A245062 -id:A245062 - cp's OEIS frontend

A052217 Numbers whose sum of digits is 3.

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

A052223 Numbers whose sum of digits is 9.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 108, 117, 126, 135, 144, 153, 162, 171, 180, 207, 216, 225, 234, 243, 252, 261, 270, 306, 315, 324, 333, 342, 351, 360, 405, 414, 423, 432, 441, 450, 504, 513, 522, 531, 540, 603, 612, 621, 630, 702, 711, 720, 801, 810

Offset: 1

Henry Bottomley, Feb 01 2000

Any term of this sequence with an 11 appended cannot have 11 as prime factor. See A075154. [Lekraj Beedassy, Sep 27 2009]
A007953(a(n)) = 9; number of repdigits = #{9,333,1^9} = A242627(9) = 3. - Reinhard Zumkeller, Jul 17 2014
A010872(a(n)) = A010878(a(n)) = 0. - Ilya Gutkovskiy, Jun 04 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A007953.
Row n=9 of A245062.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.

a052223 n = a052223_list !! (n-1)
a052223_list = filter ((== 9) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1500] | &+Intseq(n) eq 9 ]; // Vincenzo Librandi, Mar 08 2013

Select[Range[1500], Total[IntegerDigits[#]] == 9 &] (* Vincenzo Librandi, Mar 08 2013 *)

More terms from Larry Reeves (Larryr(AT)acm.org), Sep 05 2000

Offset changed by Bruno Berselli, Mar 07 2013

A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 92, 94, 97, 101, 102, 103, 106, 107, 109, 113, 114, 116, 118, 122, 124, 127, 131, 134, 136, 137, 138

Offset: 1

Omar E. Pol, Apr 06 2014

nonn

All odd primes are in the sequence because the parts of the symmetric representation of sigma(prime(i)) are [m, m], where m = (1 + prime(i))/2, for i >= 2.
There are no odd composite numbers in this sequence.
First differs from A173708 at a(13).
Since sigma(p*q) >= 1 + p + q + p*q for odd p and q, the symmetric representation of sigma(p*q) has more parts than the two extremal ones of size (p*q + 1)/2; therefore, the above comments are true. - Hartmut F. W. Hoft, Jul 16 2014
From Hartmut F. W. Hoft, Sep 16 2015: (Start)
The following two statements are equivalent:
  (1) The symmetric representation of sigma(n) has two parts, and
  (2) n = q * p where q is in A174973, p is prime, and 2 * q < p.
For a proof see the link and also the link in A071561.
This characterization allows for much faster computation of numbers in the sequence - function a239929F[] in the Mathematica section - than computations based on Dyck paths. The function a239929Stalk[] gives rise to the associated irregular triangle whose columns are indexed by A174973 and whose rows are indexed by A065091, the odd primes. (End)
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime:
(a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p].
(b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p].
(c) each number k is of the form 2^m * p with p prime [by definition].
m = 1: (a) A100484 even semiprimes (except 4 and 6)
       (b) A161344 (except 4, 6 and 8)
       (c) A001747 (except 2, 4 and 6)
m = 2: (a) A270298
       (b) A161424 (except 16, 20, 24, 28 and 32)
       (c) A001749 (except 8, 12, 20 and 28)
m = 3: (a) A270301
       (b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)
       (c) sequence not in OEIS
b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End)

			From _Hartmut F. W. Hoft_, Sep 16 2015: (Start)
a(23) = 52 = 2^2 * 13 = q * p with q = 4 in A174973 and 8 < 13 = p.
a(59) = 136 = 2^3 * 17 = q * p with q = 8 in A174973 and 16 < 17 = p.
The first six columns of the irregular triangle through prime 37:
   1    2    4    6    8   12 ...
  -------------------------------
   3
   5   10
   7   14
  11   22   44
  13   26   52   78
  17   34   68  102  136
  19   38   76  114  152
  23   46   92  138  184
  29   58  116  174  232  348
  31   62  124  186  248  372
  37   74  148  222  296  444
  ...
(End)

