A061909 -id:A061909 - cp's OEIS frontend

A351650 Integers m such that digsum(m) divides digsum(m^2) where digsum = sum of digits = A007953.

1, 2, 3, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 24, 27, 30, 31, 33, 36, 42, 45, 46, 54, 55, 63, 72, 74, 81, 90, 92, 99, 100, 101, 102, 103, 108, 110, 111, 112, 113, 117, 120, 121, 122, 123, 126, 128, 130, 132, 135, 144, 145, 153, 162, 171, 180, 189, 190, 191, 198

Offset: 1

Bernard Schott, Feb 16 2022

This is a generalization of a problem proposed by French site Diophante in link.
The smallest term k such that the corresponding quotient = n is A280012(n).
The quotient is 1 iff m is in A058369 \ {0}.
If k is in A061909, then digsum(k^2) = digsum(k)^2.
If k is a term, 10*k is also a term.
There are infinitely many m such that both m and m+1 are in the sequence, for example subsequence A002283 \ {0}.
Corresponding quotients are in A351651.

			digit sum of 42 = 4+2 = 6; then 42^2 = 1764, digit sum of 1764 = 1+7+6+4 = 18; as 6 divides 18, 42 is a term.

Diophante, A1730 - Des chiffres à sommer pour un entier (in French).

Cf. A004159, A007953, A351651.

Subsequences: A002283, A011557, A052268, A058369, A061909, A093136, A093138, A254066, A280012.

Select[Range[200], Divisible[Total[IntegerDigits[#^2]], Total[IntegerDigits[#]]] &] (* Amiram Eldar, Feb 16 2022 *)

is(n)=sumdigits(n^2)%sumdigits(n) == 0 \\ David A. Corneth, Feb 16 2022

def sd(n): return sum(map(int, str(n)))
def ok(n): return sd(n**2)%sd(n) == 0
print([m for m in range(1, 200) if ok(m)]) # Michael S. Branicky, Feb 16 2022

A004159(a(n)) = A007953(a(n)) * A351651(n).

More terms from David A. Corneth, Feb 16 2022

A061911 Square root of the sum of the digits of k^2 when this sum is a square.

Original entry on oeis.org

1, 2, 3, 3, 3, 1, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 4, 3, 5, 4, 5, 5, 4, 5, 3, 5, 4, 1, 2, 3, 4, 4, 3, 2, 3, 4, 5, 3, 4, 5, 4, 4, 4, 4, 4, 3, 5, 5, 5, 4, 5, 3, 5, 4, 5, 5, 2, 3, 4, 4, 3, 4, 5, 4, 5, 4, 4, 5, 4, 4, 4, 3, 5, 5, 6, 4, 5, 5, 5, 5, 5, 5, 3, 4, 4, 4, 5, 3, 4, 3, 5, 4, 5, 4, 5, 4, 3

Offset: 1

Asher Auel, May 17 2001

			6^2 = 36 and 3+6 = 9 is a square, thus 3 is in the sequence. 13^2 = 169 and 1+6+9 = 16 is a square, thus 4 is in the sequence.

Cf. A007953, A004159, A061909, A061910, A061912.

readlib(issqr): f := []: for n from 1 to 200 do if issqr(convert(convert(n^2,base,10),`+`)) then f := [op(f),sqrt(convert(convert(n^2,base,10),`+`))] fi; od; f;

Select[Table[Sqrt[Total[IntegerDigits[n^2]]],{n,350}],IntegerQ] (* Jayanta Basu, May 06 2013 *)

a(n) = sqrt(A004159(A061910(n))) = sqrt(A007953((A061910(n))^2)). - Zak Seidov, Jul 04 2012

A069263 (Sum of digits of n)^3 = sum of digits of n^3.

Original entry on oeis.org

1, 2, 10, 11, 20, 100, 101, 110, 111, 200, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 2000, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11011, 11100, 20000, 100000

Offset: 1

Joseph L. Pe, Apr 19 2002

			Let s(n) = sum of digits of n. Then s(101)^3 = 8 and s(101^3) = s(1030301) = 8, so 101 belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..12456 (terms < 10^20)

Cf. A061909.

f[n_] := Apply[Plus, IntegerDigits[n]]; Select[Range[10^5], f[ #^3] == f[ # ]^3 &]

is(n)=sumdigits(n^3)==sumdigits(n)^3 \\ Charles R Greathouse IV, Dec 19 2016

A169952 Second entry in row n of triangle in A169950.

Original entry on oeis.org

1, 2, 5, 8, 13, 20, 33, 48, 75, 100, 145, 204, 293, 396, 559, 746, 1027, 1340, 1809, 2342, 3177, 4050, 5369, 6920, 9013, 11360, 14837, 18718, 24081, 29952, 38219, 47662, 60549, 74618, 93847, 115960, 145319, 177548, 221675, 270334, 335123, 406290, 500915

Offset: 1

N. J. A. Sloane, Aug 01 2010

nonn

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..99 (terms 1..64 from Andrew Howroyd)
Index entries for sequences related to carryless arithmetic

Related to thickness: A169940-A169954, A061909.

