A080151 -id:A080151 - cp's OEIS frontend

A003132 Sum of squares of digits of n.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49

Offset: 0

N. J. A. Sloane

It is easy to show that a(n) < 81*(log_10(n)+1). - Stefan Steinerberger, Mar 25 2006
It is known that a(0)=0 and a(1)=1 are the only fixed points of this map. For more information about iterations of this map, see A007770, A099645 and A000216 ff. - M. F. Hasler, May 24 2009
Also known as the "Happy number map", since happy numbers A007770 are those whose trajectory under iterations of this map ends at 1. - M. F. Hasler, Jun 03 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Hugo Steinhaus, One Hundred Problems in Elementary Mathematics, Dover New York, 1979, republication of English translation of Sto Zadań, Basic Books, New York, 1964. Chapter I.2, An interesting property of numbers, pp. 11-12 (available on Google Books).

T. D. Noe, Table of n, a(n) for n = 0..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29; [preprint from author's web page, PostScript format].
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
B. M. Stewart, Sums of functions of digits, Canad. J. Math., 12 (1960), 374-389.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for Colombian or self numbers and related sequences

Cf. A052034, A052035.
Cf. A007953, A055017, A076313, A076314.
Concerning iterations of this map, see A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4, this is the only nontrivial limit cycle), A000221 (starting with 5), A008460 (starting with 6), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009
Cf. A080151, A051885 (record values and where they occur).
Cf. A257588, A332919.

a003132 0 = 0
a003132 x = d ^ 2 + a003132 x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015, Aug 07 2012, Jul 10 2011

[0] cat [&+[d^2: d in Intseq(n)]: n in [1..80]]; // Bruno Berselli, Feb 01 2013

A003132 := proc(n) local d; add(d^2,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Oct 16 2010

Table[Sum[DigitCount[n][[i]]*i^2, {i, 1, 9}], {n, 0, 40}] (* Stefan Steinerberger, Mar 25 2006 *)
Total/@(IntegerDigits[Range[0,80]]^2) (* Harvey P. Dale, Jun 20 2011 *)

A003132(n)=norml2(digits(n)) \\ M. F. Hasler, May 24 2009, updated Apr 12 2015

def A003132(n): return sum(int(d)**2 for d in str(n)) # Chai Wah Wu, Apr 02 2021

a(n) = n^2 - 20*n*floor(n/10) + 81*(Sum_{k>0} floor(n/10^k)^2) + 20*Sum_{k>0} floor(n/10^k)*(floor(n/10^k) - floor(n/10^(k+1))). - Hieronymus Fischer, Jun 17 2007
a(10n+k) = a(n)+k^2, 0 <= k < 10. - Hieronymus Fischer, Jun 17 2007
a(n) = A007953(A048377(n)) - A007953(n). - Reinhard Zumkeller, Jul 10 2011

More terms from Stefan Steinerberger, Mar 25 2006
Terms checked using the given PARI code, M. F. Hasler, May 24 2009
Replaced the Maple program with a version which works also for arguments with >2 digits, R. J. Mathar, Oct 16 2010
Added ref to Porges. Steinhaus also treated iterations of this function in his Polish book Sto zadań, but I don't have access to it. - Don Knuth, Sep 07 2015

A051885 Smallest number whose sum of digits is n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 8999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 89999, 99999, 199999, 299999, 399999, 499999

