A069530 -id:A069530 - cp's OEIS frontend

A088400 a(n)=A069530(n)/n.

29, 19, 0, 14, 13, 0, 8, 7, 0, 29, 19, 0, 5, 4, 0, 8, 7, 0, 2, 19, 0, 14, 4, 0, 17, 7, 0, 2, 1, 0, 5, 4, 0, 8, 7, 0, 2, 1, 0, 14, 4, 0, 8, 7, 0, 2, 1, 0, 5, 13, 0, 8, 7, 0, 11, 1, 0, 5, 4, 0, 17, 7, 0, 2, 1, 0, 5, 4, 0, 8, 16, 0, 2, 1, 0, 5, 4, 0, 8, 7, 0, 2, 1, 0, 5, 4, 0, 8, 7, 0, 2, 1, 0, 5, 4, 0, 26, 25

Offset: 1

Ray Chandler, Sep 28 2003

R. J. Mathar, Table of n, a(n) for n = 1..1110

Cf. A069530.

read("transforms"):
A088400 := proc(n)
    local m ;
    if modp(n,3) = 0 then
         0 ;
    else
        for m from 1 do
            if digsum(m*n) = 11 then
                return m ;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Aug 06 2019

A052217 Numbers whose sum of digits is 3.

Original entry on oeis.org

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

Offset: 1

Henry Bottomley, Feb 01 2000

From Joshua S.M. Weiner, Oct 19 2012: (Start)
Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.
0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)
A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014
Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

Cf. A069521 to A069530, A069532 to A069537.
Cf. A007953, A218043 (subsequence).
Row n=3 of A245062.
Other digit sums: A011557 (1), A052216 (2), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Other bases: A014311 (binary), A226636 (ternary), A179243 (Zeckendorf).
Cf. A242614, A242627.
Cf. A003056, A002262 (triangular coordinates), A056556, A056557, A056558 (tetrahedral coordinates).

a052217 n = a052217_list !! (n-1)
a052217_list = filter ((== 3) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0,3],n],{n,6}],1],Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)
Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

apply( {A052217_row(n,s,t=-1)=vector(n*(n+1)\2,k,t++>s&&t=!s++;10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))
print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A052217(n): return 10**((m:=integer_nthroot(6*n,3)[0])-(a:=n<=comb(m+2,3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)),2)) # Chai Wah Wu, Dec 11 2024

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

Offset changed from 0 to 1 by Vincenzo Librandi, Mar 07 2013

A069537 Multiples of 2 whose digit sum is 2.

Original entry on oeis.org

2, 20, 110, 200, 1010, 1100, 2000, 10010, 10100, 11000, 20000, 100010, 100100, 101000, 110000, 200000, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000, 100000010, 100000100, 100001000

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002024, A002260, A088404 (half).
Cf. A069521 to A069530, A069532, A069534, A069536.
Subsequence of A005349.
Row n=2 of A245062.

a(n) = my(r,s=sqrtint((n-1)<<1,&r), x=s+(r>s), y=if(r>s,r-s,r+s)>>1); 10^x + 10^y; \\ Kevin Ryde, Jul 17 2025

from itertools import product
def agen():
  digits = 1
  while True:
    for i in range(digits-2): yield int("1"+"0"*(digits-3-i)+"1"+"0"*i+"0")
    yield int("2"+"0"*(digits-1))
    digits += 1
g = agen()
print([next(g) for i in range(32)]) # Michael S. Branicky, Feb 20 2021

a(n) = 10^A002024(n-1) + 10^A002260(n-1) for n >= 2. - Kevin Ryde, Jul 17 2025

Corrected and extended by Ray Chandler, Sep 28 2003

A063997 Multiples of 4 whose digits add to 4.

Original entry on oeis.org

4, 40, 112, 220, 400, 1012, 1120, 1300, 2020, 2200, 3100, 4000, 10012, 10120, 10300, 11020, 11200, 12100, 13000, 20020, 20200, 21100, 22000, 30100, 31000, 40000, 100012, 100120, 100300, 101020, 101200, 102100, 103000, 110020, 110200, 111100

Offset: 1

Lisa O. Coulter (lcoulter(AT)stetson.edu), Sep 06 2001

			4 is an element of the sequence, since 4 is a multiple of 4 the sum of whose digits is 4; 220 is an element of the sequence, since 220 = 4*55 and 2 + 2+ 0 = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 218 terms from Harry J. Smith)

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217.

