible by the current dividing prime, 47 in this case. The current additive prime is 3011, so a(449) = 3021. Coincidently 3021 is divisible by the next dividing prime 51, so a(450) = 57. This is the shortest possible gap between divisions by different primes.

Scott R. Shannon, Image of the first 100000 terms.

Cf. A350877, A351101, A000040, A350615.

cp's OEIS Frontend

A350615 Indices of 1's in A350877.

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A350616 Indices of odd terms in A350877.

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A350618 Terms in A350877 that immediately follow an odd term.

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A350620 a(n) = smallest k such that A350877(k) = n, or -1 if n does not appear in A350877.

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A350617 Odd-valued terms in A350877.

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A350619 Record high values in A350877.

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A350621 a(n) is the smallest k such that A350877(k) = prime(n), or -1 if prime(n) does not appear in A350877.

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A350833 Run lengths of even terms in A350877 (half if even, add next prime if odd).

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Formula

A351101 Variation of the Sisyphus sequence A350877: the same rules apply except that each time a(n) is divided by a prime the dividing prime is incremented to the next prime while the prime being added to each term is reset to 2.

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A351102 Variation of the Sisyphus sequence A350877: the same rules apply except that each time a(n) is divided by a prime the dividing prime is incremented to the next prime.

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