Offset: 0

Felice Russo, Dec 15 1999

This is also the list of lunar triangular numbers: A087052 with duplicates removed. - N. J. A. Sloane, Jan 25 2011
Numbers n such that A061486(n) = n. - Amarnath Murthy, May 06 2001
The product of digits incremented by 1 is the same as the number incremented by 1. If a(n) = abcd...(a,b,c,d, etc. are digits of a(n)) {a(n) + 1} = (a+1)*(b+1)(c+1)*(d+1)*..., e.g., 299 + 1 = (2+1)*(9+1)*(9+1) = 300. - Amarnath Murthy, Jul 29 2003
A138471(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008
a(n+1) = A108971(A179988(n)). - Reinhard Zumkeller, Aug 09 2010, Jul 10 2011
Positions of records in A003132: A080151(n) = A003132(a(n)). - Reinhard Zumkeller, Jul 10 2011
a(n) = A242614(n,1). - Reinhard Zumkeller, Jul 16 2014
A254524(a(n)) = 1. - Reinhard Zumkeller, Oct 09 2015
The slowest strictly increasing sequence of nonnegative integers such that, for any two terms, calculating the difference of their decimal representations requires no borrowing. - Rick L. Shepherd, Aug 11 2017

Iain Fox, Table of n, a(n) for n = 0..9000 (first 101 terms from Reinhard Zumkeller)
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
A. Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000.
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A061104, A061105, A061486, A007953, A067043, A087052.
Numbers of form i*b^j-1 (i=1..b-1, j >= 0) for bases b = 2 through 9: A000225, A062318, A180516, A181287, A181288, A181303, A165804, A140576. - N. J. A. Sloane, Jan 25 2011
Cf. A002283.
Cf. A254524.

a051885 n = (m + 1) * 10^n' - 1 where (n',m) = divMod n 9
-- Reinhard Zumkeller, Jul 10 2011

Magma
```
[i*10^j-1: i in [1..9], j in [0..5]];
```

b:=10; t1:=[]; for j from 0 to 15 do for i from 1 to b-1 do t1:=[op(t1), i*b^j-1]; od: od: t1; # N. J. A. Sloane, Jan 25 2011

a[n_] := (Mod[n, 9] + 1)*10^Floor[n/9] - 1; Table[a[n], {n, 0, 49}](* Jean-François Alcover, Dec 01 2011, after Henry Bottomley *)

A051885(n) = (n%9+1)*10^(n\9)-1  \\ M. F. Hasler, Jun 17 2012

first(n) = Vec(x*(x^2 + x + 1)*(x^6 + x^3 + 1)/((x - 1)*(10*x^9 - 1)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

def A051885(n): return ((n % 9)+1)*10**(n//9)-1 # Chai Wah Wu, Apr 04 2021

These are the numbers i*10^j-1 (i=1..9, j >= 0). - N. J. A. Sloane, Jan 25 2011
a(n) = ((n mod 9) + 1)*10^floor(n/9) - 1 = a(n-1) + 10^floor((n-1)/9). - Henry Bottomley, Apr 24 2001
a(n) = A037124(n+1) - 1. - Reinhard Zumkeller, Jan 03 2008, Jul 10 2011
G.f.: x*(x^2+x+1)*(x^6+x^3+1) / ((x-1)*(10*x^9-1)). - Colin Barker, Feb 01 2013

More terms from James Sellers, Dec 16 1999

Offset fixed by Reinhard Zumkeller, Jul 10 2011

A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.

Original entry on oeis.org

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321

Offset: 1

N. J. A. Sloane

Only the first nine terms of this sequence are palindromes. - Bui Quang Tuan, Mar 30 2015

Not all of the terms are Demlo numbers as defined by Kaprekar, i.e., concat(L,M,R) with M and L+R repdigits using the same digit. For example, a(10), a(19), a(28) are not, but a(k) for k = 11, 12, ..., 18 are. - M. F. Hasler, Nov 18 2017

			From _José de Jesús Camacho Medina_, Apr 01 2016: (Start)
n=1: ....................... 1 = 9 / 9;
n=2: ..................... 121 = 1089 / 9;
n=3: ................... 12321 = 110889 / 9;
n=4: ................. 1234321 = 11108889 / 9;
n=5: ............... 123454321 = 1111088889 / 9;
n=6: ............. 12345654321 = 111110888889 / 9;
n=7: ........... 1234567654321 = 11111108888889 / 9;
n=8: ......... 123456787654321 = 1111111088888889 / 9;
n=9: ....... 12345678987654321 = 111111110888888889 / 9.        (End)
a(11) = concat(L = 1234567901, R = 20987654321), with L + R = 22222222222 = 2*(10^11-1)/9, of same length as R. - _M. F. Hasler_, Nov 23 2017

D. R. Kaprekar, On Wonderful Demlo numbers, Math. Stud., 6 (1938), 68.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. [Mentions this sequence.]
K. R. Gunjikar and D. R. Kaprekar, Theory of Demlo numbers, J. Univ. Bombay, Vol. VIII, Part 3, Nov. 1939, pp. 3-9. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Demlo Number
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275, A080151.