Row n=4 of A245062.

Select[4Range[120000],Total[IntegerDigits[#]]==4&] (* Harvey P. Dale, May 07 2011 *)

SumDE(x,y)= { local(s); s=0; while (x>9 && sHarry J. Smith, Sep 05 2009

More terms from Ray Chandler, Sep 28 2003

A069532 Smallest even number with digit sum n.

Original entry on oeis.org

10, 2, 12, 4, 14, 6, 16, 8, 18, 28, 38, 48, 58, 68, 78, 88, 98, 198, 298, 398, 498, 598, 698, 798, 898, 998, 1998, 2998, 3998, 4998, 5998, 6998, 7998, 8998, 9998, 19998, 29998, 39998, 49998, 59998, 69998, 79998, 89998, 99998, 199998, 299998, 399998, 499998

Offset: 1

Amarnath Murthy, Apr 01 2002

Chai Wah Wu, Table of n, a(n) for n = 1..8999
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A069521 to A069530.
Cf. A000918 (smallest even number with bit sum n), A051885 (smallest number with digit sum n).
Cf. A077491.

t={}; Do[i=2; While[Total[IntegerDigits[i]]!=n,i=i+2]; AppendTo[t,i],{n,48}]; t (* Jayanta Basu, May 18 2013 *)

a(n) = {my(k = 2); while(sumdigits(k) != n, k+=2); k;} \\ Michel Marcus, Mar 18 2016

From Chai Wah Wu, Sep 15 2020: (Start)
a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n > 17.
G.f.: 2*x*(45*x^16 - 45*x^15 + 45*x^14 - 45*x^13 + 45*x^12 - 45*x^11 + 45*x^10 - 45*x^9 + 5*x^8 - 4*x^7 + 5*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 5*x^2 - 4*x + 5)/((x - 1)*(10*x^9 - 1)). (End)
a(n) = 2 * A077491(n). - Alois P. Heinz, Sep 15 2020

More terms from Ray Chandler, Jul 28 2003

A062768 Multiples of 6 such that the sum of the digits is equal to 6.

Original entry on oeis.org

6, 24, 42, 60, 114, 132, 150, 204, 222, 240, 312, 330, 402, 420, 510, 600, 1014, 1032, 1050, 1104, 1122, 1140, 1212, 1230, 1302, 1320, 1410, 1500, 2004, 2022, 2040, 2112, 2130, 2202, 2220, 2310, 2400, 3012, 3030, 3102, 3120, 3210, 3300, 4002, 4020, 4110

Offset: 1

Lisa O Coulter (lisa_coulter(AT)my-deja.com), Jul 17 2001

Even numbers with sum of digits equal to 6 are Harshad numbers (A005349). - Davide Rotondo, Sep 04 2020

			60 is a member of the sequence since 60 / 6 = 10 and 6 + 0 = 6; 114 is also an element since 114 is divisible by 6 and 1 + 1+ 4 = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A063416, A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997, A069540.
Subsequence of A005349.
Row n=6 of A245062.

: var stk: stack; end; minarg := 0; maxarg := 900; n := 6; for k := minarg to maxarg do m := k*n; s := itoa(m); for j := 0 to length(s) - 1 do stack_push(stk,atoi(s[j..j])); end; if sum(stack2array(stk)) = n then write(m," "); end; end;.

Select[ Range[ 6, 4200, 6 ], Plus @@ IntegerDigits[ # ] == 6 & ]

More terms from Klaus Brockhaus, Jul 20 2001

A063416 Multiples of 7 whose sum of digits is equal to 7.

Original entry on oeis.org

7, 70, 133, 322, 511, 700, 1015, 1141, 1204, 1330, 2023, 2212, 2401, 3031, 3220, 4102, 5110, 7000, 10024, 10150, 10213, 10402, 11032, 11221, 11410, 12040, 12103, 13111, 13300, 15001, 20041, 20104, 20230, 21112, 21301, 22120, 23002, 24010

Offset: 1

Klaus Brockhaus, Jul 20 2001

Numbers are all 7 mod 63.