Hartmut F. W. Hoft, Proof of Characterization Theorem

Column 2 of A240062.
Cf. A000203, A006254, A065091, A071561, A174973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A238443, A239660, A239663, A239665, A239931-A239934, A244050, A245062, A262626.
Cf. A000040, A001747, A001749, A091999, A100484, A161344, A161424, A162528, A270298, A270301. - Hartmut F. W. Hoft, Dec 06 2016

isA174973 := proc(n)
    option remember;
    local k,dvs;
    dvs := sort(convert(numtheory[divisors](n),list)) ;
    for k from 2 to nops(dvs) do
        if op(k,dvs) > 2*op(k-1,dvs) then
            return false;
        end if;
    end do:
    true ;
end proc:
A174973 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA174973(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
isA239929 := proc(n)
    local i,p,j,a73;
    for i from 1 do
        p := ithprime(i+1) ;
        if p > n then
            return false;
        end if;
        for j from 1 do
            a73 := A174973(j) ;
            if a73 > n then
                break;
            end if;
            if p > 2*a73 and n = p*a73 then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 200 do
    if isA239929(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 04 2018

(* sequence of numbers k for m <= k <= n having exactly two parts *)
(* Function a237270[] is defined in A237270 *)
a239929[m_, n_]:=Select[Range[m, n], Length[a237270[#]]==2&]
a239929[1, 260] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
(* test for membership in A174973 *)
a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}]
a174973[n_]:=Select[Range[n], a174973Q]
(* compute numbers satisfying the condition *)
a239929Stalk[start_, bound_]:=Module[{p=NextPrime[2 start], list={}}, While[start p<=bound, AppendTo[list, start p]; p=NextPrime[p]]; list]
a239929F[n_]:=Sort[Flatten[Map[a239929Stalk[#, n]&, a174973[n]]]]
a239929F[138] (* data *)(* Hartmut F. W. Hoft, Sep 16 2015 *)

Entries b(i, j) in the irregular triangle with rows indexed by i>=1 and columns indexed by j>=1 (alternate indexing of the example):

b(i,j) = A000040(i+1) * A174973(j) where A000040(i+1) > 2 * A174973(j). - Hartmut F. W. Hoft, Dec 06 2016

Extended beyond a(56) by Michel Marcus, Apr 07 2014

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

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

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

A279769 Numbers n such that the sum of digits of 9n is 18.

Original entry on oeis.org

11, 21, 22, 31, 32, 33, 41, 42, 43, 44, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 121, 122, 131, 132, 133, 141

Offset: 1

M. F. Hasler, Dec 18 2016

Differs from A084854 from a(55) = 110 on.
Numbers n such that A008591(n) is a term of A235228. - Felix Fröhlich, Dec 18 2016
The digital sum of 9n is always a multiple of 9, and never zero. For most numbers < 100, the digital sum is equal to 9, but for example in the range [91..110] all numbers except 100 have their digital sum equal to 18. The b-file / graph gives a hint on the "asymptotic" distribution / density of this set. After a "flat" range like that at [91..110] there comes a record gap. Sizes [and upper ends] of record gaps are: 10 [a(2) = 21], 11 [a(56) = 121, a(119) = 231, a(188) = 341, ..., a(553) = 891, a(616) = 1001], 21 [a(671) = 1121], 31 [a(1331) = 2231], ..., 91 [a(4339) = 8891], 101 [a(4621) = 10001], 121 [a(4841) = 11121], 231 [a(9176) = 22231], ..., 891 [a(24217) = 88891], 1001 [a(25213) = 100001], 1121 [a(25928) = 111121], 2231 [a(47510) = 222231], ..., 8891 [a(108577) = 888891], 10001 [a(111574) = 1000001], 11121 [a(113576) = 1111121], 22231 [a(202511) = 2222231], ..., 88891 [a(416215) = 8888891], ... - M. F. Hasler, Dec 22 2016

M. F. Hasler, Table of n, a(n) for n = 1..25212

Cf. A007953 (digital sum), A008591, A084854.
Cf. A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), 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).

Select[Range@ 141, Total@ IntegerDigits[9 #] == 18 &]

is(n) = sumdigits(9*n)==18 \\ Felix Fröhlich, Dec 18 2016

a(n) = A235228(n)/9.

cp's OEIS Frontend

A052217 Numbers whose sum of digits is 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Python

Python

Formula

Extensions

A052223 Numbers whose sum of digits is 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Extensions

A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A069537 Multiples of 2 whose digit sum is 2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A063997 Multiples of 4 whose digits add to 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

ARIBAS

Mathematica

Extensions

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

Views

Author