[((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Jul 26 2011

A002477 := proc(n)
    (10^n-1)^2/81 ;
end proc:
seq(A002477(n),n=1..12) ; # R. J. Mathar, Aug 06 2019

Table[FromDigits[PadRight[{},n,1]]^2,{n,15}] (* Harvey P. Dale, Oct 16 2012 *)
(10^Range[20] - 1)^2/81 (* Paolo Xausa, Aug 03 2025 *)

A002477(n):=((10^n - 1)/9)^2$
makelist(A002477(n),n,1,10); /* Martin Ettl, Nov 12 2012 */

a(n) = (10^n\9)^2 \\ Charles R Greathouse IV, Jul 25 2011

G.f.: x*(1+10*x) / ((1-x)*(1-10*x)*(1-100*x)). - Simon Plouffe in his 1992 dissertation
a(n+1) = 100*a(n) + 20*A000042(n) + 1; a(1) = 1. - Reinhard Zumkeller, May 31 2010
a(n) = A000042(n)^2.
a(n) = A075412(n)/9 = A178630(n)/18 = A178631(n)/27 = A075415(n)/36 = A178632(n)/45 = A178633(n)/54 = A178634(n)/63 = A178635(n)/72 = A059988(n)/81. - Reinhard Zumkeller, May 31 2010
a(n+2) = -1000*a(n)+110*a(n+1)+11. - Alexander R. Povolotsky, Jun 06 2014
E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81. - Stefano Spezia, May 23 2025

Minor edits from N. J. A. Sloane, Aug 18 2009

Further edits from Reinhard Zumkeller, May 12 2010

A080160 Squares that are digit sums of Wonderful Demlo numbers A002477.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 324, 441, 576, 729, 900, 1089, 1296, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 9801, 10404, 11025, 11664

Offset: 1

N. J. A. Sloane, Jun 19 2005

These are the squares in A080151.

x^2 is a term iff x mod 81 is in {0, 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}. - Robert Israel, Jul 30 2025

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

Cf. A002477, A080151, A080161, A080162.

Cf. A007953.

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 b(n) else fi end: seq(a(n),n=0..2000); # Emeric Deutsch, Jun 19 2005

A080151[n_] := (9^2)*(n/9 - FractionalPart[n/9] + FractionalPart[n/9]^2)
A080151[Select[Range[10000], IntegerQ[Sqrt[A080151[#]]] &]]
(* Enrique Pérez Herrero, Nov 05 2022 *)

More terms from Emeric Deutsch, Jun 19 2005

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 *)

A080150 Let m = Wonderful Demlo number A002477(n); a(n) = square of the sum of digits of m.

Original entry on oeis.org

1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 6724, 7225, 8100, 9409, 11236, 13689, 16900, 21025, 26244, 26569, 27556, 29241, 31684, 34969, 39204, 44521, 51076, 59049, 59536, 61009, 63504, 67081, 71824, 77841, 85264, 94249, 104976, 105625

Offset: 1

Eric W. Weisstein, Jan 31 2003

These numbers are themselves squares, their square roots being in A080151.

Eric Weisstein's World of Mathematics, Demlo Number

Cf. A080151, A002477.

a(n) = (A080151(n))^2.

Name corrected by Robert Israel, Aug 06 2019

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A003132 Sum of squares of digits of n.

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A051885 Smallest number whose sum of digits is n.

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A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.

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A080160 Squares that are digit sums of Wonderful Demlo numbers A002477.

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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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