A353989 -id:A353989 - cp's OEIS frontend

A353245 a(n) = A353989(n) AND A353989(n+1), where AND is the binary AND operator.

1, 2, 2, 2, 8, 8, 4, 4, 6, 5, 1, 9, 5, 4, 16, 16, 18, 2, 1, 1, 24, 24, 8, 5, 6, 24, 1, 32, 8, 4, 32, 32, 34, 2, 17, 40, 40, 44, 6, 5, 5, 32, 48, 48, 52, 50, 17, 17, 21, 52, 56, 48, 49, 2, 2, 24, 21, 64, 8, 30, 29, 73, 65, 64, 64, 64, 64, 72, 76, 64, 80, 80, 68, 72, 80, 2, 1, 1, 80, 88, 92, 14

Offset: 1

Scott R. Shannon, May 15 2022

nonn

See A353989 for further details. A graph of the terms displays a repetitive structure below the line y = n. See the linked images.

			a(2) = 2 as A353989(2) = 3 = 11_2 and A353989(3) = 6 = 110_2, and the binary AND of 11_2 and 110_2 = 10_2 = 2.

Scott R. Shannon, Image of the first 10000 terms. The green line is y = n.
Scott R. Shannon, Image of the first 100000 terms.

Cf. A353989, A129760, A064413.

A352763 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 6, 9, 18, 12, 3, 24, 32, 10, 5, 40, 16, 14, 48, 15, 80, 34, 17, 68, 26, 36, 27, 96, 20, 35, 28, 64, 22, 33, 30, 65, 50, 72, 21, 42, 69, 138, 52, 66, 44, 82, 41, 656, 38, 88, 128, 46, 144, 39, 192, 45, 130, 13, 208, 256, 54, 129, 60, 194, 56, 7, 112, 132, 11, 176, 70, 25, 100, 136, 51

Offset: 1

Scott R. Shannon, May 15 2022

nonn

This sequence is similar to the EKG sequence A064413 with the additional restriction that no term can have a 1-bit in common with the previous term in their binary expansions. These restrictions lead to numerous terms being much larger than their preceding term, while the smaller terms overall show similar behavior to A109812. See the linked image. Unlike A064413 the primes do not occur in their natural order and the term following a prime can be a very large multiple of the prime.

In the first 50000 terms the fixed points are 1, 2, 105, 135, 225, 2157, 3972, 7009, 8531, although it is likely more exist. In the same range the lowest unseen number is 383; the sequence is conjectured to be a permutation of the positive integers.

			a(5) = 6 as a(4) = 8, 6 = 110_2, 8 = 1000_2, and 6 is the smallest unused number that shares a common factor with 8 but has no 1-bit in common with 8 in their binary expansions.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 10000 terms for values below 15000. The green line is y = n. The maximum value in this range is a(8045) = 959024.

Cf. A353989, A109812, A064413, A152458, A353712, A352633.

A354087 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and whose binary expansion has a single 1-bit in common with the binary expansion of a(n-1).

Original entry on oeis.org

1, 3, 6, 2, 10, 8, 12, 4, 14, 18, 15, 20, 5, 25, 35, 21, 9, 24, 16, 22, 11, 33, 27, 48, 26, 13, 52, 32, 34, 30, 36, 28, 7, 42, 49, 77, 56, 38, 19, 133, 57, 69, 46, 66, 39, 65, 45, 50, 40, 54, 68, 44, 70, 58, 72, 60, 74, 64, 76, 80, 55, 88, 96, 51, 78, 81, 102, 130, 62, 132, 63, 129, 43, 86, 104, 82

Offset: 1

Scott R. Shannon, May 17 2022

This sequence is similar to the EKG sequence A064413 with the additional restriction that each term must share a single 1-bit in common with the previous term in their binary expansions. These restrictions lead to numerous terms being significantly larger than their preceding term, while the smaller terms overall show similar behavior to A109812. See the linked image. Unlike A064413 the primes do not occur in their natural order and both the proceeding and following terms of the primes can be large multiples of the prime.

