A130181
Largest k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k, or 0 if no such k exists.
Original entry on oeis.org
486, 1215, 4374, 4672, 12862, 12649, 23408, 32761, 47477, 56852, 59048, 90746, 116864, 112346, 139472, 149705, 190512, 234247, 254015, 0, 322322, 331775, 391238, 446512, 454951, 546121, 530145, 316250, 613927, 763795, 786664, 809936
Offset: 1
For n = 2 the largest such k is 1215: 1215^2 = 1476225 and 1+4+7+6+2+2+5 = 27; 1215^3 = 1793613375and 1+7+9+3+6+1+3+3+7+5 = 45; 27*45 = 1215. Hence a(2) = 1215.
A126783
Smallest k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k, or 0 if no such k exists.
Original entry on oeis.org
80, 80, 70, 3905, 4004, 700, 19278, 32761, 5600, 8100, 24940, 10600, 56330, 68040, 81760, 149705, 116180, 126360, 123580, 0, 65500, 311003, 205030, 114400, 454951, 317350, 312170, 296270, 359380, 332750, 699785, 723338, 498150, 499130, 901368
Offset: 1
For n = 2 the smallest such k is 80: 80^2 = 6400 and 6+4+0+0 = 10; 80^3 = 512000 and 5+1+2+0+0+0 = 8; 10*8 = 80. Hence a(2) = 80.
For n = 3 the smallest such k is 70: 70^3 = 343000 and 3+4+3+0+0+0 = 10; 70^4 = 24010000 and 2+4+0+1+0+0+0+0 = 7; 10*7 = 70. Hence a(3) = 70.
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P:=proc(n) local a,i,j,k,w,x; for a from 1 by 1 to n do for i from 1 by 1 to n*n do w:=0;k:=i^a;j:=0;x:=i^(a+1); while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; while x>0 do j:=j+x-(trunc(x/10)*10); x:=trunc(x/10); od; if (i=w*j and i>1) then print(i); break; fi; od; od; end: P(1000);
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