A352808 -id:A352808 - cp's OEIS frontend

A353731 a(n) = index where prime(n) appears in A352808.

2, 6, 4, 11, 18, 19, 16, 24, 45, 52, 63, 29, 30, 78, 122, 71, 123, 148, 46, 72, 53, 162, 96, 66, 100, 161, 243, 212, 194, 186, 394, 75, 86, 110, 114, 177, 204, 111, 254, 295, 273, 274, 494, 124, 167, 386, 338, 642, 339, 410, 347, 650, 351, 762, 159, 180, 195, 250, 191, 208, 225, 270, 284, 595, 340, 689, 290, 238, 732, 730, 258, 651, 696, 624, 749, 1062, 271, 311, 300, 326, 666, 625, 1006, 634, 1007, 1050, 269, 738, 759, 970, 898, 1443, 1122, 1123, 1139, 1570, 1483

Offset: 1

N. J. A. Sloane, May 17 2022

nonn

Here prime(n) means the n-th prime in 2, 3, 5, 7, ..., not the n-th term of A352808 that happens to be a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A352808, A353732.

from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    A352808lst = [0, 1]; A352808set = {0, 1}
    mink = p = 2
    for n in count(2):
        ahalf, k = A352808lst[n//2], mink
        while k in A352808set or k&ahalf: k += 1
        A352808lst.append(k); A352808set.add(k)
        while mink in A352808set: mink += 1
        while p in A352808set: yield A352808lst.index(p); p = nextprime(p)
print(list(islice(agen(), 76))) # Michael S. Branicky, May 17 2022

a(n) = k if A352808(k) = prime(n). - Michael S. Branicky, May 17 2022

A353732 Inverse permutation to A352808: a(n) = index where n appears in A352808.

Original entry on oeis.org

0, 1, 2, 6, 3, 4, 10, 11, 5, 7, 8, 18, 12, 19, 32, 44, 9, 16, 14, 24, 13, 17, 15, 45, 20, 21, 50, 56, 51, 52, 62, 63, 22, 25, 26, 27, 28, 29, 33, 40, 23, 30, 34, 78, 48, 70, 79, 122, 36, 64, 38, 84, 37, 71, 82, 98, 83, 85, 94, 123, 92, 148, 134, 210, 31, 41, 35, 46, 42, 47, 43, 72, 49, 53, 57, 73, 54, 76, 128, 162, 39, 65, 58, 96, 55, 68, 60

Offset: 0

N. J. A. Sloane, May 17 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A353808, A353731.

from itertools import count, islice
def agen(): # generator of terms
    A352808lst = [0, 1]; A352808set = {0, 1}; yield from [0, 1]
    mink = 2
    for n in count(2):
        ahalf, k = A352808lst[n//2], mink
        while k in A352808set or k&ahalf: k += 1
        A352808lst.append(k); A352808set.add(k)
        while mink in A352808set: yield A352808lst.index(mink); mink += 1
print(list(islice(agen(), 87))) # Michael S. Branicky, May 17 2022

A354142 a(n) = smallest number missing from A352808 after A352808(n) has been found.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 6, 6, 6, 6, 7, 11, 11, 11, 11, 11, 11, 11, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 23, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 30, 30, 30, 30, 30, 30, 31, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43

Offset: 0

N. J. A. Sloane, May 18 2022

nonn

			A352808 (with offset 0) begins 0,1,2,4,5,8; after A352808(0) = 0 has been found, the smallest missing number is 1, so a(0) = 1; after A352808(5) = 8 has been found, the smallest missing number is 3, so a(5) = 3.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A352808.

from itertools import count, islice
def agen(): # generator of terms
    A352808lst, A352808set, mink = [0], {0}, 1
    for n in count(1):
        yield mink
        ahalf, k = A352808lst[n//2], mink
        while k in A352808set or k&ahalf: k += 1
        A352808lst.append(k); A352808set.add(k)
        while mink in A352808set: mink += 1
print(list(islice(agen(), 78))) # Michael S. Branicky, May 18 2022

A354141 Indices of terms in A352808 that are powers of 2.

