A350877 -id:A350877 - cp's OEIS frontend

A362107 Exponents of powers of 2 in Sisyphus sequence A350877 that start a maximal chain of powers of 2 ending at a 1.

0, 3, 3, 4, 11, 32

Offset: 1

N. J. A. Sloane, Apr 13 2023

See A362108 for the indices of these terms in A350877.

			Terms 730-742 in A350877 are 505, 2^11, 2^10,..., 8, 4, 2, 1, corresponding to a(5) = 11 and A362108(5) = 731.

Cf. A350877, A350615, A362108.

A362108 Index of 2^k in A350877, where k = A362107(n).

Original entry on oeis.org

1, 5, 9, 16, 731, 513152096

Offset: 1

N. J. A. Sloane, Apr 13 2023

			Terms 730-742 in A350877 are 505, 2^11, 2^10,..., 8, 4, 2, 1, corresponding to A362107(5) = 11 and a(5) = 731.

f. A350877, A350615, A362107.

A362105 Numbers that take a record number of steps to appear in the Sisyphus sequence A350877.

Original entry on oeis.org

1, 2, 5, 13, 25, 27, 36, 97, 115

Offset: 1

N. J. A. Sloane, Apr 12 2023, based on a suggestion from Siobhan Roberts

It is conjectured that every number will eventually appear in A350877. The terms in this sequence are the numbers that are the slowest to appear. For example, 115 does not appear until term 45274461582754.

Cf. A350877, A362106. Records in A350620.

A362106 Index where A362105(n) first appears in the Sisyphus sequence A350877.

Original entry on oeis.org

1, 7, 71, 345, 1154, 161336, 77534485877, 17282073747557, 45274461582754

Offset: 1

N. J. A. Sloane, Apr 12 2023, based on a suggestion from Siobhan Roberts

			For example, A362105(9) = 115 does not appear in A350877 until term 45274461582754, so a(9) = 45274461582754.

Cf. A350877, A362105. Indices of records in A350620.

A347296 a(1) = 1; for n >= 1, if a(n) is even then a(n+1) = a(n) / 2, otherwise, say a(n) is the k-th odd term in the sequence, a(n+1) = a(n) + a(k).

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 4, 2, 1, 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1, 6, 3, 10, 5, 13, 17, 19, 20, 10, 5, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 15, 18, 9, 19, 24, 12, 6, 3, 16, 8, 4, 2, 1, 18, 9, 28, 14, 7, 27, 37, 42, 21, 31, 36, 18, 9, 21, 27, 30, 15, 26, 13, 28

Offset: 1

Rémy Sigrist, Jan 23 2022

nonn

This sequence is a variant of A350877; here we add terms of the sequence, there prime numbers.

			a(1) = 1.
a(2) = a(1) + a(1) = 2 as a(1) is the 1st odd term.
a(3) = a(2) / 2 = 1 as a(2) is even.
a(4) = a(3) + a(2) = 3 as a(3) is the 2nd odd term.
a(5) = a(4) + a(3) = 4 as a(4) is the 3rd odd term.

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

Cf. A347297, A350877.

k=0; for (n=1, #a=vector(75), print1 (a[n]=if (n==1, 1, a[n-1]%2==0, a[n-1]/2, a[n-1]+a[k++])", "))

A347297 a(1) = 1; for n >= 1, if a(n) is even then a(n+1) = a(n) / 2, otherwise, say a(n) is the k-th odd term in the sequence, a(n+1) = a(n) + k.

Original entry on oeis.org

1, 2, 1, 3, 6, 3, 7, 12, 6, 3, 9, 16, 8, 4, 2, 1, 9, 18, 9, 19, 30, 15, 27, 40, 20, 10, 5, 19, 34, 17, 33, 50, 25, 43, 62, 31, 51, 72, 36, 18, 9, 31, 54, 27, 51, 76, 38, 19, 45, 72, 36, 18, 9, 37, 66, 33, 63, 94, 47, 79, 112, 56, 28, 14, 7, 41, 76, 38, 19, 55

Offset: 1

Rémy Sigrist, Jan 23 2022

nonn

This sequence is a variant of A350877; here we add positive integers, there prime numbers.

			a(1) = 1.
a(2) = a(1) + 1 = 2 as a(1) is the 1st odd term in the sequence.
a(3) = a(2) / 2 = 1 as a(2) is even.
a(4) = a(3) + 2 = 3 as a(3) is the 2nd odd term in the sequence.
a(5) = a(4) + 3 = 6 as a(4) is the 3rd odd term in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^16, showing records in red, labeling the indices of 1's in blue, appearances of powers of 2 in gold, and terms instigated by even predecessors in green.