In the first 100000 terms the fixed points are 1, 3, 30, 38, 350, 1603, 1936, 10176, 11976, 46123, 58471, 89870, although it is likely more exist. In the same range the lowest unseen number is 1019; the sequence is conjectured to be a permutation of the positive integers.

			a(6) = 8 as a(5) = 10, 8 = 1000_2, 10 = 1010_2, and 8 is the smallest unused number that shares a common factor with 10 and has a single 1-bit in common with 10 in their binary expansions. Note that 4 satisfies the first criterion but not the second.

Scott R. Shannon, Image of the first 100000 terms for values less than 200000. The green line is y = n.

Cf. A353989, A064413, A353245, A152458, A353712, A352633.

A359799 a(1) = 1, a(2) = 3; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with |a(n-1) - a(n-2)| while the difference |a(n) - a(n-1)| is distinct from all previous differences |a(i) - a(i-1)|, i=2..n-1.

Original entry on oeis.org

1, 3, 6, 12, 2, 10, 14, 26, 4, 11, 28, 17, 22, 35, 65, 5, 20, 36, 8, 32, 9, 23, 42, 76, 18, 38, 56, 15, 41, 16, 25, 54, 87, 21, 48, 27, 63, 24, 66, 7, 59, 13, 44, 93, 49, 84, 30, 62, 100, 19, 69, 106, 37, 90, 212, 34, 74, 122, 33, 89, 46, 129, 249, 39, 86, 141, 40, 101, 183, 50, 95, 159, 52

Offset: 1

Scott R. Shannon, Mar 07 2023

nonn

In the first 100000 terms the only fixed point is a(1) = 1; it is unknown if more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(5) = 2 as |a(4) - a(3)| = |12 - 6| = 6, and 2 is the smallest unused number that shares a factor with 6 while the difference |2 - a(4)| = |2 - 12| = 10 is distinct from all previous differences.

Scott R. Shannon, Image for n = 1..100000. The green line is a(n) = n.

Cf. A361314, A337136, A354755, A354727, A354687, A354753, A353989, A354087, A352763.

A354755 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 3, 9, 6, 8, 8, 10, 10, 12, 9, 15, 10, 16, 12, 15, 15, 18, 14, 20, 15, 21, 18, 20, 20, 22, 22, 24, 21, 27, 24, 26, 26, 28, 21, 35, 20, 38, 19, 57, 3, 60, 2, 62, 4, 64, 6, 63, 9, 66, 8, 70, 7, 77, 11, 88, 2, 78, 3, 84, 2, 80, 5, 60, 32, 62, 31, 93, 3, 99, 6, 92, 8, 96, 10, 85, 25

Offset: 1

Scott R. Shannon, Jun 06 2022

In the first 500000 terms the fixed points are 1,2,4,6,2388,2390,2392,2394; it is likely no more exist. In the same range many numbers do not appear, the lowest five being 59,67,73,89,97. It is possible these and many other numbers never appear although this is unknown.

			a(7) = 3 as a(6) = 6, and 3 is the smallest number that shares a factor with 6 and whose sum with the previous term, 6 + 3 = 9, has not appeared. Note 2 shares a factor with 6 but 6 + 2 = 8, and a sum of 8 has already occurred with a(4) + a(5) = 4 + 4 = 8, so 2 cannot be chosen.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n) n = 1..2^14, showing records in red and a(n) = 2 in blue, highlighting fixed points in gold.
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A064413, A354727, A354687, A354753, A353989, A354087, A352763

nn = 120; c[] = 0; a[1] = c[1] = 1; a[2] = j = 2; c[3] = 2; Do[k = 2; While[Nand[c[j + k] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[n], c[j + k]}, {k, n}]; j = k, {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jun 15 2022 *)

lista(nn) = my(va = vector(nn), vs = vector(nn-2)); va[1] = 1; va[2] = 2; for (n=3, nn, my(k=2); while ((gcd(k, va[n-1]) == 1) || #select(x->(x==k+va[n-1]), vs), k++); va[n] = k; vs[n-2] = k+va[n-1];); va; \\ Michel Marcus, Jun 15 2022

