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 1-5 of 5 results.

A116136 Numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 4.

Original entry on oeis.org

9, 99, 183, 328, 528, 715, 999, 6099, 9999, 13224, 40495, 99999, 106755, 453288, 999999, 2066115, 2975208, 9999999, 22145328, 28027683, 99999999, 110213248, 110667555, 147928995, 178838403, 226123528, 275074575, 333052608, 378698224, 445332888, 446245635, 518348515
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with itself gives the product of two numbers which differ by 2.

Examples

			8315420899//8315420896 = 9118892968 * 9118892972, where // denotes concatenation.

Links

Crossrefs

Cf. A116129, A115431, A116099, A115426, A116267.
Cf. also A102567, A116154, A116130, A116286.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A116136_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(1,a,all_roots=True)):
            if a*(b+3) <= k**2-1 < a*(a+2):
                yield (k**2-1)//a
A116136_list = list(islice(A116136_gen(),40)) # Chai Wah Wu, Feb 19 2024

Extensions

Edited by N. J. A. Sloane, Apr 15 2007
a(29)-a(32) from Chai Wah Wu, Feb 19 2024

A116130 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 5.

Original entry on oeis.org

8, 98, 590, 738, 830, 998, 1080, 4508, 9998, 20660, 29754, 99998, 980300, 999998, 6694218, 9999998, 49826988, 99999998, 117738578, 131505858, 132231404, 176445054, 177285320, 247979808, 252028388, 335180054, 336337790
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with itself gives the product of two numbers which differ by 3.

Examples

			7531357568//7531357564 = 8678339452 * 8678339457, where // denotes concatenation.

Crossrefs

Cf. A116124, A116129, A116131, A116099, A116261, A116163, A116136, A116125, A116287.

Extensions

Edited by N. J. A. Sloane, Apr 13 2007

A116260 n times n+4 gives the concatenation of two numbers m and m-4.

Original entry on oeis.org

9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 36363636362, 45454545453, 54545454544, 63636363635, 72727272726, 81818181817, 90909090908, 99999999999, 999999999999, 9999999999999
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			99999999 * 100000003 = 100000001//99999997, where // denotes concatenation.

Links

Crossrefs

Cf. A116129, A116254, A116259, A116261, A116267.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A116260_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(0,a,all_roots=True))+[a]:
            if a*(b+4) <= k**2 < a*(a+3):
                yield k-2
A116260_list = list(islice(A116260_gen(),20)) # Chai Wah Wu, Feb 19 2024

Formula

10^d-1 for d>0 are terms. - Chai Wah Wu, Feb 19 2024

A116098 Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 6.

Original entry on oeis.org

11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 100000000001, 1000000000001
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Aug 13 2018: (Start)
Contained in, and apparently identical, to A116129.
Numbers k such that k*(10^d+1) is a square, where k-9 has d decimal digits.
(End)

Examples

			100000001//99999992 = 99999998 * 100000004, where // denotes
concatenation.

Links

Crossrefs

Cf. A116097, A116099, A116106, A054214, A116128, A116129, A116130, A116136, A116229, A116260.

Programs

  • Maple
    g:= proc(d) local r,c,a,b;
   r:= mul(t[1],t=select(s -> s[2]::odd, ifactors(10^d+1)[2]))
   c:= ceil((10^(d-1)+9)/r);
   a:= isqrt(c);
   if a^2 < c then a:= a+1 fi;
   c:= floor((10^d+8)/r);
   b:= isqrt(c);
   if b^2 > c then b:= b-1 fi;
   seq(r*y^2, y = a..b)
end proc:
seq(g(d),d=1..60); # Robert Israel, Aug 13 2018

A116128 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 3.

Original entry on oeis.org

5303944, 6677714, 2070936216988528558, 2969428172738875624, 6685545813563350444, 8013829604553736451395958429212, 9724110515510343256451213152382
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A116122, A116129, A116143, A116259.
Showing 1-5 of 5 results.

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

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