A116436 -id:A116436 - cp's OEIS frontend

A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

111, 101, 87, 23, 21, 83, 21, 21, 27, 101, 87, 29, 23, 21, 33, 21, 83, 21, 39, 101, 87, 23, 21, 21, 21, 83, 101, 87, 59, 23, 21, 99, 57, 21, 27, 21, 101, 87, 29, 23, 21, 83, 69, 21, 71, 21, 101, 87, 33, 23, 21, 21, 83, 21, 101, 87, 23, 21, 27, 21, 39, 21, 83, 101, 87, 29, 23, 21, 21, 107, 21, 101

Offset: 1

Bernard Schott, Feb 07 2022

Except for a(1) = 111, which is unique, all terms appear infinitely many times and belong to this set of fifteen integers: {21, 23, 27, 29, 33, 39, 57, 59, 69, 71, 83, 87, 99, 101, 107}; see A329914.

The corresponding indices where these integers appear the first time are respectively: 5, 4, 9, 12, 15, 19, 33, 29, 43, 45, 6, 3, 32, 2, 70.

			A116436(1) = 1 and 111 * 1 = 1.1.1, hence a(1) = 111.
A116436(2) = 11 and 101 * 11 = 1.11.1, hence a(2) = 101.
A116436(32) = 112359550561797752809 and 99 * 112359550561797752809 = 1.112359550561797752809.1 hence a(32) = 99 (see Penguin reference).

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

Cf. A116436, A329914, A329915.

M such that k*M=1M1 for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), A351238 (k=87), A351239 (k=101).

A116436(k) = {local(l, d, lb, ub); d=divisors(10^(k+1)+1); l=[]; lb=10^(k-1); ub=10*lb; for(i=1, #d, if(d[i]>=lb&&d[i]A116436
a(n) = {my(v6=[], i=1); while (#v6 < n, v6 = concat(v6, A116436(i)); i++); my(x= v6[n]); my(k=1); while (eval(Str(1, x, 1)) % x, k++); eval(Str(1, x, 1))/x;} \\ Michel Marcus, Feb 10 2022

a(n) * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

A116437 Numbers k which when sandwiched between two 2's give a multiple of k.

Original entry on oeis.org

1, 2, 11, 13, 14, 22, 26, 77, 91, 137, 146, 274, 9091, 19802, 909091, 5882353, 10989011, 12987013, 13986014, 15037594, 21978022, 25974026, 52631579, 76923077, 90909091, 198019802, 1652892562, 4347826087, 8695652174, 9090909091, 13698630137, 14598540146, 27397260274

Offset: 1

Giovanni Resta, Feb 15 2006

			77 belongs since 2772 is a multiple of 77 (77*36).

Michael S. Branicky, Table of n, a(n) for n = 1..2921

Cf. A116436, A116438, A116439, A116440, A116441, A116442, A116443, A116444.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[2, 10] (* Ray Chandler, May 11 2007 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 2]
    for k in count(2):
        t = 2*(10**(k+1) + 1)
        yield from (t//i for i in range(200, 20, -1) if t%i == 0)
print(list(islice(agen(), 33))) # Michael S. Branicky, Mar 26 2023

a(30) and beyond from Michael S. Branicky, Mar 26 2023

A116438 Numbers k which when sandwiched between two 3's give a multiple of k.

Original entry on oeis.org

1, 3, 11, 13, 21, 33, 39, 77, 91, 137, 219, 411, 9091, 29703, 909091, 5882353, 10989011, 12145749, 12987013, 14354067, 20979021, 22556391, 32967033, 38961039, 52631579, 76923077, 90909091, 297029703, 1185770751, 2479338843, 4347826087, 9090909091, 13698630137

Offset: 1

Giovanni Resta, Feb 15 2006

			219 belongs since 32193 is a multiple of 219 (219*147).