A169947 = Import["https://oeis.org/A169947/b169947.txt", "Table"][[All, 2]];
Join[{1}, Differences[A169947]] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A169947(n) - A169947(n-1) for n>1. - Andrew Howroyd, Jul 09 2017

a(16)-a(29) from Nathaniel Johnston, Nov 15 2010

Terms a(30) and beyond from Andrew Howroyd, Jul 09 2017

A225301 Number of solutions to rev(x^2) = rev(x)^2 with at most n digits, where the function rev(x) reverses the digits of x.

Original entry on oeis.org

4, 10, 25, 64, 154, 363, 820, 1811, 3873, 8161, 16682, 33757, 66865, 130938, 251983, 480794, 903982, 1685564, 3106009, 5677864, 10276936, 18464659, 32891188, 58169965, 102136773, 178096365, 308593320, 531191385, 909227947, 1546356486, 2617639293

Offset: 1

David Radcliffe, May 05 2013

Numbers (other than 0) that end in zero are excluded.

			For n = 2 the a(2) = 10 solutions are 0, 1, 2, 3, 11, 12, 13, 21, 22, 31.

David Radcliffe, Python code for calculating a(n).

Cf. A085305, A061909.

Equals one more than the partial sums of A098701.

A169948 Fourth entry in row n of triangle in A169945.

Original entry on oeis.org

1, 2, 6, 14, 29, 52, 96, 160, 277, 450, 712, 1086, 1657, 2448, 3636, 5280, 7635, 10840, 15392, 21372, 29655, 40580, 55282, 74620, 100651, 134232, 178922, 236488, 312019, 408550, 534288, 692978, 897931, 1156256, 1485650, 1897704, 2421635, 3071608, 3894042

Offset: 2

N. J. A. Sloane, Aug 01 2010

nonn

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Index entries for sequences related to carryless arithmetic

Related to thickness: A169940-A169954, A061909.

Cf. A143823, A196723.

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]];
c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n - 1, sn], 0] + c[n-1, s]]];
A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
a[n_] := a[n] = A196723[n + 1] - A143823[n + 1];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 40}] (* Jean-François Alcover, Aug 27 2019, after Alois P. Heinz in A196723 and A143823 *)

a(n) = A196723(n+1) - A143823(n+1). - Andrew Howroyd, Jul 09 2017

a(15)-a(28) from Nathaniel Johnston, Nov 12 2010

a(29)-a(40) from Andrew Howroyd, Jul 09 2017

A169953 Third entry in row n of triangle in A169950.

Original entry on oeis.org

1, 1, 4, 8, 15, 23, 44, 64, 117, 173, 262, 374, 571, 791, 1188, 1644, 2355, 3205, 4552, 5980, 8283, 10925, 14702, 19338, 26031, 33581, 44690, 57566, 75531, 96531, 125738, 158690, 204953, 258325, 329394, 412054, 523931, 649973, 822434, 1018332, 1274909

Offset: 2

N. J. A. Sloane, Aug 01 2010

nonn

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Index entries for sequences related to carryless arithmetic

Related to thickness: A169940-A169954, A061909.

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]]; c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n-1, sn], 0] + c[n-1, s]]];
A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
a[n_] := A196723[n+1] - A196723[n] - A143823[n+1] + A143823[n];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 42}] (* Jean-François Alcover, Sep 07 2019, after Alois P. Heinz in A196723 and A143823 *)

a(n) = A169948(n)-A169948(n-1) for n>2. - Andrew Howroyd, Jul 09 2017

a(15)-a(28) and definition corrected by Nathaniel Johnston, Nov 15 2010

Offset corrected and a(30)-a(42) from Andrew Howroyd, Jul 09 2017

A351651 a(n) is the quotient obtained when digsum(m^2) is divided by digsum(m), with digsum = sum of digits = A007953 and m = A351650(n).

Original entry on oeis.org

1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 1, 1, 2, 1, 3, 2, 2, 2, 1, 2, 1, 1, 2, 3, 4, 2, 2, 3, 4, 5, 3, 3, 4, 5, 3, 3, 2, 4, 3, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 3, 4, 3, 3, 2, 3, 4, 5, 3, 2, 4, 5, 2, 2, 3, 3, 3, 3, 2, 2, 2, 3, 2, 1, 3, 4, 3, 4, 5

Offset: 1

Bernard Schott, Feb 16 2022

All positive integers are terms of this sequence (see A280012).
a(n) = 1 iff m = A351650(n) is a term of A058369 \ {0}.
a(n) = digsum(n) if m = A351650(n) is a term of A061909 \ {0}.

