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 1-3 of 3 results.

A023110 Squares which remain squares when the last digit is removed.

Original entry on oeis.org

0, 1, 4, 9, 16, 49, 169, 256, 361, 1444, 3249, 18496, 64009, 237169, 364816, 519841, 2079364, 4678569, 26666896, 92294449, 341991049, 526060096, 749609641, 2998438564, 6746486769, 38453641216, 133088524969, 493150849009, 758578289296, 1080936581761
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

This A023110 = A031149^2 is the base 10 version of A001541^2 = A055792 (base 2), A001075^2 = A055793 (base 3), A004275^2 = A055808 (base 4), A204520^2 = A055812 (base 5), A204518^2 = A055851 (base 6), A204516^2 = A055859 (base 7), A204514^2 = A055872 (base 8) and A204502^2 = A204503 (base 9). - M. F. Hasler, Sep 28 2014
For the first 4 terms the square has only one digit. It is understood that deleting this digit yields 0. - Colin Barker, Dec 31 2017

References

  • R. K. Guy, Neg and Reg, preprint, Jan 2012.

Links

Crossrefs

Cf. A023111.
Cf. A031150, A053784, A031149, A055792, A055793, A055808, A055812, A055851, A055859, A055872.
Cf. A001541, A001075, A004275, A204520, A204518, A204516, A204514, A204502, A204503.

Programs

  • Maple
    count:= 1: A[1]:= 0:
for n from 0 while count < 35 do
  for t in [1,4,6,9] do
    if issqr(10*n^2+t) then
       count:= count+1;
       A[count]:= 10*n^2+t;
    fi
  od
od:
seq(A[i],i=1..count); # Robert Israel, Sep 28 2014
  • Mathematica
    fQ[n_] := IntegerQ@ Sqrt@ Quotient[n^2, 10]; Select[ Range@ 1000000, fQ]^2 (* Robert G. Wilson v, Jan 15 2011 *)
  • PARI
    for(n=0,1e7, issquare(n^2\10) & print1(n^2",")) \\  M. F. Hasler, Jan 16 2012

Formula

Appears to satisfy a(n)=1444*a(n-7)+a(n-14)-76*sqrt(a(n-7)*a(n-14)) for n >= 16. For n = 15, 14, 13, ... this would require a(1) = 16, a(0) = 49, a(-1) = 169, ... - Henry Bottomley, May 08 2001; edited by Robert Israel, Sep 28 2014
a(n) = A031149(n)^2. - M. F. Hasler, Sep 28 2014
Conjectures from Colin Barker, Dec 31 2017: (Start)
G.f.: x^2*(1 + 4*x + 9*x^2 + 16*x^3 + 49*x^4 + 169*x^5 + 256*x^6 - 1082*x^7 - 4328*x^8 - 9738*x^9 - 4592*x^10 - 6698*x^11 - 6698*x^12 - 4592*x^13 + 361*x^14 + 1444*x^15 + 3249*x^16 + 256*x^17 + 169*x^18 + 49*x^19 + 16*x^20) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 - 1442*x^7 + x^14)).
a(n) = 1443*a(n-7) - 1443*a(n-14) + a(n-21) for n>22.
(End)

Extensions

More terms from M. F. Hasler, Jan 16 2012

A031150 Appending a digit to n^2 gives another perfect square.

Original entry on oeis.org

1, 2, 4, 5, 6, 12, 18, 43, 80, 154, 191, 228, 456, 684, 1633, 3038, 5848, 7253, 8658, 17316, 25974, 62011, 115364, 222070, 275423, 328776, 657552, 986328, 2354785, 4380794, 8432812, 10458821, 12484830, 24969660, 37454490
Offset: 1

Views

Author

Patrick De Geest

Keywords

Comments

Square root of 'Squares from A023110 with last digit removed'.
One could include an initial '0', and even list it with multiplicity 3 or 4, since 00, 01, 04 and 09 are all perfect squares: In analogy to corresponding sequences for other bases, this sequence could be defined as sqrt(floor[A023110/10]), see A204512 [base 8], A204517 (base 7), A204519 (base 6), A204521 [base 5], A001353 [base 3], A001542 [base 2]. (For bases 4 and 9, the corresponding sequence contains all integers.) - M. F. Hasler, Jan 16 2012

Examples

			5^2 = 25 and 16^2 = 256, so 5 is in the sequence.
115364^2 = 13308852496, 364813^2 = 133088524969.

References

  • R. K. Guy, Neg and Reg, preprint, Jan 2012.

Links

Crossrefs

Cf. A023110, A030686, A030687, A053784.
See A202303 for the resulting squares.

Programs

  • Maple
    for i from 1 to 150000 do if (floor(sqrt(10 * i^2 + 9)) > floor(sqrt(10 * i^2))) then print(i) end if end do;
  • Mathematica
    CoefficientList[Series[(x^10 + 2 x^9 + 4 x^8 + 5 x^7 + 18 x^6 + 12 x^5 + 6 x^4 + 5 x^3 + 4 x^2 + 2 x + 1)/(x^14 - 38 x^7 + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{0,0,0,0,0,0,38,0,0,0,0,0,0,-1},{1,2,4,5,6,12,18,43,80,154,191,228,456,684},40] (* Harvey P. Dale, Jun 09 2017 *)

Formula

G.f.: x*(x^10+2*x^9+4*x^8+5*x^7+18*x^6+12*x^5+6*x^4+5*x^3+4*x^2+2*x+1) / (x^14-38*x^7+1). - Colin Barker, Jan 30 2013

A053783 (1+e)-harmonic numbers: harmonic mean of (1+e)-divisors is an integer.

Original entry on oeis.org

1, 6, 28, 140, 728, 1638, 2184, 3640, 8008, 8190, 10920, 18620, 23808, 23895, 27846, 37128, 47790, 55860, 69160, 148960, 166656, 189810, 237510, 242060, 316680, 334530, 359600, 406215, 446880, 484120, 525690, 669060, 726180, 1029952, 1078800, 1089270, 1099170
Offset: 1

Views

Author

Naohiro Nomoto, Apr 14 2001

Keywords

Comments

If k = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of k.

Links

Crossrefs

Cf. A001599, A049599, A051378, A053784.

Programs

  • Mathematica
    f[p_, e_] := (DivisorSigma[0, e] + 1)/(p^e + DivisorSum[e, p^(e - #) &]); aQ[n_] := IntegerQ[n * Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^5], aQ] (* Amiram Eldar, Sep 07 2019 *)

Extensions

More terms from Amiram Eldar, Sep 07 2019
Showing 1-3 of 3 results.

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