A052018 -id:A052018 - cp's OEIS frontend

A352461 Numbers k whose decimal expansion starts with the sum of digits of k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063, 1072, 1081, 1090, 1109, 1118, 1127, 1136, 1145, 1154, 1163, 1172, 1181, 1190, 1209, 1218, 1227, 1236

Offset: 1

Eric Angelini and Carole Dubois, Mar 17 2022

			    10 is a term because   "10" starts with  "1", which is the sum 1 + 0.
   119 is a term because  "119" starts with "11", which is the sum 1 + 1 + 9.
  1018 is a term because "1018" starts with "10", which is the sum 1 + 0 + 1 + 8.

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

Cf. A052018, A352462.

q[n_] := Module[{d = IntegerDigits[n], s, ds, nds}, s = Plus @@ d; ds = IntegerDigits[s]; nds = Length[ds]; ds == d[[1 ;; nds]]]; Select[Range[1240], q] (* Amiram Eldar, Mar 17 2022 *)
Select[Range[1236],StringStartsQ[ToString[#],ToString[Plus@@IntegerDigits[#]]]&] (* Ivan N. Ianakiev, Mar 18 2022 *)
Select[Range[1300],SequencePosition[IntegerDigits[#],IntegerDigits[Total[IntegerDigits[#]]]][[;;,1]]=={1}&] (* Harvey P. Dale, Nov 03 2024 *)

def ok(n): s = str(n); return s.startswith(str(sum(map(int, s))))
print([k for k in range(1, 1237) if ok(k)]) # Michael S. Branicky, Mar 17 2022

A352462 Numbers k with the property that k ends in the sum of the digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1810, 1811, 1812, 1813, 1814, 1815, 1816, 1817, 1818, 1819, 2710, 2711, 2712, 2713, 2714, 2715, 2716, 2717, 2718, 2719, 3610, 3611, 3612, 3613, 3614, 3615, 3616, 3617, 3618, 3619, 4510, 4511, 4512, 4513, 4514, 4515, 4516

Offset: 1

Eric Angelini and Carole Dubois, Mar 17 2022

			a(10) = 910, which ends in 10 = 9 + 1 + 0.
a(20) = 1810, which ends in 10 = 1 + 8 + 1 + 0.
a(39) = 2719, which ends in 19 = 2 + 7 + 1 + 9.

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

Cf. A052018, A352461.

q[n_] := Module[{d = IntegerDigits[n], s, ds, nds}, s = Plus @@ d; s = IntegerDigits[s]; nds = Length[ds]; ds == d[[-nds ;; -1]]]; Select[Range[4516], q] (* Amiram Eldar, Mar 17 2022 *)
Select[Range[4516],StringEndsQ[ToString[#],ToString[Plus@@IntegerDigits[#]]]&] (* Ivan N. Ianakiev, Mar 18 2022 *)
sdkQ[n_]:=Module[{idn=IntegerDigits[n],d},d=IntegerDigits[Total[idn]];d==Take[idn,-Length[d]]]; Select[Range[5000],sdkQ] (* Harvey P. Dale, Apr 19 2022 *)

def ok(n): s = str(n); return s.endswith(str(sum(map(int, s))))
print([k for k in range(1, 4517) if ok(k)]) # Michael S. Branicky, Mar 17 2022

A162016 Numbers k such that the sum of the decimal digits of k is a substring of k and of k^3.

Original entry on oeis.org

1, 4, 5, 6, 9, 10, 40, 50, 60, 90, 100, 400, 500, 600, 900, 910, 1000, 1009, 1018, 1027, 1145, 1158, 1178, 1198, 1245, 1345, 1363, 1427, 1509, 1609, 1672, 1736, 1809, 1810, 1814, 1836, 2177, 2710, 2712, 3610, 3612, 4000, 4125, 4510, 4514, 5000, 5134, 5144, 5410

Offset: 1

Claudio Meller, Jun 24 2009

			1158 is a term because its digital sum is 1+1+5+8 = 15 and "15" is a substring of 1158 and of 1158^3 = 1552836312.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A052018, A162017.

def sd(n): return sum(map(int, str(n)))
def ok(n): s = str(sd(n)); return s in str(n) and s in str(n**3)
print([k for k in range(1, 5411) if ok(k)]) # Michael S. Branicky, Jan 31 2022

a(47) and beyond from Michael S. Branicky, Jan 31 2022

A162017 Numbers k such that the sum of the decimal digits of k is a substring of k, of k^2 and of k^3.

