A226971
Numbers k such that the sum of digits of k^7 is equal to k.
Original entry on oeis.org
0, 1, 18, 27, 31, 34, 43, 53, 58, 68
Offset: 1
a(3) = 18 because 18^7 = 612220032 and 6+1+2+2+2+0+0+3+2 = 18.
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[n: n in [0..80] | &+Intseq(n^7) eq n]; // Vincenzo Librandi, Feb 23 2015
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Select[Range[0, 100], #==Total[IntegerDigits[#^7]]&]
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isok(k)=sumdigits(k^7)==k \\ Patrick De Geest, Dec 13 2024
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[n for n in (0..70) if sum((n^7).digits()) == n] # Bruno Berselli, Feb 23 2015
A355370
Irregular triangle read by rows in which row n lists the numbers that divide the sum of the digits of their n-th powers.
Original entry on oeis.org
1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 9, 1, 2, 3, 8, 9, 17, 18, 26, 27, 1, 3, 6, 7, 9, 22, 25, 28, 36, 1, 3, 9, 28, 35, 36, 46, 1, 2, 3, 7, 9, 18, 23, 45, 54, 64, 1, 3, 6, 9, 12, 15, 18, 27, 31, 34, 43, 53, 58, 68, 1, 3, 5, 6, 9, 15, 27, 46, 54, 63
Offset: 0
Triangle begins:
n=0: 1;
n=1: 1, 2, 3, 4, 5, 6, 7, 8, 9;
n=2: 1, 2, 3, 9;
n=3: 1, 2, 3, 8, 9, 17, 18, 26, 27;
n=4: 1, 3, 6, 7, 9, 22, 25, 28, 36;
n=5: 1, 3, 9, 28, 35, 36, 46;
n=6: 1, 2, 3, 7, 9, 18, 23, 45, 54, 64;
n=7: 1, 3, 6, 9, 12, 15, 18, 27, 31, 34, 43, 53, 58, 68;
n=8: 1, 3, 5, 6, 9, 15, 27, 46, 54, 63;
n=9: 1, 2, 3, 6, 7, 9, 16, 27, 36, 54, 71, 81;
n=10: 1, 3, 5, 6, 9, 18, 36, 82, 85, 94, 97, 106, 117;
...
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def ok(k, n): return sum(map(int, str(k**n)))%k==0
def row(n):
d, lim = 1, 1
while lim < n*9*d: d, lim = d+1, lim*10
yield from [k for k in range(1, lim+1) if ok(k, n)]
print([an for n in range(9) for an in row(n)]) # Michael S. Branicky, Jul 06 2022
A375343
Numbers which are the sixth powers of their digit sum.
Original entry on oeis.org
0, 1, 34012224, 8303765625, 24794911296, 68719476736
Offset: 1
68719476736 = (6+8+7+1+9+4+7+6+7+3+6)^6 = 64^6.
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for (k=0, sqrtnint(10^13,6), if (k^6 == sumdigits(k^6)^6, print1(k^6, ", ")); )
A379767
Triangle read by rows: row n lists numbers which are the n-th powers of their digit sum.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 81, 0, 1, 512, 4913, 5832, 17576, 19683, 0, 1, 2401, 234256, 390625, 614656, 1679616, 0, 1, 17210368, 52521875, 60466176, 205962976, 0, 1, 34012224, 8303765625, 24794911296, 68719476736, 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432
Offset: 1
Triangle begins:
1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
2 | 0, 1, 81;
3 | 0, 1, 512, 4913, 5832, 17576, 19683;
4 | 0, 1, 2401, 234256, 390625, 614656, 1679616;
5 | 0, 1, 17210368, 52521875, 60466176, 205962976;
6 | 0, 1, 34012224, 8303765625, 24794911296, 68719476736;
7 | 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432;
8 | 0, 1, 20047612231936, 72301961339136, 248155780267521;
9 | 0, 1, 3904305912313344, 45848500718449031, 150094635296999121;
...
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R(n) = for(j=2,oo, if((j*9)^n <10^j, return(j)));
row(n) = my(L=List()); for (k=0, sqrtnint(10^R(n),n), if (k^n == sumdigits(k^n)^n, listput(L, k^n))); Vec(L)
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