A343132 a(n) is the quotient obtained when integer A343131(n) = k is divided by A061486(k).

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, 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, 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

Offset: 1

Author

Bernard Schott, Apr 07 2021

Keywords

• nonn
• base

Comments

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

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).

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).

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).

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.

Examples

			For A343131(7) = 7, A061486(7) = 7 and a(7) = 7/7 = 1.
For A343131(17) = 42, A061486(42) = 4+2 + 4*2 = 14 and a(17) = 42/14 = 3.
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.
		

Links

• Eric Weisstein's World of Mathematics, Symmetric polynomial.
• Wikipedia, Elementary symmetric polynomial.

Crossrefs

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

Programs

PARI

sympol(X, n) = my(s=0); forvec(i=vector(n, j, [1, #X]), s+=prod(k=1, n, X[i[k]]), 2); s ;
f(n) = my(d=digits(n)); sum(k=1, #d, sympol(d, k));
lista(nn) = {for (n=1, nn, my(q = n/f(n)); if (denominator(q) == 1, print1(q, ", ")););} \\ Michel Marcus, Apr 08 2021

Formula

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