Original entry on oeis.org

1, 5, 6, 10, 50, 60, 100, 500, 600, 910, 1000, 1009, 1018, 1027, 1145, 1810, 2710, 3610, 4510, 5000, 5410, 6000, 6310, 7210, 7212, 8110, 9010, 9100, 9925, 10000, 10009, 10018, 10027, 10036, 10045, 10054, 10063, 10072, 10081, 10090, 10108, 10117, 10126, 10135

Offset: 1

Claudio Meller, Jun 24 2009

			1145 is a term because its digital sum is 1+1+4+5 = 11 and "11" is a substring of 1145, of 1145^2 = 1311025 and of 1145^3 = 1501123625.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A052018, A162016.

sddQ[n_]:=With[{d=IntegerDigits[Total[IntegerDigits[n]]]},SequenceCount[IntegerDigits[n],d]>0&&SequenceCount[IntegerDigits[n^2],d]>0&&SequenceCount[IntegerDigits[n^3],d]>0]; Select[Range[11000],sddQ] (* Harvey P. Dale, Jun 16 2025 *)

def sd(n): return sum(map(int, str(n)))
def ok(n): s = str(sd(n)); return all(s in str(t) for t in [n, n**2, n**3])
print([k for k in range(1, 10140) if ok(k)]) # Michael S. Branicky, Jan 31 2022

a(21) and beyond from Michael S. Branicky, Jan 31 2022

A202272 Numbers n with k digits such that each sum of 1, 2, ..., k digits of n is a substring of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 200, 300, 400, 500, 600, 700, 800, 910, 1000, 1009, 1090, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 9010, 9100, 10000, 10009, 10090, 10900, 20000, 30000, 40000, 50000, 60000, 70000

Offset: 1

Jaroslav Krizek, Jan 06 2012

Conjecture: sequence contains only
- numbers of the form k*10^j (j>=0, k>=0),
- numbers of the form 10X, where X = string of k digits containing one digit 9 and k-1 digits 0 (k>=1),
- numbers of form 9Y10Z; where Y = string of k digits 0 (k>=0), Z = string of j digits 0 (j>=0).
Subsequence of A052018.

			109 is a term because all possible sums of digits 0, 1, 1, 9, 9, 10 are its substrings.

Cf. A052018 (numbers k such that the sum of the digits of k is substring of k).

A344486 a(n) is the least k such that the sum of digits of k is a substring of n and the sum of digits of n is a substring of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 29, 39, 49, 59, 69, 79, 89, 99, 10, 2, 399, 499, 599, 699, 799, 899, 999, 101, 11, 3, 4999, 5999, 6999, 7999, 8999, 9999, 102, 111, 12, 4, 59999, 69999, 79999, 89999, 99999, 103, 112, 112, 13, 5, 699999, 799999, 899999, 999999

Offset: 0

Rémy Sigrist, May 21 2021

The sequence is well defined:
- for any number n with sum of digits d,
- by necessity, d <= n,
- the number k obtained by concatenating n-d 1's in front of d meets the requirements.

			For n = 11:
- the sum of digits of 11 is 2,
- the sum of digits of a(n) must equal 1 or 11,
    - the numbers whose sum of digits is 1 are the powers of 10,
    - 2 cannot be a substring of a power of 10,
    - the first number with sum of digits 11 is 29,
    - 2 is a substring of 29,
- so a(11) = 29.

Rémy Sigrist, Perl program for A344486

Cf. A007953, A052018, A344487.

Perl
```
See Links section.
```

a(10 * n) = a(n).

A162015 Numbers n with property that the sum of the digits of n is substring of n and of n^2.

