A067179 -id:A067179 - cp's OEIS frontend

A348300 a(n) is the largest number that is the digit sum of the square of an n-digit number.

13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, 193, 211, 229, 244, 262, 277, 297, 310, 331, 343, 360, 378, 396

Offset: 1

Bernardo Recamán and Freddy Barrera, Oct 10 2021

18*n-a(n) appears to be nondecreasing. - Chai Wah Wu, Nov 18 2021
According to new data 18*n-a(n) sometimes decreases. - David A. Corneth, Feb 21 2024
a(n) is the digit sum of the square of the last n-digit integer in A067179. - Zhao Hui Du, Mar 04 2024
a(n) appears to be approximately equal to 16.5*n. - Zhining Yang, Mar 12 2024
a(n) modulo 9 is either 0, 1, 4 or 7. - Chai Wah Wu, Apr 04 2024

			a(3) = 46 because 46 is the largest digital sum encountered among the squares (that of 937) of all 3-digit numbers. Such maximal digital sum can be achieved by more than one square (squares of 836 and 883 also have digital sum 46). Largest of these is A348303.

Cf. A004159, A348303, A370522.

Cf. A371728.

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

def A348300(n): return max(sum(int(d) for d in str(m**2)) for m in range(10**(n-1),10**n)) # Chai Wah Wu, Jun 26 2024

def A348300(n):
    return max(sum((k^2).digits()) for k in (10^(n-1)..10^n-1))

a(n) = Max_{k=10^(n-1)..10^n-1} A004159(k).

a(11) from Chai Wah Wu, Nov 18 2021
a(12)-a(13) from Martin Ehrenstein, Nov 20 2021
a(14)-a(24) from Zhao Hui Du, Feb 23 2024
Name edited by Jon E. Schoenfield, Mar 10 2024

A067178 Smallest square whose sum of digits is A056991(n).

Original entry on oeis.org

1, 4, 16, 9, 64, 49, 169, 576, 289, 1849, 4489, 3969, 17956, 6889, 27889, 69696, 98596, 97969, 499849, 1887876, 698896, 2778889, 4999696, 9696996, 19998784, 46689889, 66699889, 79869969, 277788889, 478996996, 876988996, 3679999569, 1749999889

Offset: 1

Amarnath Murthy, Jan 09 2002

The smallest square giving a possible digit sum.

Giovanni Resta, Table of n, a(n) for n = 1..94
Amarnath Murthy, Fabricating perfect squares with a given valid digits sum, page 154 in "Generalized partitions and New Ideas on Number Theory and Smarandache Sequences" by A. Murthy and C. Ashbacher

Cf. A056991, A067179.

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 24 2003

Offset corrected. - R. J. Mathar, Aug 26 2009

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

A369955 a(n) is the least integer m such that the sum of the digits of m^2 is 9*(k+n) where k is the number of digits of m.

Original entry on oeis.org

3, 63, 3114, 8937, 94863, 5477133, 82395381, 706399164, 9380293167, 99497231067, 4472135831667, 62441868958167, 836594274358167, 9983486364492063, 435866837461509417, 707106074079263583, 77453069648658793167, 754718284918279954614, 8882505274864168010583

Offset: 0

Zhining Yang, Feb 06 2024

3|a(n).

			a(2)=3114 because 3114 is the least 4-digit integer whose square has digit sum 9*(4+2) = 9*6 = 54: 3114^2 = 9696996 and 9+6+9+6+9+9+6 = 54.

Shouen Wang, Chinese BBS: How many of these A's are there?

Cf. A004159, A067179, A369953.

n=0;For[k=0,k<10^8/3,k++,If[Total[IntegerDigits[9k^2]]==9*(n+Ceiling@Log10@(3k)),Print[{n,3k}];n++]]

a(n) = my(m=1); while (sumdigits(m^2) != 9*(#Str(m)+n), m++); m; \\ Michel Marcus, Feb 10 2024

def sd(n):
    return sum(int(d) for d in str(n*n))
n=0
for k in range(0,10**8,3):
    if sd(k)==9*(len(str(k))+n):
        print([n,k])
        n+=1

a(9)-a(18) from Zhao Hui Du, Feb 19 2024

A358702 a(n) is the least k > 0 such that the sum of the decimal digits of k^2 is n, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 0, 7, 0, 0, 13, 0, 24, 17, 0, 0, 43, 0, 0, 67, 0, 63, 134, 0, 0, 83, 0, 0, 167, 0, 264, 314, 0, 0, 313, 0, 0, 707, 0, 1374, 836, 0, 0, 1667, 0, 0, 2236, 0, 3114, 4472, 0, 0, 6833, 0, 0, 8167, 0, 8937, 16667, 0, 0, 21886, 0, 0, 29614

Offset: 1

Hugo Pfoertner, Dec 04 2022

The nonzero terms are A067179.

Cf. A056991, A231897 (similar for binary weight).

cp's OEIS Frontend

A348300 a(n) is the largest number that is the digit sum of the square of an n-digit number.

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A067178 Smallest square whose sum of digits is A056991(n).

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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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A369955 a(n) is the least integer m such that the sum of the digits of m^2 is 9*(k+n) where k is the number of digits of m.

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A358702 a(n) is the least k > 0 such that the sum of the decimal digits of k^2 is n, or 0 if no such k exists.

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