cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A116436 Numbers m which when sandwiched between two 1's give a multiple of m.

Original entry on oeis.org

1, 11, 13, 77, 91, 137, 9091, 909091, 5882353, 10989011, 12987013, 52631579, 76923077, 90909091, 4347826087, 9090909091, 13698630137, 909090909091, 3448275862069, 10989010989011, 12987012987013, 76923076923077, 90909090909091, 9090909090909091, 909090909090909091
Offset: 1

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Comments

All k-digit numbers that divide 10^{k+1} + 1. - Franklin T. Adams-Watters, Apr 23 2008
Notice the infinite pattern m = (90..90..90)91 with 1m1/m = 21, e.g., 1911/91 = 190911/9091 = 19090911/909091 = 21 (see A095372). - Zak Seidov, Apr 22 2008
Corresponding numbers k such that k * a(n) = 1.a(n).1 where '.' stands for concatenation are in A351320. - Bernard Schott, Feb 07 2022

Examples

			77 is a member since 1771 is a multiple of 77 (77*23).

Links

Crossrefs

Subsequence of A116437, A116438, A116439, A116440, A116441, A116442, A116443, A116444.
Cf. A136296, A329914, A329915, A351320.
Some subsequences, M such that k*M=1M1 for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), A351238 (k=87), A351239 (k=101).

Programs

  • Mathematica
    f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[1, 14] (* Ray Chandler, May 11 2007 *)
  • PARI
    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(i))); l
\\ Franklin T. Adams-Watters, Apr 22 2008
  • Python
    from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield 1
    for k in count(2):
        t = 10**(k+1) + 1
        yield from (t//i for i in range(100, 10, -1) if t%i == 0)
print(list(islice(agen(), 25))) # Michael S. Branicky, Mar 26 2023 following Franklin T. Adams-Watters but removing factorization

Formula

A351320(n) * a(n) = 1.a(n).1 where "." stands for concatenation. - Bernard Schott, Feb 07 2022

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Giovanni Resta, Feb 15 2006

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

a(39) and beyond from Michael S. Branicky, Mar 26 2023
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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