A282764
9*n analog to Keith numbers.
Original entry on oeis.org
9, 17, 48, 55, 96, 120, 124, 131, 244, 426, 787, 1893, 5307, 5364, 5600, 10083, 31085, 46733, 52700, 53456, 56857, 56920, 109620, 110313, 110376, 374016, 2989245, 4081505, 5173765, 13017112, 17242512, 34346372, 34638676
Offset: 1
9*17 = 153:
1 + 5 + 3 = 9;
5 + 3 + 9 = 17.
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with(numtheory): P:=proc(q, h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do a:=w*n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]
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Select[Range[10^6], Function[n, Module[{d = IntegerDigits[9 n], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)
A282767
n/3 analog of Keith numbers.
Original entry on oeis.org
45, 609, 1218, 1827, 3213, 21309, 28206, 29319, 31917, 39333, 47337, 78666, 102090, 117999, 204180, 406437, 302867592, 4507146801, 5440407522
Offset: 1
609/3 = 203:
2 + 0 + 3 = 5;
0 + 3 + 5 = 8;
3 + 5 + 8 = 16;
5 + 8 + 16 = 29;
8 + 16 + 29 = 53;
16 + 29 + 53 = 98;
29 + 53 + 98 = 180;
53 + 98 + 180 = 331;
98 + 180 + 331 = 609.
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with(numtheory): P:=proc(q,h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1/w by 1/w to q do a:=w*n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]
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With[{n = 3}, Select[Range[10 n, 10^6, n], Function[k, Last@ NestWhile[Append[Rest@ #, Total@ #] &, IntegerDigits[k/n], Total@ # <= k &] == k]]] (* Michael De Vlieger, Feb 27 2017 *)
A284493
Analog of Keith numbers based on digits of sum of anti-divisors.
Original entry on oeis.org
18, 26, 40, 93, 95, 122, 227, 5640, 8910, 15481, 56028, 117056, 282103, 394608, 2059983, 3775282, 3804607, 5005918, 10390740, 31753906, 42117745, 67170923, 98908536, 176337241
Offset: 1
Sum of the anti-divisors of 18 is 28: 2 + 8 = 10, 8 + 10 = 18.
Sum of the anti-divisors of 93 is 140: 1 + 4 + 0 = 5, 4 + 0 + 5 = 9, 0 + 5 + 9 = 14, 5 + 9 + 14 = 28, 9 + 14 + 28 = 51, 14 + 28 + 51 = 93.
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with(numtheory): P:=proc(q,h) local a,b,j,k,n,t,v; v:=array(1..h);
for n from 10^6 to q do k:=0; j:=n; while j mod 2 <> 1 do
k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
b:=ilog10(a)+1; if b>1 then for k from 1 to b do
v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1;
v[t]:=add(v[k], k=1..b); while v[t]
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