Michael S. Branicky, Table of n, a(n) for n = 1..3170

Cf. A116436, A116437, A116439, A116440, A116441, A116442, A116443, A116444.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[3, 10] (* Ray Chandler, May 11 2007 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 3]
    for k in count(2):
        t = 3*(10**(k+1) + 1)
        yield from (t//i for i in range(300, 30, -1) if t%i == 0)
print(list(islice(agen(), 33))) # Michael S. Branicky, Mar 26 2023

a(31) and beyond from Michael S. Branicky, Mar 26 2023

A116439 Numbers k which when sandwiched between two 4's give a multiple of k.

Original entry on oeis.org

1, 2, 4, 11, 13, 14, 22, 26, 28, 44, 52, 77, 91, 137, 146, 274, 292, 548, 9091, 19802, 39604, 909091, 5882353, 10989011, 12987013, 13986014, 15037594, 16194332, 19138756, 21978022, 25974026, 27972028, 30075188, 43956044, 51948052, 52631579, 76923077, 90909091

Offset: 1

Giovanni Resta, Feb 15 2006

			91 belongs since 4914 is a multiple of 91 (91*54).

Michael S. Branicky, Table of n, a(n) for n = 1..4735

Cf. A116436, A116437, A116438, A116440, A116441, A116442, A116443, A116444.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[4, 9] (* Ray Chandler, May 11 2007 *)
Select[Range[52000000],Divisible[FromDigits[Join[{4}, IntegerDigits[#],{4}]],#]&]  (* Harvey P. Dale, Mar 14 2011 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 2, 4]
    for k in count(2):
        t = 4*(10**(k+1) + 1)
        yield from (t//i for i in range(400, 40, -1) if t%i == 0)
print(list(islice(agen(), 38))) # Michael S. Branicky, Mar 26 2023

a(36) and beyond from Michael S. Branicky, Mar 26 2023

A116440 Numbers k which when sandwiched between two 5's give a multiple of k.

Original entry on oeis.org

1, 5, 11, 13, 35, 55, 65, 77, 91, 137, 365, 685, 9091, 49505, 909091, 5882353, 10989011, 12987013, 20242915, 23923445, 34965035, 37593985, 52631579, 54945055, 64935065, 76923077, 90909091, 495049505, 1976284585, 4132231405, 4347826087, 9090909091, 13698630137

Offset: 1

Giovanni Resta, Feb 15 2006

All terms must be odd. - Harvey P. Dale, Jul 29 2015

			137 belongs since 51375 is a multiple 137 (137*375).

Michael S. Branicky, Table of n, a(n) for n = 1..3294

Cf. A116436, A116437, A116438, A116439, A116441, A116442, A116443, A116444.

a:=proc(n) local nn: nn:=convert(n,base,10): if type((5+10*n+5*10^(nops(nn)+1))/n, integer)=true then n else fi end: seq(a(n),n=1..10000); # Emeric Deutsch, Feb 28 2006

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[5, 10] (* Ray Chandler, May 11 2007 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 5]
    for k in count(2):
        t = 5*(10**(k+1) + 1)
        yield from (t//i for i in range(500, 50, -1) if t%i == 0)
print(list(islice(agen(), 33))) # Michael S. Branicky, Mar 26 2023

a(31) and beyond from Michael S. Branicky, Mar 26 2023

A116441 Numbers k which when sandwiched between two 6's give a multiple of k.

Original entry on oeis.org

1, 2, 3, 6, 11, 13, 14, 21, 22, 26, 33, 39, 42, 66, 77, 78, 91, 137, 146, 219, 274, 411, 438, 822, 9091, 19802, 29703, 59406, 909091, 5882353, 10989011, 12145749, 12987013, 13986014, 14354067, 15037594, 20979021, 21978022, 22556391, 24291498, 25974026, 28708134

Offset: 1

Giovanni Resta, Feb 15 2006

			39 belongs to the sequence since 6396 is a multiple of 39 (39*164).

