A339144 -id:A339144 - cp's OEIS frontend

A339403 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.

0, -1, 2, 24, 37, 26, 34, 35, 57, 9, -1, 12, 11, 45, 193, 228, 28, 51, 23, 44, 841, 11, 27, 18, 3, 626, 5, 22, 16, 46716, 56, 41, 33, 32, 6, 7, 21, 4, 3, 24, 592, 31, 7, 619, 19, 13, 38, 2, 117, 5, 463, 17, 34, 308, 33, 36, 30, 8, 31, 4, 23, 21, 648, 124, 921, 903, 386, 395, 4, 334, 755, 31, 563

Offset: 0

Scott R. Shannon, Dec 03 2020

This is a variation of A339144 where, instead of the n*a(n) containing n+a(n) as a substring, it contains the reverse of the string n+a(n), including any leading zeros.
Based on a search limit of 5x10^9 up to n = 100000 the values of n for which no a(n) is found are n = 10^k, with k>=0, and 17500. A test of 175000 and 1750000 also found no a(n) indicating that all values of the form 17500*10^k may have no term for a(n).
It is found that when n = 200*10^k, with k>=0, the corresponding value for a(n) is significantly larger than neighboring terms. As an example a(20000) = 666843331, which is the largest term up to n = 100000.
Unlike A339144, which contains multiple consecutive terms with the same value of a(n), in this sequence the largest consecutive run of the same a(n) in the first 100000 terms is only two. The first term of these pairs occurs at n = 110, 121, 2717, 4368, 7916, 10100, 11211, 13231, 17271, 44573, 63529.

			a(3) = 24 as 3*24 = 72 which contains reverse(3+24) = reverse(27) = 72 as a substring.
a(6) = 34 as 6*34 = 204 which contains reverse(6+34) = reverse(40) = 04 as a substring. Note the leading zero is included.
a(29) = 46716 as 29*46716 = 1354764 which contains reverse(29+4671) = reverse(46745) = 54764 as a substring.
a(110) = 11 as 110*11 = 1210 which contains reverse(110+11) = reverse(121) = 121 as a substring. This is the first of two consecutive terms with a(n) = 11.
a(20000) = 666843331 as 20000*666843331 = 13336866620000 which contains reverse(20000+666843331) = reverse(666863331) = 133368666 as a substring. This is the largest value in the first 100000 terms.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A339144, A339107, A063427, A063428, A328095, A333923, A332580.

isok(n, k) = #strsplit(Str(n*k), concat(Vecrev(Str(n+k)))) > 1;
ispt(n) = my(t); ispower(n,,&t) && (t==10);
a(n) = {if ((n==1) || (n==10) || ispt(n), return (-1)); my(k=0); while (! isok(n, k), k++); k;} \\ Michel Marcus, Jan 22 2021

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.

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, 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.

A355790 Numbers that can be written as the product of two divisors greater than 1 such that the number is contained in the string concatenation of the divisors.

Original entry on oeis.org

64, 95, 110, 210, 325, 510, 624, 640, 664, 950, 995, 1010, 1100, 1110, 3250, 3325, 5134, 6240, 6400, 6640, 6664, 7125, 7616, 8145, 9500, 9950, 9995, 11000, 11100, 11110, 20100, 21052, 21175, 25100, 26208, 32500, 33250, 33325, 35126, 50100, 51020, 51204, 51340, 57125, 62400, 64000, 65114

Offset: 1

Scott R. Shannon, Jul 17 2022

			64 is a term as 64 = 16 * 4 and "16" + "4" = "164" contains "64".
65114 is a term as 65114 = 4651 * 14 and "4651" + "14" = "465114" contains "65114".
See the attached text file for other examples.

Scott R. Shannon, Divisor product of the first 232 terms. These are all the numbers up to 50000000.

Cf. A355791 (base-2), A030190, A210959, A027750, A355852, A339144, A341035, A342127, A027748, A048991, A330027, A077293.

from sympy import divisors
def ok(n):
    s, divs = str(n), divisors(n)[1:-1]
    return any(s in str(d)+str(n//d) for d in divs)
print([k for k in range(1, 10**5) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

Original entry on oeis.org

1, 10, 98, 767, 111, 122, 2, 11, 100, 889, 110, 4490, 400, 560, 1096, 124, 20, 129, 70, 502, 93, 171, 212, 361, 512, 26, 21, 36, 54, 14, 1011, 139, 99, 59, 550, 684, 345, 102, 1021, 1999, 2871, 137, 892, 89, 126, 875, 510, 994, 586, 2012, 662, 1836, 201, 405, 388, 2007, 2798, 1641, 50, 340

