A343132 - OEIS

%I #24 Jul 19 2021 01:23:54

%S 1,1,1,1,1,1,1,1,1,10,1,10,1,10,1,10,3,1,10,1,10,1,10,1,10,1,10,1,100,

%T 6,5,24,11,10,1,100,9,8,42,13,13,10,1,100,43,16,22,10,1,100,30,9,2,10,

%U 1,100,4,3,10,1,100,31,6,5,10,1,100,15,10,1,100,13,11,10,1,100,10,1,1000

%N a(n) is the quotient obtained when integer A343131(n) = k is divided by A061486(k).

%C The first 9 terms corresponding to the 1-digit numbers k = u are the quotients u/u = 1.

%C The next 19 terms from a(10) = 10 to a(28) = 1 corresponding to 2-digit numbers k = du are the quotients du/(d+u + d*u).

%C The next 50 terms from a(29) = 100 to a(78) = 1 corresponding to 3-digit numbers k = hdu (in A328864) are the quotients hdu/f_3(h,d,u) where f_3(h,d,u) = (h+d+u) + (h*d+d*u+u*h) + (h*d*u).

%C The next 87 terms, from a(79) = 1000 to a(165) = 1, corresponding to 4-digit numbers k = thdu are the quotients thdu/f_4(t,h,d,u) where f_4(t,h,d,u) = (t+h+d+u) + (t*h+t*d+t*u+h*d+h*u+d*u) + (t*h*d+t*h*u+t*d*u+h*d*u) + (t*h*d*u).

%C When A343131(n) = z*10^q = A037124(r) is a number that contains only one nonzero digit z, then A061486(A037124(r)) = this nonzero digit z and a(n) = 10^q.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SymmetricPolynomial.html">Symmetric polynomial</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial">Elementary symmetric polynomial</a>.

%F a(n) = A343131(n)/A061486(A343131(n)).

%e For A343131(7) = 7, A061486(7) = 7 and a(7) = 7/7 = 1.

%e For A343131(17) = 42, A061486(42) = 4+2 + 4*2 = 14 and a(17) = 42/14 = 3.

%e For A343131(58) = 573, A061486(573) = 5+7+3 + 5*7+7*3+3*5 + 5*7*3 = 191 and a(58) = 573/191 = 3.

%o (PARI) sympol(X, n) = my(s=0); forvec(i=vector(n, j, [1, #X]), s+=prod(k=1, n, X[i[k]]), 2); s ;

%o f(n) = my(d=digits(n)); sum(k=1, #d, sympol(d, k));

%o lista(nn) = {for (n=1, nn, my(q = n/f(n)); if (denominator(q) == 1, print1(q, ", ")););} \\ _Michel Marcus_, Apr 08 2021

%Y Cf. A005349, A007602, A038366, A061486, A328864, A343131.

%K nonn,base

%O 1,10

%A _Bernard Schott_, Apr 07 2021