Original entry on oeis.org

1, 2, 3, 5, 9, 22, 31, 61, 121, 247, 479, 951, 1862, 3802, 7431, 15180, 29723, 59766, 118893, 239999, 475573, 959341, 1902293, 3835229, 7609175, 15268473, 30436701, 61001391

Offset: 1

N. J. A. Sloane, May 18 2022

Every power of 2 will eventually appear in A353730, so the sequence is infinite.

Rémy Sigrist, C++ program

Cf. A352808, A353734.

from itertools import count, islice
def ispow2(k): return bin(k).count("1") == 1
def agen(): # generator of terms
    A352808lst = [0, 1]; A352808set = {0, 1}
    k, mink, p = 1, 2, 2
    for n in count(2):
        if ispow2(k): yield n-1
        ahalf, k = A352808lst[n//2], mink
        while k in A352808set or k&ahalf: k += 1
        A352808lst.append(k); A352808set.add(k)
        while mink in A352808set: mink += 1
print(list(islice(agen(), 8))) # Michael S. Branicky, May 18 2022
(C++) See Links section.

a(16)-a(22) from Michael S. Branicky, May 19 2022

a(23)-a(28) from Rémy Sigrist, May 21 2022

A353733 a(0)=0, a(1)=1; for k >= 1, a(2k+1) and a(2k+2) are the two smallest numbers not yet in the sequence whose binary expansions have no 1's in common with the binary expansion of a(k).

Original entry on oeis.org

0, 1, 2, 4, 6, 5, 8, 3, 9, 16, 17, 10, 18, 7, 19, 12, 20, 22, 32, 11, 13, 14, 34, 21, 33, 36, 37, 24, 40, 44, 64, 35, 48, 41, 42, 65, 72, 15, 23, 52, 68, 50, 66, 49, 80, 25, 28, 74, 96, 26, 30, 27, 67, 82, 88, 38, 39, 69, 70, 81, 83, 29, 31, 76, 84, 71, 73, 86

Offset: 0

N. J. A. Sloane, May 17 2022

A variant of A352808.

This is a permutation of the nonnegative numbers (the proof is similar to that for A352808).

			For k=2, after a(2) = 2 = 10_2, we get a(5) = 5 = 101_2 and a(6) = 8 = 1000_2 since 101_2, 1000_2 have no 1's in common with 10_2.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A352808.

from itertools import count, islice
def agen(): # generator of terms
    alst = [0, 1]; aset = {0, 1}; yield from alst
    mink = 2
    for n in count(2):
        ahalf, k = alst[(n-1)//2], mink
        while k in aset or k&ahalf: k += 1
        alst.append(k); aset.add(k); yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 68))) # Michael S. Branicky, May 17 2022

More terms from Michael S. Branicky, May 17 2022

A354143 a(n) = smallest number missing from A353733 after A353733(n) has been found.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 23, 25, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 43, 43, 43, 43, 43, 43

Offset: 0

N. J. A. Sloane, May 18 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A353733, A352808, A354142.

from itertools import count, islice
def agen(): # generator of terms
    A353733lst, A353733set, mink = [0], {0}, 1
    for n in count(1):
        yield mink; A353733half, k = A353733lst[(n-1)//2], mink
        while k in A353733set or k&A353733half: k += 1
        A353733lst.append(k); A353733set.add(k)
        while mink in A353733set: mink += 1
print(list(islice(agen(), 68))) # Michael S. Branicky, May 18 2022

cp's OEIS Frontend

A353731 a(n) = index where prime(n) appears in A352808.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A353732 Inverse permutation to A352808: a(n) = index where n appears in A352808.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A354142 a(n) = smallest number missing from A352808 after A352808(n) has been found.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A354141 Indices of terms in A352808 that are powers of 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Extensions

A353733 a(0)=0, a(1)=1; for k >= 1, a(2*k+1) and a(2*k+2) are the two smallest numbers not yet in the sequence whose binary expansions have no 1's in common with the binary expansion of a(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions

A354143 a(n) = smallest number missing from A353733 after A353733(n) has been found.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A353733 a(0)=0, a(1)=1; for k >= 1, a(2k+1) and a(2k+2) are the two smallest numbers not yet in the sequence whose binary expansions have no 1's in common with the binary expansion of a(k).