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%I #36 Jun 16 2026 18:42:22
%S 1,2,3,4,5,6,7,8,9,1,9,1,8,17,18,26,27,1,7,22,25,28,36,1,28,35,36,46,
%T 1,18,45,54,64,1,18,27,31,34,43,53,58,68,1,46,54,63,1,54,71,81,1,82,
%U 85,94,97,106,117,1,98,107,108,1,108,1,20,40,86,103,104,106,107,126,134,135
%N Irregular triangle in which row n lists k > 0 such that the sum of digits of k^n equals k.
%C Each row begins with 1 and has length A046019(n).
%H T. D. Noe, <a href="/A152147/b152147.txt">Rows n = 1..1000 of triangle, flattened</a>
%H V. Oxman and M. Stupel, <a href="https://resolve.cambridge.org/core/journals/mathematical-gazette/article/abs/10838-a-number-whose-square-root-is-the-sum-of-its-digits/96817045BDBCFA5751FC72562B862D49">A number whose square root is the sum of its digits</a>, The Mathematical Gazette 108 (573) (2024), 513-514.
%e 1, 2, 3, 4, 5, 6, 7, 8, 9;
%e 1, 9;
%e 1, 8, 17, 18, 26, 27; (A046459, with 0)
%e 1, 7, 22, 25, 28, 36; (A055575 " )
%e 1, 28, 35, 36, 46; (A055576 " )
%e 1, 18, 45, 54, 64; (A055577 " )
%e 1, 18, 27, 31, 34, 43, 53, 58, 68; (A226971 " )
%e 1, 46, 54, 63;
%e 1, 54, 71, 81;
%e 1, 82, 85, 94, 97, 106, 117;
%e 1, 98, 107, 108;
%e 1, 108;
%e 1, 20, 40, 86, 103, 104, 106, 107, 126, 134, 135, 146;
%e 1, 91, 118, 127, 135, 154; etc.
%o (Python)
%o def ok(k, r): return sum(map(int, str(k**r))) == k
%o def agen(rows, startrow=1, withzero=0):
%o for r in range(startrow, rows + startrow):
%o d, lim = 1, 1
%o while lim < r*9*d: d, lim = d+1, lim*10
%o yield from [k for k in range(1-withzero, lim+1) if ok(k, r)]
%o print([an for an in agen(13)]) # _Michael S. Branicky_, May 23 2021
%o (PARI) M152147=Map(); {A152147(n,k)=if(k<2, k, mapisdefined(M152147,[n,k], &k), k, !mapisdefined(M152147,[n,0]), my(L=(n+1)*(5+log(n+1)*3), t=A152147(n,k-1)); until(t++>L|| sumdigits(t^n)==t,); mapput(M152147, [n, if(t>L, t=0, k)], t); t)} \\ _M. F. Hasler_, Jun 12 2026
%Y Cf. A046000, A046017, A046471, A133509.
%K base,tabf,nonn
%O 1,2
%A _T. D. Noe_, Nov 26 2008