Original entry on oeis.org

1, 5, 6, 10, 50, 60, 100, 500, 600, 910, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1090, 1128, 1145, 1309, 1454, 1463, 1654, 1781, 1810, 2127, 2499, 2710, 3610, 3613, 4105, 4175, 4510, 5000, 5410, 6000, 6123, 6133, 6310, 7152, 7210, 7212, 8110, 8131, 9010

Offset: 1

Claudio Meller, Jun 24 2009

1454 is in the list because its digital sum is 1+4+5+4 = 14 and "14" is a substring of 1454 and of 1454^2 = 2114116

Harvey P. Dale, Table of n, a(n) for n = 1..2500

A052018

sdssQ[n_]:=Module[{idn1=IntegerDigits[n],idn2=IntegerDigits[n^2],idnt}, idnt= IntegerDigits[ Total[idn1]];SequenceCount[idn1,idnt]>0 && SequenceCount[idn2,idnt]>0]; Select[Range[10000],sdssQ] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Jul 29 2015 *)

A384508 Nonnegative integers k such that the digits of k include the digits of the digital sum of k as a (not necessarily contiguous) subsequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 300, 400, 500, 600, 700, 800, 900, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1000

Offset: 1

Felix Huber, Jun 26 2025

			196 is a term because 1 + 9 + 6 = 16 and 1, 6 is a subsequence of 1, 9, 6.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A052018, A005349, A007953, A046829.

A384508:=proc(n)
    option remember;
    local k,L;	
    if n=1 then
        0
    else
        for k from procname(n-1)+1 do
            L:=convert(k,'base',10);
            if ArrayTools:-IsSubsequence(convert(add(L),'base',10),L) then
                return k
            fi
        od
    fi;
end proc;
seq(A384508(n),n=1..58);

A121939 Palindromic numbers that contain the sum of their digits as a substring.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 919, 1881, 8118, 9229, 10801, 11711, 12621, 13531, 14441, 15351, 16261, 17171, 18081, 21612, 25152, 27972, 28882, 29792, 31513, 33133, 41014, 41114, 41214, 41314, 41414, 41514, 41614, 41714, 41814, 41914, 51315

Offset: 1

Tanya Khovanova, Sep 03 2006

Cf. Palindromic subsequence of A052018 - Sum of digits of n is substring of n.

Select[Table[n, {n, 100000}], Length[StringPosition[ToString[ # ], ToString[Plus @@ IntegerDigits[ # ]]]] > 0 && ToString[ # ] == StringReverse[ToString[ # ]] &]

A302768 Numbers k whose sum and product of digits are substrings of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 200, 300, 400, 500, 600, 700, 800, 900, 910, 911, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063, 1072, 1081, 1090, 1108, 1109, 1118, 1181, 1190, 1209, 1236, 1290, 1309, 1390, 1409, 1490

Offset: 1

Paolo P. Lava, Apr 13 2018

First term greater than 9 without digit 0 is 119.

First term greater than 9 without digits 0 and 1 is 3499236.

			911 => 9 + 1 + 1 = 11 and 9 * 1 * 1 = 9;
1309 => 1 + 3 + 0 + 9 = 13 and 1 * 3 * 0 * 9 = 0;
3499236 => 3 + 4 + 9 + 9 + 2 + 3 + 6 = 36 and 3 * 4 * 9 * 9 * 2 * 3 * 6 = 34992.

Cf. A007953, A007954, A052018, A227510.

select(n->searchtext(convert(convert(convert(n, base, 10), `+`),string),x.n)*searchtext(convert(convert(convert(n, base, 10), `*`),string),x.n)>0,[$1..1500]);

Select[Range@ 1500, With[{d = IntegerDigits[#]}, AllTrue[IntegerDigits@ {Total@ d, Times @@ d}, SequenceCount[d, #] > 0 &]] &] (* Michael De Vlieger, Apr 21 2018 *)

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A352461 Numbers k whose decimal expansion starts with the sum of digits of k.

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A352462 Numbers k with the property that k ends in the sum of the digits of k.

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A162016 Numbers k such that the sum of the decimal digits of k is a substring of k and of k^3.

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A162017 Numbers k such that the sum of the decimal digits of k is a substring of k, of k^2 and of k^3.

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A202272 Numbers n with k digits such that each sum of 1, 2, ..., k digits of n is a substring of n.

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A344486 a(n) is the least k such that the sum of digits of k is a substring of n and the sum of digits of n is a substring of k.

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Perl

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A162015 Numbers n with property that the sum of the digits of n is substring of n and of n^2.

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Mathematica

A384508 Nonnegative integers k such that the digits of k include the digits of the digital sum of k as a (not necessarily contiguous) subsequence.

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Maple

A121939 Palindromic numbers that contain the sum of their digits as a substring.

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