cp's OEIS Frontend

A375727 a(n) is the least number that is a Smith number in all bases 2 to n but not in base n+1.

%I A375727 #18 Aug 30 2024 06:25:53
%S A375727 15,475,1023,222,1924475,26910204,191999912,240365505,10127638898,
%T A375727 15357419170
%N A375727 a(n) is the least number that is a Smith number in all bases 2 to n but not in base n+1.
%e A375727 a(3) = 475 = 5^2 * 19.
%e A375727 In base 2, 475 = 111011011_2 with digit sum 7, 5 = 101_2 with digit sum 2, 19 = 10011_2 with digit sum 3, and 7 = 2 * 2 + 3.
%e A375727 In base 3, 475 = 122121_3 with digit sum 9, 5 = 12_3 with digit sum 3, 19 = 201_3 with digit sum 3, and 9 = 2 * 3 + 3.
%e A375727 In base 4, 475 = 13123_4 with digit sum 10, 5 = 11_4 with digit sum 2, 19 = 103_4 with digit sum 4, and 10 <> 2 * 2 + 4.
%p A375727 f:= proc(n) local F, t,b;
%p A375727      if isprime(n) then return 0 fi;
%p A375727      F:= ifactors(n)[2];
%p A375727      for b from 2 while convert(convert(n,base,b),`+`) = add(t[2]*convert(convert(t[1],base,b),`+`), t = F) do od:
%p A375727      if b = 2 then 0 else b-1 fi
%p A375727 end proc:
%p A375727 N:= 7: # for a(2) .. a(N)
%p A375727 V:= Vector(N): count:= 0:
%p A375727 for n from 4 while count < N-1 do
%p A375727   v:= f(n);
%p A375727   if v > 0 and V[v] = 0 then V[v]:= n; count:= count+1 fi;
%p A375727 od:
%p A375727 convert(V[2..N],list);
%o A375727 (Python)
%o A375727 from sympy.ntheory import digits
%o A375727 from sympy import factorint, isprime
%o A375727 from itertools import count, islice
%o A375727 def sd(n, base=10): return sum(digits(n, base)[1:])
%o A375727 def f(n, factors):
%o A375727     for b in count(2):
%o A375727         if sd(n, base=b) != sum(sd(p, base=b)*factors[p] for p in factors):
%o A375727             break
%o A375727     return b-1
%o A375727 def agen(): # generator of terms
%o A375727     adict, n = dict(), 2
%o A375727     for k in count(1):
%o A375727         if isprime(k): continue
%o A375727         v = f(k, factorint(k))
%o A375727         if v not in adict: adict[v] = k
%o A375727         while n in adict: yield adict[n]; n += 1
%o A375727 print(list(islice(agen(), 4))) # _Michael S. Branicky_, Aug 25 2024
%o A375727 (PARI) smith(k) = {my(f = factor(k), p = f[,1], e = f[,2], b = 2); while(sumdigits(k, b) == sum(i = 1, #p, e[i] * sumdigits(p[i], b)), b++); b-1;}
%o A375727 lista(len) = {my(m = len + 1, v = vector(m), c = 0, i); forcomposite(k = 1, , i = smith(k); if(i <= m && v[i] == 0, c++; v[i] = k; if(c == m, break))); vecextract(v, "^1");} \\ _Amiram Eldar_, Aug 29 2024
%Y A375727 Cf. A006753, A278909.
%K A375727 nonn,base,more
%O A375727 2,1
%A A375727 _Robert Israel_, Aug 25 2024
%E A375727 a(8)-a(9) from _Michael S. Branicky_, Aug 26 2024
%E A375727 a(10)-a(11) from _Amiram Eldar_, Aug 29 2024