			133 = 19*7 and 1+3+3 = 7, so 133 is a term of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 700 terms from Harry J. Smith)

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997, A069540, A062768.

Row n=7 of A245062.

: var stk: stack; end; minarg := 0; maxarg := 5000; n := 7; for k := minarg to maxarg do m := k*n; s := itoa(m); for j := 0 to length(s) - 1 do stack_push(stk,atoi(s[j..j])); end; if sum(stack2array(stk)) = n then write(m," "); end; end;.

Select[Range[7, 25000, 7], Plus @@ IntegerDigits[ # ] == 7 &]

forstep(m=0, 70000, 7, if(vecsum(digits(m))==7, print1(m, ", "))) \\ Harry J. Smith, Aug 20 2009

A069540 Multiples of 5 with digit sum 5.

Original entry on oeis.org

5, 50, 140, 230, 320, 410, 500, 1040, 1130, 1220, 1310, 1400, 2030, 2120, 2210, 2300, 3020, 3110, 3200, 4010, 4100, 5000, 10040, 10130, 10220, 10310, 10400, 11030, 11120, 11210, 11300, 12020, 12110, 12200, 13010, 13100, 14000, 20030, 20120

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997.

Row n=5 of A245062.

Select[5*Range[5000],Total[IntegerDigits[#]]==5&] (* Harvey P. Dale, Nov 08 2017 *)

Corrected and extended by Ray Chandler, Sep 28 2003

A069534 Smallest multiple of 5 with digit sum n.

Original entry on oeis.org

10, 20, 30, 40, 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 195, 295, 395, 495, 595, 695, 795, 895, 995, 1995, 2995, 3995, 4995, 5995, 6995, 7995, 8995, 9995, 19995, 29995, 39995, 49995, 59995, 69995, 79995, 89995, 99995, 199995, 299995, 399995, 499995, 599995

Offset: 1

Amarnath Murthy, Apr 01 2002

a(6) onwards the pattern is evident.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A069521 to A069530, A069532, A069533.

Cf. A077492.

t={}; Do[i=5; While[Total[IntegerDigits[i]]!=n,i=i+5]; AppendTo[t,i],{n,46}]; t (* Jayanta Basu, May 19 2013 *)
With[{f=5*Range[200000]},Flatten[Table[Select[f,Total[IntegerDigits[#]] == n&,1],{n,50}]]] (* Harvey P. Dale, Dec 31 2013 *)

A069534(n)=(((n+4)%9+1)*10^((n+4)\9)-5)*10^(n<5) \\ M. F. Hasler, Sep 16 2016

a(n) = ((n+4)%9+1)*10^floor((n+4)/9)-5 for all n > 4, where % is the binary mod/remainder operator. - M. F. Hasler, Sep 16 2016
From Chai Wah Wu, Sep 15 2020: (Start)
a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n > 14.
G.f.: 5*x*(72*x^13 - 18*x^12 - 18*x^11 - 18*x^10 - 18*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 - 7*x^4 + 2*x^3 + 2*x^2 + 2*x + 2)/((x - 1)*(10*x^9 - 1)). (End)
a(n) = 5 * A077492(n). - Alois P. Heinz, Sep 15 2020

More terms from Ray Chandler, Jul 28 2003

A069543 Multiples of 8 with digit sum 8.

Original entry on oeis.org

8, 80, 152, 224, 440, 512, 800, 1016, 1160, 1232, 1304, 1520, 2024, 2240, 2312, 2600, 3032, 3104, 3320, 4040, 4112, 4400, 5120, 6200, 8000, 10016, 10160, 10232, 10304, 10520, 11024, 11240, 11312, 11600, 12032, 12104, 12320, 13040, 13112, 13400

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997, A069540, A062768, A063416.

Row n=8 of A245062.

More terms from Ray Chandler, Sep 28 2003

cp's OEIS Frontend

A088400 a(n)=A069530(n)/n.

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A052217 Numbers whose sum of digits is 3.

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A069537 Multiples of 2 whose digit sum is 2.

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A063997 Multiples of 4 whose digits add to 4.

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A069532 Smallest even number with digit sum n.

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A062768 Multiples of 6 such that the sum of the digits is equal to 6.

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A063416 Multiples of 7 whose sum of digits is equal to 7.

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