A038206 -id:A038206 - cp's OEIS frontend

A104113 Numbers which when chopped into one, two or more parts, added and squared result in the same number.

0, 1, 81, 100, 1296, 2025, 3025, 6724, 8281, 9801, 10000, 55225, 88209, 136161, 136900, 143641, 171396, 431649, 455625, 494209, 571536, 627264, 826281, 842724, 893025, 929296, 980100, 982081, 998001, 1000000, 1679616, 2896804, 3175524, 4941729, 7441984

Offset: 1

Bodo Zinser, Mar 05 2005

Every term is congruent to 0 or 1 modulo 9. - Andrea Tarantini, Sep 27 2021

			1296 is a term since (1+29+6)^2 = 36^2 = 1296.

John Drake, Table of n, a(n) for n = 1..408 (terms 1..80 from Mehrad Mahmoudian, terms 81..225 from Giovanni Resta)
Mehrad Mahmoudian, R code to produce the sequence
Mehrad Mahmoudian, Decompositions for a(1)-a(80)
Project Euler, Problem 719: Number Splitting (2020)

Cf. A038206, A102766.

Join[{0},Select[Select[Range@3000^2,Mod[#,9]<2&],(n=#;MemberQ[(Total/@(FromDigits/@#&/@Union[DeleteCases[SplitBy[#,#==-1&],{-1}]&/@(Insert[IntegerDigits@n,-1,#]&/@(List/@#&/@Rest@Subsets[Range@IntegerLength@n]))]))^2,#])&]] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

def expr(t, d): # can you express target t with digits d, only adding +'s
    if t < 0: return False
    if t == int(d): return True
    return any(expr(t-int(d[:i]), d[i:]) for i in range(1, len(d)))
def aupto(limit):
    alst, k, k2 = [], 0, 0
    while k2 <= limit:
        if expr(k, str(k2)):
            alst.append(k2)
        k, k2 = k+1, k2 + 2*k + 1
    return alst
print(aupto(7500000)) # Michael S. Branicky, Sep 27 2021

a(n) = A038206(n)^2. - Andrea Tarantini, Sep 27 2021

a(30) and beyond from Mehrad Mahmoudian, Dec 16 2019

A038211 Can express a(n) with the digits of a(n)^4 in order, only adding plus signs.

Original entry on oeis.org

0, 1, 7, 10, 19, 22, 25, 27, 28, 34, 36, 37, 45, 52, 54, 55, 58, 61, 63, 64, 67, 72, 73, 76, 79, 81, 82, 85, 91, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 127, 133, 135, 142, 144, 145, 148, 153, 154, 157, 162, 163, 166, 169, 172, 175, 178, 180, 181

Offset: 1

Erich Friedman

All terms == 0, 1, 4 or 7 (mod 9). - Robert Israel, Sep 06 2020

			a(6)=22 is in the sequence because 22^4=234256 and 2+3+4+2+5+6=22.
a(8)=27 is in the sequence because 27^4=531441 and 5+3+14+4+1=27.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5000 terms from Robert Israel)

Cf. A001477, A038206, A038209.

F:= proc(n,t) local d; option remember;
   if n = t then return true fi;
   if n < t then return false fi;
   for d from 1 to min(ilog10(t)+1,ilog10(n)+1) do
     if procname(floor(n/10^d), t - (n mod 10^d)) then return true fi
   od;
   false
end proc:
select(t -> F(t^4,t), [$0..1000]); # Robert Israel, Sep 06 2020

Offset changed by Robert Israel, Sep 06 2020

A305706 a(n) = smallest m such that the sum of digits of n^m is greater than n, or 0 if no such m exists.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 2, 2, 2, 3, 0, 4, 3, 2, 2, 3, 3, 2, 4, 3, 14, 4, 5, 4, 4, 5, 4, 5, 6, 6, 14, 5, 6, 5, 6, 6, 6, 5, 5, 6, 14, 7, 6, 8, 6, 7, 6, 8, 7, 7, 16, 6, 6, 8, 7, 8, 7, 7, 9, 7, 15, 8, 8, 7, 8, 8, 9, 8, 8, 8, 18, 12, 9, 9, 8, 9, 9, 9, 9, 9, 18, 10, 11, 11, 10, 11, 11, 10, 9, 10, 17, 11, 11, 10, 11, 10, 12, 12, 10, 11, 0

Offset: 0

Max Alekseyev, Jun 08 2018

a(n) = smallest m such that A007953(n^m) > n, or 0 if no such m exists.

If n is a term of A178501, then a(n) = 0. - Felix Fröhlich, Jun 10 2018

Cf. A007953, A038206, A178501, A305707.

a(n) = if(sumdigits(n) < 2, return(0), my(m=2); while(1, if(sumdigits(n^m) > n, return(m)); m++)) \\ Felix Fröhlich, Jun 10 2018

A305707 Numbers n such that for every k = 1, 2, ..., A305706(n)-1, it is possible to insert plus signs into the decimal representation of n^k to make sum equal n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 17, 45, 91, 100, 675, 945, 964, 990, 991, 1000, 1296, 1702, 2728, 4879, 5050, 5149, 5292, 7777, 8938, 9325, 9765, 9901, 9909, 9918, 9945, 9955, 9964, 10000, 10512, 12222, 12727, 17271, 41149, 42643, 48790, 50050, 59284, 72612, 75331, 77778, 81118, 87571, 93574, 95121, 99226, 99630, 99631, 99703, 99901, 99909, 99918, 99945, 99955, 99964, 99991, 100000, 104878, 117343, 329967, 461539

Offset: 1

Max Alekseyev, Jun 09 2018

It is not possible to insert pluses in the decimal representation of n^A305706(n) to make the sum equal n.

Terms starting with a(15)=45 form a subsequence of A038206.

			For n = 45, we have A305706(n) = 6, and
n^1 = 45 with 45 = n;
n^2 = 2025 with 20+25 = n;
n^3 = 91125 with 9+11+25 = n;
n^4 = 4100625 with 4+10+0+6+25 = n;
n^5 = 184528125 with 18+4+5+2+8+1+2+5 = n.
So, 45 is a term.

Cf. A038206, A305706

cp's OEIS Frontend

A104113 Numbers which when chopped into one, two or more parts, added and squared result in the same number.

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A038211 Can express a(n) with the digits of a(n)^4 in order, only adding plus signs.

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A305706 a(n) = smallest m such that the sum of digits of n^m is greater than n, or 0 if no such m exists.

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A305707 Numbers n such that for every k = 1, 2, ..., A305706(n)-1, it is possible to insert plus signs into the decimal representation of n^k to make sum equal n.

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