A339403 -id:A339403 - 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.

A340917 Integers m that have at least one divisor d such that reverse(d+m/d) is a substring of m.

Original entry on oeis.org

4, 72, 81, 94, 114, 130, 132, 148, 168, 204, 231, 236, 245, 272, 294, 414, 448, 456, 498, 518, 585, 594, 756, 792, 836, 867, 936, 988, 994, 1056, 1127, 1170, 1210, 1221, 1271, 1281, 1380, 1478, 1608, 1680, 1748, 1768, 1782, 1798, 1887, 1914, 1930, 1938, 1948, 1960

Offset: 1

Michel Marcus, Jan 26 2021

These are the resulting strings in A339403.

			204 = 6*34 contains reverse(6+34) = reverse(40) = 04 as a substring, so 204 is a term.

Cf. A339403, A340916.

q[n_] := AnyTrue[Divisors[n], SequenceCount[IntegerDigits[n], Reverse @ IntegerDigits[# + n/#]] > 0 &]; Select[Range[2000], q] (* Amiram Eldar, Jan 26 2021 *)

isok(n) = {fordiv(n, d, if (#strsplit(Str(n), concat(Vecrev(Str(d+n/d)))) > 1, return(1)); if (d^2 > n, return(0)););}

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.

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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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A340917 Integers m that have at least one divisor d such that reverse(d+m/d) is a substring of m.

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Mathematica

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