Cf. A347296, A350877.

j = q = 1; {j}~Join~Reap[Do[If[EvenQ[j], k = j/2, k = j + q; q++]; Sow[k]; j = k, {i, 69}]][[-1, -1]] (* Michael De Vlieger, Jan 24 2022 *)

print1 (v=1); for (k=1, 36, print1 (", "v+=k); while (v%2==0, print1 (", "v/=2)))

A350742 Sisyphus with squares: a(1) = 1; thereafter, if a(n-1) is even, a(n) = a(n-1)/2; otherwise a(n) = a(n-1) + k^2, where k^2 is the smallest square (starting from 1) that has not yet been added.

Original entry on oeis.org

1, 2, 1, 5, 14, 7, 23, 48, 24, 12, 6, 3, 39, 88, 44, 22, 11, 75, 156, 78, 39, 139, 260, 130, 65, 209, 378, 189, 385, 610, 305, 561, 850, 425, 749, 1110, 555, 955, 1396, 698, 349, 833, 1362, 681, 1257, 1882, 941, 1617, 2346, 1173, 1957, 2798, 1399, 2299, 3260, 1630

Offset: 1

Allan C. Wechsler, Jan 28 2022

Allan C. Wechsler's initial remarks: (Start)
This variation on A350877 was proposed by Keith F. Lynch on 2022 January 28. Because the former sequence uses prime increments, and because squares grow much faster than primes, Lynch felt that my conjecture about the former sequence (that all integers eventually appear) would not be true of this sequence. I responded that because squares nevertheless grow much more slowly than exponentials, we should still expect the same basic argument, that descents to the vicinity of 1 should happen on the order of once per "era", to apply, and I still thought that all values would appear, albeit perhaps very slowly.
But later, Gareth McCaughan did the "naive" probabilistic calculation more carefully, and I am now convinced that he and Lynch are correct and that the sequence eventually permanently exceeds any given value, most often without ever equaling that value exactly.
Primes > 2 are all odd, so in the former sequence every ascents will be immediately followed by at least one descent. But squares alternate parity, so in this sequence, ascents (after the first) will always occur in pairs, followed by at least one descent. (End)

			Given a(10) = 12, the even rule gives a(11) = 6 and a(12) = 3, and then the odd rule must govern. Because the odd rule has already acted 5 times before a(10), we must add 6^2 = 36, so a(13) = 39. Now the odd rule must act again, giving a(14) = a(13) + 7^2 = 39 + 49 = 88.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..2^16, annotating the first terms, showing records in red and terms preceded by an even number in blue.
Hans Havermann, A350742 minima in the first 150 billion terms.

A variation of the Sisyphus sequence A350877.

j = 1; q = 1; {j}~Join~Reap[Do[If[EvenQ[j], k = j/2, k = j + q^2; q++]; Sow[k]; j = k, {i, 55}]][[-1, -1]] (* Michael De Vlieger, Jan 28 2022 *)

k=0; for (n=1, 56, print1 (v=if (n==1, 1, v%2, v+k++^2, v/2)", ")) \\ Rémy Sigrist, Jan 29 2022

A351278 Indices where A351101(n) = 1.

Original entry on oeis.org

1, 184, 612, 3484, 69262, 170626, 212784, 1310354, 6835786, 14502466, 21133598, 34260294, 80011718

Offset: 1

Scott R. Shannon, Feb 06 2022

See A351101 for further details. It is likely this sequence is infinite although this is unknown.

			a(2) = 184 as A351101(182) = 30, A351101(183) = 41. The additive prime is now 13 while the dividing prime is 41. A351101(183) is divisible by 41 thus A351101(184) = 41/41 = 1. This is the first term in A351101 to return to 1.

Cf. A351101, A350877, A351102, A000040, A350615.

A358166 a(1) = 13; for n > 1, if a(n-1) is even, then a(n) = a(n-1)/2; otherwise, a(n) = a(n-1) + prime(a(n-1)).

