A343131 - OEIS

%I #34 Jan 07 2022 19:34:48

%S 1,2,3,4,5,6,7,8,9,10,19,20,29,30,39,40,42,49,50,59,60,69,70,79,80,89,

%T 90,99,100,114,115,120,121,190,199,200,207,208,210,221,260,290,299,

%U 300,301,304,330,390,399,400,420,441,448,490,499,500,572,573,590,599,600,620

%N For m >= 1, the m-digit number k = d_m||...||d_2||d_1 is a term if it is divisible by f_m(k) that is the sum of the m elementary symmetric polynomials in m variables e_i(k): f_m(k) = Sum_{i=1..m} e_i(d_1, ..., d_m).

%C Equivalently, integers k that are divisible by A061486(k); the quotients obtained when these terms k are divided by A061486(k) are in A343132.

%C The idea of this sequence comes from A328864 (1st problem of the 30th British Mathematical Olympiad in 1994) and also from the elementary symmetric polynomials in m variables (see links) that allow the generalization of this Olympiad problem to m-digit numbers.

%C -> When k = u has only one digit, then f_1(k) = e_1(u) = u and the numbers u from 1 to 9 satisfy u/e_1(u) = 1, so those are the first nine terms of this sequence.

%C -> When k = du has two digits, the numbers that are divisible by f_2(k) = (e_1(d,u) + e_2(d,u)) = (d+u) + (d*u) are the first 19 terms of A038366, from a(10) = 10 to a(28) = 99.

%C -> when k = hdu is a 3-digit number, then e_1(h,d,u) = h+d+u, e_2(h,d,u) = h*d+d*u+u*h and e_3(h,d,u) = h*d*u so f_3(k) = (h+d+u) + (hd+du+uh) + (hdu). This subsequence with 3-digit numbers A328864 has 50 terms that are here from a(29) = 100 to a(78) = 999.

%C -> When k = thdu is a 4-digit number, then e_1(t,h,d,u) = e_1(k) = t+h+d+u, e_2(t,h,d,u)= t*h+t*d+t*u+h*d+h*u+d*u, e_3(t,h,d,u) = t*h*d+t*h*u+t*d*u+h*d*u and e_4(t,h,d,u) = t*h*d*u with f_4(k) = e_1(t,h,d,u) + e_2(t,h,d,u) + e_3(t,h,d,u) + e_4(t,h,d,u). Numbers k such that k/f(k) is an integer form another subsequence with 87 terms. This subsequence with 4-digit numbers goes from a(79) = 1000 to a(165) = 9999.

%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>.

%e For k = 7, f_1(7) = 7, and 7/7 = 1, hence 7 is a term.

%e For k = 42, f_2(4,2) = 4+2 + 4*2 = 14 and 42/14 = 3, hence 42 is a term.

%e For k = 572, f_3(5,7,2) = 5+7+2 + 5*7+7*2+2*5 + 5*7*2 = 14 + 59 + 70 = 143 and 572/143 = 4, hence 572 is a term.

%e For k = 3225, f_4(3,2,2,5) = 3+2+2+5 + 3*2+3*2+3*5+2*2+2*5+2*5 + 3*2*2+3*2*5+3*2*5+2*2*5 + 3*2*2*5 = 215 and 3225/215 = 15, hence 3225 is a term.

%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)); \\ A061486

%o isok(m) = (m % f(m)) == 0; \\ _Michel Marcus_, Apr 06 2021

%Y Cf. A005349, A007602, A038366, A061486, A328864, A343132.

%K nonn,base

%O 1,2

%A _Bernard Schott_, Apr 06 2021