A004159 -id:A004159 - cp's OEIS frontend

A348303 a(n) is the largest n-digit number whose square has a digital sum equal to A348300(n).

7, 83, 937, 9417, 94863, 987917, 9893887, 99483667, 994927133, 9486778167, 99497231067, 999949483667, 9892825177313

Offset: 1

Bernardo Recamán and Freddy Barrera, Oct 10 2021

Array[#1 + Position[#2, Max[#2]][[-1, -1]] - 1 & @@ {#1, Map[Total@ IntegerDigits[#^2] &, Range[#1, #2]]} & @@ {10^(# - 1), 10^# - 1} &, 8] (* Michael De Vlieger, Oct 12 2021 *)

a(11) from Chai Wah Wu, Nov 18 2021

a(12)-a(13) from Martin Ehrenstein, Nov 20 2021

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

Original entry on oeis.org

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

A369953 a(n) is the least integer k such that the sum of the digits of k^2 is 9*n.

Original entry on oeis.org

0, 3, 24, 63, 264, 1374, 3114, 8937, 60663, 94863, 545793, 1989417, 5477133, 20736417, 82395387, 260191833, 706399164, 2428989417, 9380293167, 28105157886, 99497231067, 538479339417, 1974763271886, 4472135831667, 14106593458167, 62441868958167, 244744764757083, 836594274358167

Offset: 0

Zhining Yang, Feb 06 2024

3|a(n).

			a(3)=63 because k=63 is the least integer k such that the sum of the digits of k^2 = 3969 is 9*3 = 27 (3+9+6+9 = 27).

Zhining Yang, Table of n, a(n) for n = 0..40 (terms 19..40 from Zhao Hui Du)
Shouen Wang, Chinese BBS: How many of these A's are there?

Cf. A004159, A008591, A067179, A369955.

n=1;lst={};For[k=0,k<10^8,k+=3,If[Total[IntegerDigits[k^2]]==9*n,AppendTo[lst,k];n++]];lst

a(n) = my(k=0); while(sumdigits(k^2) != 9*n, k+=3); k; \\ Michel Marcus, Feb 17 2024

n=1
lst=[]
for k in range(0,10**8,3):
    if sum(int(d) for d in str(k*k))==9*n:
        lst.append(k)
        n=n+1
print(lst)

a(n) = A067179(4n).

a(19)-a(27) from Zhao Hui Du, Feb 09 2024

A370522 a(n) is the least n-digit number whose square has the maximum sum of digits (A348300(n)).

Original entry on oeis.org

7, 83, 836, 8937, 94863, 987917, 9893887, 99477133, 994927133, 9380293167, 99497231067, 926174913167, 9892825177313, 89324067192437, 943291047332683, 9949874270443813, 83066231922477313, 707106074079263583, 9429681807356492126, 94180040294109027313, 888142995231510436417, 8882505274864168010583

Offset: 1

Zhining Yang, Feb 21 2024

a(n) is the last n-digit term in A067179.

As the last two of the only nine known numbers whose square has a digit mean above 8.25 (see A164841), there is a high probability that a(30)=314610537013606681884298837387 and a(31)=9984988582817657883693383344833.

			a(3) = 836 because among all 3-digit numbers, 836 is the smallest whose square 698896 has the maximum sum of digits, 46 = A348300(3).

Zhining Yang, Table of n, a(n) for n = 1..24 (terms 11..24 from Zhao Hui Du)

Cf. A004159, A067179, A348300, A348303.

A348300={13,31,46,63,81,97,112,130,148,162,180};
A370522[n_]:=Do[If[Total@IntegerDigits[k^2]==A348300[[n]],Return[k];],{k,10^(n-1),10^n-1}];
Table[A370522[n],{n,8}]

def A370522(n):
    A348300=[0,13,31,46,63,81,97,112,130,148,162,180]
    for k in range(10**(n-1), 10**n):
        if sum(int(d) for d in str(k**2))==A348300[n]:
            return(k)
print([A370522(n) for n in range(1,9)])

a(11)-a(24) from Zhao Hui Du, Feb 23 2024

A056528 Sum of digits of square of sum of digits of square.

Original entry on oeis.org

1, 7, 9, 13, 13, 9, 16, 1, 9, 1, 7, 9, 13, 13, 9, 16, 10, 9, 1, 7, 9, 13, 13, 9, 16, 10, 9, 10, 16, 9, 13, 13, 9, 16, 1, 9, 10, 16, 9, 13, 13, 9, 16, 10, 9, 1, 16, 9, 13, 13, 9, 16, 10, 9, 1, 16, 9, 13, 13, 9, 16, 10, 18, 10, 16, 9, 13, 13, 9, 16, 1, 9, 10, 16, 9, 13, 13, 9, 16, 1, 9

Offset: 1

Henry Bottomley, Jun 19 2000

			a(2)=7 because sum of digits of square of 2 is 4 and sum of digits of square of 4 is 1+6=7

Cf. A004159 for sum of digits of square, A056020 where iteration settles to 1, A056020 where iteration settles to 9, A056527 where iteration settles to 13 and 16. See also A056529.

