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 1-7 of 7 results.

A370486 Terms in A116260 that are not of the form 10^d-1.

Original entry on oeis.org

36363636362, 45454545453, 54545454544, 63636363635, 72727272726, 81818181817, 90909090908, 428571428571428571427, 571428571428571428570, 714285714285714285713, 857142857142857142856, 363636363636363636363636363636362, 454545454545454545454545454545453
Offset: 1

Views

Author

Chai Wah Wu, Feb 19 2024

Keywords

Comments

Since 10^d-1 for d>0 are terms of A116260, this sequence only includes terms of A116260 not of this form to highlight these nontrivial terms.

Links

Crossrefs

Cf. A116260.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A370486_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)):
            if a*(b+4) <= k**2 < a*(a+3):
                yield k-2
A370486_list = list(islice(A370486_gen(),20))

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

A116267 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

8, 98, 426, 571, 725, 844, 998, 7808, 9998, 36363, 63634, 99998, 326732, 673265, 999998, 4545452, 5454545, 9999998, 47058821, 52941176, 99999998, 331983805, 332667332, 384615384, 422892896, 475524475, 524475522, 577107101, 615384613, 667332665, 668016192, 719964244
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Links

Crossrefs

Cf. A116136, A116260, A116266, A116268, A116275.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A116267_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
A116267_list = list(islice(A116267_gen(),20)) # Chai Wah Wu, Feb 19 2024

Extensions

a(29)-a(32) from Chai Wah Wu, Feb 19 2024

A116129 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 4.

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)
Contains, and appears to be identical to, A116098.
Numbers k such that (10^d+1)*k is a square, where k-4 has d digits. (End)

Examples

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

Crossrefs

Cf. A116098, A054214, A116128, A116130, A116136, 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)+4)/r);
   a:= isqrt(c);
   if a^2 < c then a:= a+1 fi;
   c:= floor((10^d+3)/r);
   b:= isqrt(c);
   if b^2 > c then b:= b-1 fi;
   seq(r*y^2, y = a..b)
end proc:
map(g, [$1..60]); # Robert Israel, Aug 13 2018

A116254 k times k+4 gives the concatenation of two numbers m and m-5.

Original entry on oeis.org

89, 9077, 9899, 733672, 998999, 88225293, 99989999, 8900869206, 9296908810, 9604060395, 9999899999, 326666333265, 673333666732, 700730927006, 972603739725, 999998999999, 34519562953735, 39737862788836, 49917309624954, 50082690375043, 60262137211161, 65480437046262
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Links

Crossrefs

Cf. A054214, A116248, A116253, A116255, A116260.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A116254_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)):
            m = (k**2+1)//a
            if a*(b+4) <= k**2+1 < a*(a+3):
                yield k-2
A116254_list = list(islice(A116254_gen(),40)) # Chai Wah Wu, Feb 19 2024

Extensions

a(19)-a(22) from Chai Wah Wu, Feb 19 2024

A116261 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-4.

Original entry on oeis.org

7, 97, 766, 857, 909, 997, 3284, 6712, 9997, 45451, 54545, 99997, 990099, 999997, 8181818, 9999997, 70588232, 99999997, 343130553, 362637362, 363636360, 420053630, 421052631, 497975708, 502024288, 578947365, 579946366
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A028557, A116130, A116255, A116260, A116262, A116268.

A116259 n times n+3 gives the concatenation of two numbers m and m-4.

Original entry on oeis.org

7282817, 8171727, 4550754022124826466, 5449245977875173532, 8176518705147900805, 8951999555715882882349355522888, 9861090464806791973258446431979
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			8171727 * 8171730 = 6677714//6677710, where // denotes
concatenation.

Crossrefs

Cf. A116128, A116253, A116260, A116274.
Showing 1-7 of 7 results.

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