Michael S. Branicky, Table of n, a(n) for n = 1..6573

Cf. A116436, A116437, A116438, A116439, A116440, A116442, A116443, A116444.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[6, 8] (* Ray Chandler, May 11 2007 *)
Select[Range[23000000],Divisible[FromDigits[Join[{6},IntegerDigits[#],{6}]],#]&]  (* Harvey P. Dale, Jan 12 2011 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 2, 3, 6]
    for k in count(2):
        t = 6*(10**(k+1) + 1)
        yield from (t//i for i in range(600, 60, -1) if t%i == 0)
print(list(islice(agen(), 42))) # Michael S. Branicky, Mar 26 2023

a(40) and beyond from Michael S. Branicky, Mar 26 2023

A116442 Numbers n which when sandwiched between two 7's give a multiple of n.

Original entry on oeis.org

1, 7, 11, 13, 49, 77, 91, 137, 511, 959, 9091, 69307, 909091, 5882353, 10989011, 12987013, 28340081, 33492823, 48951049, 52631579, 76923077, 90909091, 693069307, 2766798419, 4347826087, 5785123967, 9090909091, 13698630137, 51094890511, 95890410959, 909090909091

Offset: 1

Giovanni Resta, Feb 15 2006

			511 belongs since 75117 is a multiple 511 (511*147).

Michael S. Branicky, Table of n, a(n) for n = 1..3001

Cf. A116436, A116437, A116438, A116439, A116440, A116441, A116443, A116444.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[7, 11] (* Ray Chandler, May 11 2007 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 7]
    for k in count(2):
        t = 7*(10**(k+1) + 1)
        yield from (t//i for i in range(700, 70, -1) if t%i == 0)
print(list(islice(agen(), 32))) # Michael S. Branicky, Mar 26 2023

a(29) and beyond from Michael S. Branicky, Mar 26 2023

A116443 Numbers k which when sandwiched between two 8's give a multiple of k.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 14, 22, 26, 28, 44, 52, 56, 77, 88, 91, 137, 146, 274, 292, 548, 584, 9091, 19802, 39604, 79208, 909091, 5882353, 10989011, 12987013, 13986014, 15037594, 16194332, 19138756, 21978022, 25974026, 27972028, 30075188, 32388664, 38277512, 43956044

Offset: 1

Giovanni Resta, Feb 15 2006

			91 belongs since 8918 is a multiple of 91 (91*98 = 8918).

Michael S. Branicky, Table of n, a(n) for n = 1..6504

Cf. A116436, A116437, A116438, A116439, A116440, A116441, A116442, A116444.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[8, 8] f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[9, 8] (* Ray Chandler, May 11 2007 *)
Select[Range[301*10^5],Divisible[FromDigits[Join[{8},IntegerDigits[#],{8}]],#]&] (* Harvey P. Dale, Aug 27 2019 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 2, 4, 8]
    for k in count(2):
        t = 8*(10**(k+1) + 1)
        yield from (t//i for i in range(800, 80, -1) if t%i == 0)
print(list(islice(agen(), 41))) # Michael S. Branicky, Mar 26 2023

a(39) and beyond from Michael S. Branicky, Mar 26 2023

A116444 Numbers k which when sandwiched between two 9's give a multiple of k.

Original entry on oeis.org

1, 3, 9, 11, 13, 21, 33, 39, 63, 77, 91, 99, 137, 219, 411, 657, 9091, 29703, 89109, 909091, 5882353, 10989011, 12145749, 12987013, 14354067, 20979021, 22556391, 32967033, 36437247, 38961039, 43062201, 52631579, 62937063, 67669173, 76923077, 90909091, 98901099

Offset: 1

Giovanni Resta, Feb 15 2006

			411 belongs since 94119 is a multiple of 411 (411*229).

Michael S. Branicky, Table of n, a(n) for n = 1..5016

Cf. A116436, A116437, A116438, A116439, A116440, A116441, A116442, A116443.

