A333410 -id:A333410 - cp's OEIS frontend

A341035 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n), in both forward and reverse directions, as a substring. If no such number exists then a(n) = -1.

-1, -1, -1, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 35, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 50, 50, 50, 50, 50, 55, 50, 51, 52, 53, 54, 60, 60, 60, 60, 65, 50, 50, 65, 65, 70, 70, 70

Offset: 1

Scott R. Shannon, Feb 03 2021

Based on a search limit of 5*10^9 up to n = 300000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which no a(n) exists.

The longest run of consecutive terms with the same value in the first 300000 terms is the run of 5's at the beginning of the sequence, ten in all. This is likely the longest run for all numbers.

			a(5) = 5 as 5+5 = 10 which contains both 5-5 = 0 and reverse(0) = 0 as a substring.
a(15) = 10 as 15+10 = 25 which contains both 15-10 = 5 and reverse(5) = 5 as a substring.
a(61) = 50 as 61+50 = 111 which contains both 51-50 = 11 and reverse(11) = 11 as a substring.
a(71) = 50 as 71+50 = 121 which contains both 71-50 = 21 and reverse(21) = 12 as a substring.
a(1902) = 1829 as 1902+1829 = 3731 which contains both 1902-1829 = 73 and reverse(73) = 37 as a substring.

Scott R. Shannon, Image of the terms for n=5..100000.

Cf. A341034 (forward), A341028 (reverse), A339403, A339144, A328095, A333410, A332703.

A341028 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) in reverse as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, -1, -1, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 15, 15, 9, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 9, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 9, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 9, 50, 50, 50, 50, 55, 41, 51, 52, 53, 54, 9, 60, 60, 60, 65, 50, 32, 52, 53, 54, 70, 9

Offset: 1

Scott R. Shannon, Feb 02 2021

Based on a search limit of 5*10^9 up to n = 300000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which a(n) = -1.
The longest run of consecutive terms with the same value in the first 300000 terms is the run of 5's at the beginning of the sequence, ten in all. This is likely the longest run for all numbers.
Numerous patterns exist in the values of a(n), e.g., when a(n) consists of all 9's and n is not a power of 10 then n is palindromic.

			a(5) = 5 as 5+5 = 10 which contains reverse(5-5) = reverse(0) = 0 as a substring.
a(6) = 5 as 6+5 = 11 which contains reverse(6-5) = reverse(1) = 1 as a substring.
a(15) = 10 as 15+10 = 25 which contains reverse(15-10) = reverse(5) = 5 as a substring.
a(22) = 9 as 22+9 = 31 which contains reverse(22-9) = reverse(13) = 31 as a substring.

Scott R. Shannon, Image of the terms for n = 5..300000. It is possible the frame-like pattern continues to larger and larger scales resulting in a fractal structure.

Cf. A341034 (forward), A341035 (forward and reverse), A339403, A339144, A328095, A333410, A332703.

A341034 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, -1, -1, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 35, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50

Offset: 1

Scott R. Shannon, Feb 03 2021

Based on a search limit of 5*10^9 up to n = 200000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which no a(n) exists.

The sequence contains long runs of consecutive terms with the same value, resulting in the image for the values having a staircase-like pattern. In the first 200000 terms the longest run is 88890 terms, starting from a(61110), all of which have a(n) = 50000.

			a(5) = 5 as 5+5 = 10 which contains 5-5 = 0 as a substring.
a(6) = 5 as 6+5 = 11 which contains 6-5 = 1 as a substring.
a(15) = 10 as 15+10 = 25 which contains 15-10 = 5 as a substring.
a(35) = 29 as 35+29 = 64 which contains 35-29 = 6 as a substring.

Scott R. Shannon, Line graph of the terms for n=5..6111.

Cf. A341028 (reverse), A341035 (forward and reverse), A339403, A339144, A328095, A333410, A332703.

A369899 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is a substring of the sum of all previous terms.

Original entry on oeis.org

1, 2, 3, 6, 12, 4, 8, 36, 7, 9, 88, 17, 19, 21, 23, 5, 26, 28, 15, 30, 60, 20, 40, 48, 52, 58, 38, 67, 43, 78, 64, 92, 10, 103, 11, 14, 115, 27, 13, 31, 34, 37, 41, 45, 50, 51, 16, 18, 63, 69, 68, 83, 91, 201, 22, 33, 66, 32, 236, 260, 86, 29, 75, 305, 35, 39, 42, 47, 351, 386, 25, 80, 360, 72

Offset: 1

Scott R. Shannon, Feb 05 2024

The fixed points begin 1, 2, 3, 94, 1420, 1423, 1425, 1426, 1427, 8592, although it is likely there are infinitely more. The sequence is conjectured to be a permutation of the positive numbers.

			a(6) = 4 as the sum of all previous terms is 1 + 2 + 3 + 6 + 12 = 24, and 4 is the smallest unused number that is a substring of "24".

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 50000 terms.

Cf. A370046 (base 2), A363186, A333410.

from itertools import islice
def agen(): # generator of terms
    s, mink, aset = 3, 3, {1, 2}
    yield from [1, 2]
    while True:
        an, ss = mink, str(s)
        while an in aset or not str(an) in ss: an += 1
        aset.add(an); s += an; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Feb 08 2024

A370046 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number whose binary value is a substring of the binary value of the sum of all previous terms.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 9, 5, 18, 17, 10, 7, 19, 14, 16, 11, 20, 13, 24, 22, 15, 32, 36, 34, 25, 23, 37, 27, 21, 26, 64, 69, 40, 43, 29, 30, 35, 39, 44, 28, 42, 53, 129, 72, 38, 31, 81, 45, 50, 46, 47, 49, 74, 41, 54, 55, 51, 52, 57, 58, 128, 68, 70, 140, 77, 60, 139, 85, 33, 75, 61, 59, 62, 48

Offset: 1

Scott R. Shannon, Feb 08 2024

The fixed points begin 1, 2, 3, 16, 39, 42, 50, 79, 120, 361, although it is likely there are infinitely more. The sequence is conjectured to be a permutation of the positive numbers.

			a(7) = 12 as the sum of all previous terms is 1 + 2 + 3 + 6 + 4 + 8 = 24 = 11000_2 and 12 = 1100_2 is the smallest unused number that is a substring of "11000".

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 50000 terms.

Cf. A317788, A369899 (base 10), A363186, A333410.

from itertools import islice
def agen(): # generator of terms
    s, mink, aset = 3, 3, {1, 2}
    yield from [1, 2]
    while True:
        an, ss = mink, bin(s)[2:]
        while an in aset or not bin(an)[2:] in ss: an += 1
        aset.add(an); s += an; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, Feb 08 2024

a(n) = A317788(n) for any n >= 3. - Rémy Sigrist, Feb 09 2024

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A341035 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n), in both forward and reverse directions, as a substring. If no such number exists then a(n) = -1.

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A341028 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) in reverse as a substring. If no such number exists then a(n) = -1.

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A341034 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) as a substring. If no such number exists then a(n) = -1.

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A369899 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is a substring of the sum of all previous terms.

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A370046 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number whose binary value is a substring of the binary value of the sum of all previous terms.

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