			A351650(8) = 13, then digsum(13) = 1+3 = 4 while digsum(13^2) = digsum(169) = 1+6+9 = 16; hence, a(8) = 16/4 = 4.

Diophante, A1730 - Des chiffres à sommer pour un entier (in French).

Cf. A002283, A004159, A007953, A058369, A061909, A254066, A280012, A351650.

Select[Total[IntegerDigits[#^2]]/Total[IntegerDigits[#]]& /@ Range[300], IntegerQ] (* Amiram Eldar, Feb 16 2022 *)

lista(nn) = {my(list = List(), q); for (n=1, nn, if (denominator(q=sumdigits(n^2)/sumdigits(n))==1, listput(list, q));); Vec(list);} \\ Michel Marcus, Feb 16 2022

a(n) = A004159(A351650(n)) / A007953(A351650(n)).

More terms from Michel Marcus, Feb 16 2022

A098690 Number of solutions to rev(x^2)=rev(x)^2 below 10^n.

Original entry on oeis.org

3, 9, 24, 63, 153, 362, 819, 1810, 3872, 8160, 16681, 33756, 66864, 130937, 251982, 480793, 903981, 1685563, 3106008, 5677863, 10276935, 18464658, 32891187, 58169964, 102136772, 178096364, 308593319, 531191384, 909227946, 1546356485, 2617639292

Offset: 1

Martin Renner, Oct 27 2004

Partial sums of A098701. - Michel Marcus, Apr 11 2014
Excludes multiples of 10. - David Radcliffe, Aug 28 2021
Also the number of skinny numbers (A061909) with n digits, excluding 0. - David Radcliffe, Aug 28 2021

			For n = 2 the a(2) = 9 solutions are 1, 2, 3, 11, 12, 13, 21, 22, 31. - _David Radcliffe_, Aug 28 2021

Cf. A085305, A225301.

f[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Differences[Table[Length[Select[Range[10^n],f[#^2]==f[#]^2&]],{n,0,6}]] (* Geoffrey Critzer, Dec 18 2013 *)

def rev(n): return int(str(n)[::-1])
def a(n): return sum(k % 10 and rev(k**2) == rev(k)**2 for k in range(10**n)) # David Radcliffe, Aug 28 2021

a(7),a(8) from Geoffrey Critzer, Dec 18 2013

Extended using A098701 by Michel Marcus, Apr 11 2014

A159547 Smallest number b such that the number whose digits are n in base b is a skinny number.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 50, 65, 82, 2, 3, 5, 10, 17, 26, 37, 50, 65, 82, 5, 5, 9, 13, 17, 26, 37, 50, 65, 82, 10, 10, 13, 19, 25, 31, 37, 50, 65, 82, 17, 17, 17, 25, 33, 41, 49, 57, 65, 82, 26, 26, 26, 31, 41, 51, 61, 71, 81, 91, 37, 37, 37, 37, 49, 61, 73, 85, 97, 109

Offset: 1

J. Lowell, Apr 14 2009

I assume that "the number whose digits are n in base b" means the number Sum c_i b^i, where the decimal expansion of n is Sum c_i 10^i. - N. J. A. Sloane, Jun 19 2021

			a(10) = 2 because 10^2 = 100 in all bases >= 2.
a(14) = 17 because 14_16 = 20_10, so the square is 400_10 = (1,9,0)_16, but digitsum((1,9,0)_16) = 10 != digitsum((1,4)_16)^2; while in base 17, 14_17 = 21_10, so the square is 441_10 = (1,8,16)_17 and digitsum((1,8,16)_17) = 25 = digitsum((1,4)_17)^2.

Cf. A061909.

a(n) = my(d=digits(n), s); s=vecsum(d); for(b=1+vecmax(d), oo, if(s^2==sumdigits(fromdigits(d, b)^2, b), return(b))); \\ Jinyuan Wang, Jun 19 2021

a(n) <= 10 iff n is in A061909.

More terms from Jinyuan Wang, Jun 19 2021

cp's OEIS Frontend

A351650 Integers m such that digsum(m) divides digsum(m^2) where digsum = sum of digits = A007953.

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A061911 Square root of the sum of the digits of k^2 when this sum is a square.

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A069263 (Sum of digits of n)^3 = sum of digits of n^3.

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A169952 Second entry in row n of triangle in A169950.

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A225301 Number of solutions to rev(x^2) = rev(x)^2 with at most n digits, where the function rev(x) reverses the digits of x.

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A169948 Fourth entry in row n of triangle in A169945.

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A169953 Third entry in row n of triangle in A169950.

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Extensions

A351651 a(n) is the quotient obtained when digsum(m^2) is divided by digsum(m), with digsum = sum of digits = A007953 and m = A351650(n).

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