A353990 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1), does not equal a(n-1)+1, and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 9, 16, 7, 24, 35, 12, 17, 6, 25, 32, 11, 20, 33, 10, 21, 34, 13, 18, 37, 26, 69, 40, 19, 36, 65, 14, 81, 38, 73, 22, 41, 64, 15, 112, 129, 28, 67, 44, 83, 128, 23, 72, 49, 66, 29, 96, 31, 160, 27, 68, 43, 80, 39, 88, 131, 48, 71, 56, 135, 104, 133, 50, 77, 130, 53, 74, 145, 42

Offset: 1

Scott R. Shannon, May 13 2022

This sequence is similar to A093714 with the additional restriction that no term can have a 1-bit in common with the previous term in their binary expansions. This leads to the terms showing similar behavior to A109812. See the linked image.

In the first 100000 terms the fixed points are 1, 3, 5, 12, 21, 26, 44, 49, 227, 3488, 5890, 9067, 9310, 37625, 74702, although it is likely more exist. In the same range the lowest unseen number is 30686; the sequence is conjectured to be a permutation of the positive integers.

			a(4) = 8 as a(3) = 3, and 8 has not yet appeared, is coprime to 3, is not 1 more than 3, while 8 = 1000_2 and 3 = 11_2 which have no 1-bits in common.

Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A093714, A109812, A353989, A352763.

from math import gcd
from itertools import count, islice
def A353990_gen(): # generator of terms
    yield 1
    a, s, b = 1, 2, set()
    while True:
        for i in count(s):
            if not (i == a+1 or i & a or gcd(i,a) > 1 or i in b):
                yield i
                a = i
                b.add(i)
                while s in b:
                    s += 1
                break
A353990_list = list(islice(A353990_gen(),30)) # Chai Wah Wu, May 24 2022

A355621 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with a(n-1) in their binary expansions.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 5, 6, 5, 8, 1, 8, 2, 4, 5, 11, 15, 17, 12, 11, 19, 17, 15, 23, 22, 19, 22, 21, 24, 16, 10, 18, 20, 21, 29, 33, 22, 30, 33, 23, 38, 31, 42, 28, 35, 37, 38, 37, 40, 22, 41, 40, 24, 33, 35, 46, 49, 49, 50, 47, 59, 60, 55, 61, 62, 61, 64, 1, 39, 63, 69, 58, 60, 64, 3, 60, 65, 46, 67

Offset: 1

Scott R. Shannon, Jul 10 2022

The indices where a(n) = 1 in the first 500000 terms are 1, 2, 4, 11, 68, 131, 2051, 4099. It is unknown if more exist. Many terms of the sequence are close to the line a(n) = n although only the first term is a possible fixed point. In the first 500000 terms the lowest values not to appear are 7, 9, 14, 25, 26. It is likely these and other numbers never appear although this is unknown.

			a(7) = 5 as a(6) = 5 and the total number of terms in the first six terms that share a 1-bit with 5 in their binary expansions is five, namely 1, 1, 1, 3, 5.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A355625, A030190, A129760, A353989, A352763, A354606.

from itertools import count, islice
def agen():
    an, alst = 1, [1]
    for n in count(2):
        yield an
        an = sum(1 for k in alst if k&an)
        alst.append(an)
print(list(islice(agen(), 79))) # Michael S. Branicky, Jul 10 2022

A355625 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with n in their binary expansions.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 4, 0, 3, 2, 6, 2, 6, 7, 11, 0, 6, 9, 13, 6, 13, 13, 18, 6, 11, 17, 21, 16, 21, 22, 26, 0, 14, 16, 26, 14, 23, 25, 31, 12, 22, 27, 34, 27, 33, 34, 39, 19, 31, 35, 43, 36, 44, 44, 49, 36, 42, 48, 52, 47, 52, 53, 57, 0, 29, 32, 48, 30, 48, 48, 57, 25, 41, 46, 56, 47, 57, 58, 65, 34

Offset: 1

Scott R. Shannon, Jul 10 2022

The indices where a(n) = 1 in the first 500000 terms are 1, 3, 6. It is likely no more exist although this is unknown. Many terms of the sequence are close to the line a(n) = n although only the first term is a possible fixed point. In the first 500000 terms the lowest values not to appear are 5, 8, 10, 15, 20, 24, 28. It is likely these and other numbers never appear although this is unknown. All terms for n > 1 where n is a power of 2 equal 0.

