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.

A115426 Numbers k such that the concatenation of k with k+2 gives a square.

Original entry on oeis.org

7874, 8119, 69476962, 98010199, 108746354942, 449212110367, 544978035127, 870501316279, 998001001999, 1428394731903223, 1499870932756487, 1806498025502498, 1830668275445687, 1911470478658759, 2255786189655202
Offset: 1

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Comments

Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 2.
Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 4.
Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 6.

Examples

			8119//8121 = 9011^2, where // denotes concatenation.
98010199//98010200 = 99000100 * 99000102.
98010199//98010197 = 99000099 * 99000103.

Links

Crossrefs

Cf. A030465, A102567, A115427, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115437.
Cf. A116106, A116110, A116112, A116125, A116242.
Cf. A116136, A116143, A116107, A116275, A116163, A116099, A116177, A116295.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A115426_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(2,a,all_roots=True)):
            if a*(b-2) <= k**2-2 < a*(a-3):
                yield (k**2-2)//a
A115426_list = list(islice(A115426_gen(),40)) # Chai Wah Wu, Feb 20 2024

Extensions

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

A116107 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 7.

Original entry on oeis.org

52, 63716866, 48793687600063875363014809897052, 60020753655608135708762056127156, 60446518621981165303188950156776, 71135436903815748345367595855336, 72876856643103028189103298533248
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers n such that n concatenated with n-2 gives the product of two numbers which differ by 5.

Examples

			63716866//63716864 = 79822843 * 79822848, where // denotes concatenation.

Crossrefs

Cf. A116099, A116106, A115429, A116112, A116238, A115426, A116146, A116276.

Extensions

Edited by N. J. A. Sloane, Apr 15 2007

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

A116105 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 5.

Original entry on oeis.org

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

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A116097, A116104, A116106, A116118, A116236.

A116237 n times n+6 gives the concatenation of two numbers m and m-8.

Original entry on oeis.org

7, 9, 97, 99, 425, 570, 724, 843, 997, 999, 7807, 9997, 9999, 36362, 63633, 99997, 99999, 326731, 673264, 999997, 999999, 4545451, 5454544, 9999997, 9999999, 47058820, 52941175, 99999997, 99999999, 331983804, 332667331, 384615383
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A116106, A116229, A116236, A116238, A116242.
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.