A080160 -id:A080160 - cp's OEIS frontend

A108690 Square roots of the terms in A080160.

1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 21, 24, 27, 30, 33, 36, 45, 48, 51, 54, 57, 60, 63, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 99, 102, 105, 108, 111, 114, 117, 126, 129, 132, 135, 138, 141, 144, 153, 154, 155, 156, 157, 158, 159, 160

Offset: 1

N. J. A. Sloane, Jun 19 2005

Cf. A080160.

b:=n->sum(convert(((10^(n+1)-1)/9)^2,base,10)[j],j=1..2*n+1): a:=proc(n) if type(sqrt(b(n)), integer)=true then sqrt(b(n)) else fi end:seq(a(n),n=0..2000); # Emeric Deutsch, Jun 19 2005

More terms from Emeric Deutsch, Jun 19 2005

A080151 Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 163, 166, 171, 178, 187, 198, 211, 226, 243, 244, 247, 252, 259, 268, 279, 292, 307, 324, 325, 328, 333, 340, 349, 360, 373, 388, 405, 406, 409, 414, 421, 430, 441, 454, 469, 486, 487

Offset: 1

Eric W. Weisstein, Jan 31 2003

Record values in A003132. - Reinhard Zumkeller, Jul 10 2011

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Demlo Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A080150, A002477, A080160, A080161, A080162.

a n=(div n 9)*81+(mod n 9)^2
          A080151=map a [1..] \\ Chernin Nadav, Mar 06 2014

f := n -> 9*n - 81*frac(1/9*n) + 81*frac(1/9*n)^2:
map(f, [$1..100]); # Robert Israel, Aug 05 2019

(* by direct counting *)
Repunit[n_] := (-1 + 10^n)/9; A080151[n_]:=Plus @@ IntegerDigits[Repunit[n]^2];
(* by the formula *)
A080151[n_] := (9^2)*(n/9 - FractionalPart[n/9] + FractionalPart[n/9]^2)
(* or alternatively *)
A080151[n_] := 81*(Floor[n/9]+ FractionalPart[n/9]^2) (* Enrique Pérez Herrero, Nov 22 2009 *)

vector(100, n, (n\9)*81+(n%9)^2) \\ Colin Barker, Mar 05 2014

a(n) = A007953(A002477(n)).
a(n) = sqrt( A080150(n) ).
a(n) = (9^2)*(n/9 - {n/9} + {n/9}^2) = 81*(floor(n/9) + {n/9}^2), where the symbol {n} means fractional part of n. - Enrique Pérez Herrero, Nov 22 2009
a(n) = A003132(A051885(n)). - Reinhard Zumkeller, Jul 10 2011
a(9*n + k) = 81*n + k^2, with k in range 0 to 9. - Enrique Pérez Herrero, Nov 05 2022
Empirical g.f.: x*(17*x^8 + 15*x^7 + 13*x^6 + 11*x^5 + 9*x^4 + 7*x^3 + 5*x^2 + 3*x + 1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Mar 05 2014
Empirical g.f. confirmed. - Robert Israel, Aug 05 2019

A080161 Indices of Wonderful Demlo numbers A002477 whose digit sums are squares.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 51, 66, 81, 102, 123, 144, 225, 258, 291, 324, 363, 402, 441, 576, 593, 610, 627, 644, 661, 678, 695, 712, 729, 748, 767, 786, 805, 824, 843, 862, 881, 900, 1089, 1158, 1227, 1296, 1371, 1446, 1521, 1764, 1851, 1938, 2025

Offset: 1

Eric W. Weisstein, Jan 31 2003

The numbers 9*n^2 (A016766), with n > 0, are in this sequence. - Enrique Pérez Herrero, Sep 26 2020

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Demlo Number

Cf. A080160, A080162, A002477, A080151, A108690.

A080151[n_] := (9^2)*(n/9 - FractionalPart[n/9] + FractionalPart[n/9]^2)
Select[Range[10000], IntegerQ[Sqrt[A080151[#]]] &]
(* Enrique Pérez Herrero, Sep 26 2020 *)

for(k=1,10^5,issquare((k\9)*81+(k%9)^2)&&print1(k,", ")) \\ Jeppe Stig Nielsen, May 27 2023

A080162 Wonderful Demlo numbers A002477 whose digit sums are squares.

Original entry on oeis.org

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345679012345679012345679012345678987654320987654320987654320987654321

Offset: 1

Eric W. Weisstein, Jan 31 2003

The next term (a(11)) has 101 digits. - Harvey P. Dale, Jun 16 2025

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..23
Eric Weisstein's World of Mathematics, Demlo Number

Cf. A080160, A080161, A002477, A080151.

Select[LinearRecurrence[{111,-1110,1000},{1,121,12321},40],IntegerQ[Sqrt[Total[IntegerDigits[#]]]]&] (* Harvey P. Dale, Jun 16 2025 *)

for(k=1,100,my(d=((10^k-1)/9)^2); issquare(sumdigits(d)) && print1(d,", ")) \\ Jeppe Stig Nielsen, May 27 2023

A081648 Integers congruent to 0, 1, 4, 9, 16, 25, 36, 49 or 64 (mod 81) which are not squares.

Original entry on oeis.org

82, 85, 90, 97, 106, 117, 130, 145, 162, 163, 166, 171, 178, 187, 198, 211, 226, 243, 244, 247, 252, 259, 268, 279, 292, 307, 325, 328, 333, 340, 349, 360, 373, 388, 405, 406, 409, 414, 421, 430, 454, 469, 486, 487, 490, 495, 502, 511, 522, 535, 550, 567

Offset: 1

Robert G. Wilson v, Mar 26 2003

Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, New York, 1999, page 315.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A080151, A080160.

Select[ Range[567], (Mod[ #, 81] == 0 || Mod[ #, 81] == 1 || Mod[ #, 81] == 4 || Mod[ #, 81] == 9 || Mod[ #, 81] == 16 || Mod[ #, 81] == 25 || Mod[ #, 81] == 36 || Mod[ #, 81] == 49 || Mod[ #, 81] == 64) && !IntegerQ[ Sqrt[ # ]] & ]
Select[Range[600],IntegerQ[Sqrt[Mod[#,81]]]&&!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, May 13 2018 *)

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A108690 Square roots of the terms in A080160.

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A080151 Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.

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A080161 Indices of Wonderful Demlo numbers A002477 whose digit sums are squares.

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A080162 Wonderful Demlo numbers A002477 whose digit sums are squares.

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A081648 Integers congruent to 0, 1, 4, 9, 16, 25, 36, 49 or 64 (mod 81) which are not squares.

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