			a(7) = 4 as the total number of terms in the first six terms that share a 1-bit with 7 in their binary expansions is four, namely 1, 1, 2, 1.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A355621, A030190, A129760, A353989, A352763, A354606.

from itertools import count, islice
def agen():
    an, alst = 1, [1]
    for n in count(2):
        yield an
        an = sum(1 for k in alst if k&n)
        alst.append(an)
print(list(islice(agen(), 80))) # Michael S. Branicky, Jul 10 2022

A361314 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with a(n-2) + a(n-1) while the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 29, 12, 41, 53, 14, 67, 15, 16, 31, 47, 13, 20, 18, 22, 24, 26, 25, 17, 27, 28, 11, 21, 36, 30, 32, 38, 34, 39, 73, 35, 33, 42, 45, 48, 51, 44, 40, 46, 43, 89, 50, 139, 49, 52, 101, 54, 55, 109, 56, 57, 113, 58, 60, 59, 63, 61, 62, 66, 64, 65, 69, 68, 137, 70

Offset: 1

Scott R. Shannon, Mar 08 2023

nonn

In the first 100000 terms the fixed points are 1, 2, 3, 6, 9, 10, 39, 91, 112; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(23) = 20 as a(21) + a(22) = 47 + 13 = 60, and 20 is the smallest unused number that shares a factor with 60 while the sum a(22) + 20 = 13 + 20 = 33 is distinct from all previous sums. Note that 18 is unused and shares a factor with 60 but the sum a(22) + 18 = 13 + 18 = 31 is the same as a(18) + a(19) = 15 + 16 = 31. This is the first term that differs from A337136.

Scott R. Shannon, Image for n = 1..100000. The green line is a(n) = n.

Cf. A337136, A359799, A354755, A354727, A354687, A354753, A353989, A354087, A352763.

A375828 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that (a(n-1) AND a(n)) = a(n-1) if a(n-1) is prime, otherwise (a(n-1) AND a(n)) = 0, where AND is the binary AND operation.

Original entry on oeis.org

1, 2, 3, 7, 15, 16, 4, 8, 5, 13, 29, 31, 63, 64, 6, 9, 18, 12, 17, 19, 23, 55, 72, 20, 10, 21, 32, 11, 27, 36, 24, 33, 14, 48, 65, 22, 40, 66, 25, 34, 28, 35, 68, 26, 37, 39, 80, 38, 73, 75, 52, 67, 71, 79, 95, 128, 30, 96, 129, 42, 69, 50, 76, 49, 70, 41, 43, 47, 111, 144, 44, 81, 46, 145, 74, 53, 61, 125, 130, 45, 82, 132, 51, 136, 54, 137, 139, 143, 112

Offset: 1

Scott R. Shannon, Aug 30 2024

nonn

The terms form a pattern similar to that in A109812. In the first 250000 terms the fixed points are 1, 2, 3, 8, 3650, 50624, 203074. The sequence is conjectured to be a permutation of the positive integers.

			a(5) = 15 as a(4) = 7 = 111_2 is prime, and ((15 = 1111_2) AND 111_2) = 111_2 = 7.
a(6) = 16 as a(5) = 15 = 1111_2 is not prime, and ((16 = 10000_2) AND 1111_2) = 0.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.

Cf. A375829, A109812, A007088, A115510, A000040, A353989.

cp's OEIS Frontend

A353245 a(n) = A353989(n) AND A353989(n+1), where AND is the binary AND operator.

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A352763 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

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A354087 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and whose binary expansion has a single 1-bit in common with the binary expansion of a(n-1).

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A359799 a(1) = 1, a(2) = 3; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with |a(n-1) - a(n-2)| while the difference |a(n) - a(n-1)| is distinct from all previous differences |a(i) - a(i-1)|, i=2..n-1.

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A354755 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

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PARI

A353990 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1), does not equal a(n-1)+1, and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

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A355621 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with a(n-1) in their binary expansions.

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A355625 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with n in their binary expansions.

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A361314 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with a(n-2) + a(n-1) while the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

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