A116436 -id:A116436 - cp's OEIS frontend

A351237 Numbers M such that 83 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

137, 13698630137, 1369863013698630137, 136986301369863013698630137, 13698630136986301369863013698630137, 1369863013698630136986301369863013698630137, 136986301369863013698630136986301369863013698630137

Offset: 1

Bernard Schott, Feb 05 2022

There are only 15 numbers k such that there exist numbers M_k which, when 1 is placed at both ends of M_k, the number M_k is multiplied by k; 83 is the eleventh such integer, so 83 = A329914(11), and a(1) = A329915(11) = 137; hence, the terms of this sequence form the infinite set {M_83}.

Every term M = a(n) has q = 8*n-5 digits, and 10^(q+1)+1 that has q = 8*n-5 zeros in its decimal expansion is equal to 73 * M, so a(n) = M is a divisor of 10^(8*n-4)+1. Example: a(2) = 13698630137 has 11 digits and 73 * 13698630137 = 1000000000001 that has 11 zeros in its decimal expansion.

			83 * 137 = 1[137]1, hence 137 is a term.
83 * 13698630137 = 1[13698630137]1, and 13698630137 is another term.

D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.

Index entries for linear recurrences with constant coefficients, signature (100000001,-100000000).

Subsequence of A116436.
Cf. A329914, A329915.
Similar for: A095372 \ {1} (k=21), A331630 (k=23), this sequence (k=83), A351238 (k=87), A351239 (k=101).

Maple
```
seq((10^(8*n-4)+1)/73, n=1..15);
```

Table[(10^(8*n-4)+1)/73, {n, 1, 7}] (* Amiram Eldar, Feb 06 2022 *)
LinearRecurrence[{100000001,-100000000},{137,13698630137},20] (* Harvey P. Dale, Nov 01 2022 *)

a(n) = (10^(8*n-4)+1)/73 for n >= 1.

A351238 Numbers M such that 87 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

Original entry on oeis.org

13, 12987013, 12987012987013, 12987012987012987013, 12987012987012987012987013, 12987012987012987012987012987013, 12987012987012987012987012987012987013, 12987012987012987012987012987012987012987013, 12987012987012987012987012987012987012987012987013, 12987012987012987012987012987012987012987012987012987013

Offset: 1

Bernard Schott, Feb 05 2022

There are only 15 numbers k such that there exist numbers M_k which, when 1 is placed at both ends of M_k, the number M_k is multiplied by k; 87 is the twelfth such integer, so 87 = A329914(12), and a(1) = A329915(12) = 13; hence, the terms of this sequence form the infinite set {M_87}.

Every term M = a(n) has q = 6*n-4 digits, and 10^(q+1)+1 that has q = 6*n-4 zeros in its decimal expansion is equal to 77 * M, so a(n) = M is a divisor of 10^(6*n-3)+1. Example: a(2) = 12987013 has 8 digits and 77 * 12987013 = 1000000001 that has 8 zeros in its decimal expansion.

			87 * 13 = 1[13]1, hence 13 is a term.
87 * 12987013 = 1[12987013]1, and 12987013 is a term.

D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.

Index entries for linear recurrences with constant coefficients, signature (1000001,-1000000).

Subsequence of A116436.
Cf. A329914, A329915.
Similar for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), this sequence (k=87), A351239 (k=101).

Maple
```
seq((10^(6*n-3)+1)/77, n=1..15);
```

Table[(10^(6*n - 3) + 1)/77, {n, 1, 10}] (* Amiram Eldar, Feb 06 2022 *)

a(n) = (10^(6*n-3)+1)/77 for n >= 1.