Original entry on oeis.org

13, 54, 27, 130, 65, 378, 189, 1318, 659, 5592, 2796, 1398, 699, 5972, 2986, 1493, 13996, 6998, 3499, 36102, 18051, 218932, 109466, 54733, 730334, 365167, 5622764, 2811382, 1405691, 23685544, 11842772, 5921386, 2960693, 52246474, 26123237, 521463688, 260731844, 130365922, 65182961, 1364229390

Offset: 1

Sander G. Huisman, Nov 01 2022

Does this sequence become cyclic? All the sequences defined the same as this one but with 1 <= a(1) <= 12 are known to become cyclic.

a(81) = 1977693361846020549, so calculating a(82) will require calculating the 1977693361846020549th prime.

			a(1) = 13 is odd, so a(2) = 13 + prime(13) = 13 + 41 = 54.
a(2) = 54 is even, so a(3) = a(2)/2 = 54/2 = 27.
a(3) = 27 is odd, so a(4) = 27 + prime(27) = 27 + 103 = 130, etc.

Cf. A293981, A350877, A014688.

NestList[If[EvenQ[#], #/2, # + Prime[#]] &, 13, 40]

lista(nn) = my(va = vector(nn)); va[1] = 13; for (n=2, nn, if (va[n-1] % 2, va[n] = va[n-1] + prime(va[n-1]), va[n] = va[n-1]/2);); va; \\ Michel Marcus, Nov 12 2022

A382822 If a(n-1) is odd, then a(n) is the smallest even integer not yet in the sequence; if a(n-1) is even, then a(n) = a(n-1)/2 if this number is not in the sequence, otherwise a(n) = 3*a(n-1)/2; a(1)=1.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 10, 5, 14, 7, 16, 24, 36, 54, 81, 20, 30, 15, 22, 11, 26, 13, 28, 42, 21, 32, 48, 72, 108, 162, 243, 34, 17, 38, 19, 40, 60, 90, 45, 44, 66, 33, 46, 23, 50, 25, 52, 78, 39, 56, 84, 126, 63, 58, 29, 62, 31, 64, 96, 144, 216, 324, 486, 729, 68, 102, 51, 70, 35, 74, 37, 76, 114, 57

Offset: 1

Enrique Navarrete, Apr 15 2025

nonn

Less repetitive than A350877 (the Sisyphus sequence); presumably a permutation of the positive integers.
Fixed points are 1, 2, 3, 4, 20, 38, 40, 44, 58, 74, 76, 86, 88, 92, 106, ,,., which, aside from 3, all seem to be even.
Largest runs of even terms seem to be of length 6, the first one of which appears at a(62)=64.

Cf. A350877.

cp's OEIS Frontend

A362107 Exponents of powers of 2 in Sisyphus sequence A350877 that start a maximal chain of powers of 2 ending at a 1.

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A362108 Index of 2^k in A350877, where k = A362107(n).

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A362105 Numbers that take a record number of steps to appear in the Sisyphus sequence A350877.

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A362106 Index where A362105(n) first appears in the Sisyphus sequence A350877.

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A347296 a(1) = 1; for n >= 1, if a(n) is even then a(n+1) = a(n) / 2, otherwise, say a(n) is the k-th odd term in the sequence, a(n+1) = a(n) + a(k).

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A347297 a(1) = 1; for n >= 1, if a(n) is even then a(n+1) = a(n) / 2, otherwise, say a(n) is the k-th odd term in the sequence, a(n+1) = a(n) + k.

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A350742 Sisyphus with squares: a(1) = 1; thereafter, if a(n-1) is even, a(n) = a(n-1)/2; otherwise a(n) = a(n-1) + k^2, where k^2 is the smallest square (starting from 1) that has not yet been added.

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A351278 Indices where A351101(n) = 1.

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A358166 a(1) = 13; for n > 1, if a(n-1) is even, then a(n) = a(n-1)/2; otherwise, a(n) = a(n-1) + prime(a(n-1)).

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A382822 If a(n-1) is odd, then a(n) is the smallest even integer not yet in the sequence; if a(n-1) is even, then a(n) = a(n-1)/2 if this number is not in the sequence, otherwise a(n) = 3*a(n-1)/2; a(1)=1.

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