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[9, 8] (* Ray Chandler, May 11 2007 *)
Select[Range[10^6],Mod[FromDigits[Join[{9},IntegerDigits[#],{9}]],#]==0&] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Jan 20 2025 *)

for(d=1,10,fordiv(9*10^(d+1)+9,a,if(#Str(a)==d,print1(a", ")))) /* Martin Fuller, May 10 2007 */

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 3, 9]
    for k in count(2):
        t = 9*(10**(k+1) + 1)
        yield from (t//i for i in range(900, 90, -1) if t%i == 0)
print(list(islice(agen(), 38))) # Michael S. Branicky, Mar 26 2023

Corrected by Martin Fuller, May 10 2007

a(35) and beyond from Michael S. Branicky, Mar 26 2023

A331630 Numbers M such that 23 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

Original entry on oeis.org

77, 76923077, 76923076923077, 76923076923076923077, 76923076923076923076923077, 76923076923076923076923076923077, 76923076923076923076923076923076923077, 76923076923076923076923076923076923076923077

Offset: 1

Bernard Schott, Jan 23 2020

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; 23 is the second such integer, so 23 = A329914(2), and a(1) = A329915(2) = 77; hence, the terms of this sequence form the infinite set {M_23}.

Every term M = a(n) has q = 6*n-4 digits, and 10^(q+1)+1 that has 6*n-4 zeros in its decimal expansion is equal to 13 * M, so M is a divisor of 10^(6*n-3)+1. Example: a(2) = 76923077 has 8 digits and 13 * 76923077 = 1000000001 that has 8 zeros in its decimal expansion.

			23 * 77 = 1771, hence 77 is a term.
23 * 76923076923077 = 1(76923076923077)1, and 76923076923077 is another term.

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

Colin Barker, Table of n, a(n) for n = 1..150
Index entries for linear recurrences with constant coefficients, signature (1000001,-1000000).

Subsequence of A116436.

Cf. A329914, A329915, A095372 \ {1} (similar for k = 21).

Maple
```
seq((10^(6*m-3)+1)/13, m=1..15);
```

Array[(10^(6 # - 3) + 1)/13 &, 9] (* Michael De Vlieger, Jan 24 2020 *)
LinearRecurrence[{1000001,-1000000},{77,76923077},10] (* Harvey P. Dale, Mar 03 2023 *)

vector(9, n, (10^(6*n-3)+1)/13) \\ Michel Marcus, Jan 25 2020

Vec(77*x*(1 - 1000*x) / ((1 - x)*(1 - 1000000*x)) + O(x^10)) \\ Colin Barker, Jan 25 2020

apply( {A331630(n)=10^(6*n-3)\/13}, [1..9]) \\ M. F. Hasler, Jan 26 2020, following Michel Marcus

a(n) = (10^(6*n-3)+1)/13 for n >= 1.
From Colin Barker, Jan 25 2020: (Start)
G.f.: 77*x*(1 - 1000*x) / ((1 - x)*(1 - 1000000*x)).
a(n) = 1000001*a(n-1) - 1000000*a(n-2) for n>2.
a(n) = (1000 + 1000^(2*n))/13000 for n>0.
(End)
E.g.f.: exp(x)*(1000 + exp(999999*x))/13000 - 77/1000. - Stefano Spezia, Jan 26 2020

cp's OEIS Frontend

A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

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A116437 Numbers k which when sandwiched between two 2's give a multiple of k.

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A116438 Numbers k which when sandwiched between two 3's give a multiple of k.

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A116439 Numbers k which when sandwiched between two 4's give a multiple of k.

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A116440 Numbers k which when sandwiched between two 5's give a multiple of k.

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A116441 Numbers k which when sandwiched between two 6's give a multiple of k.

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A116442 Numbers n which when sandwiched between two 7's give a multiple of n.

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