Offset: 1

Scott R. Shannon and Eric Angelini, Jul 07 2023

In the first 10000 terms the smallest number that has not yet appeared is 696; it is therefore likely all numbers eventually appear although this is unknown.

			a(2) = 10 as a(1) + 10 = 1 + 10 = 11 which is a substring of "1" + "10" = "110".
a(3) = 98 as a(1) + a(2) + 98 = 1 + 10 + 98 = 109 which is a substring of "1" + "10" + "98" = "11098".
a(4) = 767 as a(1) + a(2) + a(3) + 767 = 1 + 10 + 98 + 767 = 876 which is a substring of "1" + "10" + "98" + "767" = "11098767".

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Eric Angelini, Échecs et Maths, Personal blog, July 2023.

Cf. A359482, A300000, A339144, A357082.

from itertools import islice
def agen(): # generator of terms
    s, mink, aset, concat = 1, 2, {1}, "1"
    yield from [1]
    while True:
        an = mink
        while an in aset or not str(s+an) in concat+str(an): an += 1
        aset.add(an); s += an; concat += str(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Feb 08 2024

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.

A340916 Integers m that have at least one divisor d such that d+m/d is a substring of m.

Original entry on oeis.org

4, 272, 352, 400, 414, 418, 425, 432, 448, 450, 465, 490, 504, 518, 572, 585, 598, 720, 732, 744, 756, 768, 972, 1092, 1104, 1152, 1210, 1221, 1232, 1243, 1254, 1265, 1276, 1287, 1298, 1309, 1792, 1872, 1887, 1890, 1904, 1914, 1920, 1950, 1972, 2100, 2112, 2672

Offset: 1

Michel Marcus, Jan 26 2021

These are the resulting product strings in A339144.

			272 = 4*68 contains 4+68 = 72 as a substring, so 272 is a term.

Cf. A339144, A340917.

q[n_] := AnyTrue[Divisors[n], SequenceCount[IntegerDigits[n], IntegerDigits[# + n/#]] > 0 &]; Select[Range[3000], q] (* Amiram Eldar, Jan 26 2021 *)

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

A364201 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) in binary is a substring of the concatenation of all terms a(1)..a(n) in binary.

Original entry on oeis.org

1, 2, 3, 5, 11, 7, 16, 9, 6, 18, 4, 13, 10, 15, 12, 23, 20, 8, 27, 19, 36, 26, 22, 17, 21, 31, 25, 14, 29, 28, 30, 57, 24, 32, 39, 43, 40, 34, 38, 46, 33, 35, 42, 37, 55, 44, 58, 48, 56, 52, 41, 45, 64, 63, 54, 61, 60, 49, 50, 51, 65, 47, 67, 88, 132, 73, 76, 68, 109, 59, 82, 87, 62, 98, 69, 70

Offset: 1

Scott R. Shannon, Jul 13 2023

In the first 10000 terms the smallest number that has not yet appeared is 7026; it is conjectured all numbers eventually appear.

The fixed points begin 1, 2, 3, 29, 48, 68, 96, 182, 471, 839, ... . It is likely there are infinitely more.

			a(2) = 2 as a(1) + 2 = 1 + 2 = 3 = 11_2, which is a substring of "a(1)"_2 + "2"_2 = "1" + "10" = "110".
a(4) = 5 as a(1) + a(2) + a(3) + 5 = 1 + 2 + 3 + 5 = 11 = 1011_2, which is a substring "a(1)"_2 + "a(2)"_2 + "a(3)"_2 + "5"_2 = "1" + "10" + "11" + "101" = "11011101".
a(5) = 11 as a(1) + a(2) + a(3) + a(4) + 11 = 1 + 2 + 3 + 5 + 11 = 22 = 10110_2, which is a substring "a(1)"_2 + "a(2)"_2 + "a(3)"_2 + "a(4)"_2 + "11"_2 = "1" + "10" + "11" + "101" + "1011" = "110111011011".

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A363186 (base 10), A359482, A007088, A300000, A339144, A357082.

cp's OEIS Frontend

A339403 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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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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A355790 Numbers that can be written as the product of two divisors greater than 1 such that the number is contained in the string concatenation of the divisors.

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A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

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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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A340916 Integers m that have at least one divisor d such that d+m/d is a substring of m.

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A364201 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) in binary is a substring of the concatenation of all terms a(1)..a(n) in binary.

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