A351239 Numbers M such that 101 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

Original entry on oeis.org

11, 10989011, 10989010989011, 10989010989010989011, 10989010989010989010989011, 10989010989010989010989010989011, 10989010989010989010989010989010989011, 10989010989010989010989010989010989010989011, 10989010989010989010989010989010989010989010989011

Offset: 1

Bernard Schott, Feb 05 2022

There are only 15 numbers k such that there exist numbers M_k which, when 1 is placed at both ends of M_k, the number M_k is multiplied by k; 101 is the fourteenth such integer, so 101 = A329914(14), and a(1) = A329915(14) = 11; hence, the terms of this sequence form the infinite set {M_101}.

Every term M = a(n) has q = 6*n-4 digits, and 10^(q+1)+1 that has q = 6*n-4 zeros in its decimal expansion is equal to 91 * M, so a(n) = M is a divisor of 10^(6*n-3)+1. Example: a(2) = 10989011 has 8 digits and 91 * 10989011 = 1000000001 that has 8 zeros in its decimal expansion.

			101 * 11 = 1[11]1, hence 11 is a term.
101 * 10989011 = 1[10989011]1 and 10989011 is another term.

D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.

Index entries for linear recurrences with constant coefficients, signature (1000001,-1000000).

Subsequence of A116436.
Cf. A329914, A329915.
Similar for: A095372 \ {1} (k = 21), A331630 (k = 23), A351237 (k = 83), A351238 (k = 87), this sequence (k = 101).

Maple
```
seq((10^(6*n-3)+1)/91, n=1..15);
```

Table[(10^(6*n - 3) + 1)/91, {n, 1, 9}] (* Amiram Eldar, Feb 06 2022 *)
LinearRecurrence[{1000001,-1000000},{11,10989011},10] (* Harvey P. Dale, Sep 12 2022 *)

a(n) = (10^(6*n-3)+1)/91 for n >= 1.

A136296 "Special augmented primes": primes p such that the decimal number 1p1 is divisible by p.

Original entry on oeis.org

11, 13, 137, 9091, 909091, 5882353, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091, 909090909090909090909090909090909090909090909090909090909090909091

Offset: 1

N. J. A. Sloane, Apr 20 2008

Equals A116436 INTERSECT A000040. - R. J. Mathar, Apr 24 2008
The larger terms may be only probable primes. - Franklin T. Adams-Watters, Apr 23 2008
According to the Magma Calculator (http://magma.maths.usyd.edu.au/calc/), all nine terms given for this sequence are prime. - Jon E. Schoenfield, Aug 24 2009

			11371/137 = 83, an integer, so the prime 137 is a term.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 61.

Michael S. Branicky, Table of n, a(n) for n = 1..12 (all terms with <= 1000 digits)

Prime members of A116436.

max=6; a={}; For[i=1, i<=10^max, i++, If[Mod[FromDigits[Join[{1}, IntegerDigits[Prime[i]], {1}]], Prime[i]] == 0, AppendTo[a, Prime[i]]]]; a (* Stefano Spezia, Mar 26 2023 *)

A136296k(k) = { local(l, d, lb, ub); d=factor(10^(k+1)+1)[,1]; l=[]; lb=10^(k-1); ub=10*lb; for(i=1,#d,if(d[i]>=lb&&d[i]A136296k(k))) \\ Franklin T. Adams-Watters, Apr 23 2008

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(2):
        t = 10**(k+1) + 1
        d = [t//i for i in range(100, 10, -1) if t%i == 0]
        yield from (di for di in d if isprime(di))
print(list(islice(agen(), 8))) # Michael S. Branicky, Mar 26 2023 following Franklin T. Adams-Watters but removing factorization

a(4)-a(6) from M. F. Hasler, Apr 22 2008
a(7)-a(9) from Franklin T. Adams-Watters, Apr 23 2008
a(10) from Michael S. Branicky, Mar 26 2023

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A351237 Numbers M such that 83 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

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A351238 Numbers M such that 87 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

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A351239 Numbers M such that 101 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

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Maple

Mathematica

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A136296 "Special augmented primes": primes p such that the decimal number 1p1 is divisible by p.

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