Array[Total[IntegerDigits[(Total[IntegerDigits[#^2]])^2]]&,90] (* Harvey P. Dale, Jan 17 2012 *)
Table[Nest[Total[IntegerDigits[#^2]]&,n,2],{n,90}] (* Harvey P. Dale, Mar 07 2018 *)

a(n)=A004159(A004159(n))

A056529 Number of iterations of sum of digits of square to reach 1, 9, 13 or 16.

Original entry on oeis.org

0, 3, 1, 2, 2, 1, 1, 2, 0, 1, 3, 1, 0, 1, 1, 0, 3, 1, 2, 3, 1, 1, 1, 2, 1, 3, 2, 3, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 3, 2, 3, 1, 3, 2, 3, 2, 1, 2, 2, 3, 2, 1, 3, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Jun 19 2000

			a(2)=3 because successive iterations give 2 -> 4 -> 7 -> 13 -> 16 -> 13 etc.

Cf. A004159 for sum of digits of square, A056528 for two iterations, A056020 where iteration settles to 1, A056020 where iteration settles to 9, A056527 where iteration settles to 13 and 16.

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

A117685 Squares for which the sum of the digits are cubes.

Original entry on oeis.org

0, 1, 100, 3969, 7569, 8649, 10000, 12996, 13689, 15876, 19881, 33489, 34596, 36864, 42849, 45369, 46656, 47961, 49284, 51984, 54756, 56169, 59049, 66564, 71289, 74529, 76176, 77841, 79524, 82944, 84681, 86436, 88209, 91809, 93636, 95481, 97344, 99225

Offset: 0

Luc Stevens (lms022(AT)yahoo.com), Apr 12 2006

			8649 is in the sequence because it is a square and the sum of the digits 8+6+4+9=27 is a cube.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A004159.

Select[Range[0,400]^2,IntegerQ[Power[Total[IntegerDigits[#]], (3)^-1]]&] (* Harvey P. Dale, Jan 28 2012 *)

Corrected (a(18) inserted) and extended by Harvey P. Dale, Jan 28 2012

A147523 a(n) = number of n-digit terms in A058369.

Original entry on oeis.org

3, 8, 37, 195, 1159, 6951, 42421, 261430, 1631123, 10238173, 64536808

Offset: 1

Zak Seidov, Apr 25 2009

			a(1)=3: 0,1,9; a(2)=8: 10,18,19,45,46,55,90,99.

A058369 Numbers n such that n and n^2 have same digit sum. A004159 Sum of digits of n^2. A007953 Digital sum (i.e. sum of digits) of n.

a(9) from Donovan Johnson, Dec 08 2009

a(10)-a(11) from Donovan Johnson, Jul 31 2011

A153751 Numbers k such that there are 15 digits in k^2 and for each factor f of 15 (1,3,5) the sum of digit groupings of size f is a square.

Original entry on oeis.org

10000000, 10000001, 10000002, 10000003, 10000004, 10000005, 10000010, 10000011, 10000012, 10000013, 10000020, 10000021, 10000022, 10000030, 10000031, 10000200, 10000300, 10011003, 10022000, 10035990, 10042440

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

The last term is a(2782) = 31616301. - Giovanni Resta, Jun 06 2015

			10000011^2 = 100000220000121;
1+0+0+0+0+0+2+2+0+0+0+0+1+2+1 = 9 = 3^2;
100+000+220+000+121 = 441 = 21^2;
10000+02200+00121 = 12321 = 111^2.

Giovanni Resta, Table of n, a(n) for n = 1..2782 (all terms)

Cf. A004159, A061910, A153745.

Select[Range[10^7,31622776],AllTrue[{Sqrt[Total[IntegerDigits[#^2]]],Sqrt[Total[ FromDigits/@ Partition[IntegerDigits[#^2],3]]],Sqrt[Total[FromDigits/@Partition[IntegerDigits[#^2],5]]]},IntegerQ]&] (* Harvey P. Dale, Apr 11 2023 *)

cp's OEIS Frontend

A348303 a(n) is the largest n-digit number whose square has a digital sum equal to A348300(n).

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

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A369953 a(n) is the least integer k such that the sum of the digits of k^2 is 9*n.

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A370522 a(n) is the least n-digit number whose square has the maximum sum of digits (A348300(n)).

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A056528 Sum of digits of square of sum of digits of square.

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A056529 Number of iterations of sum of digits of square to reach 1, 9, 13 or 16.

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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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A117685 Squares for which the sum of the digits are cubes.

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