%I #37 Nov 04 2024 12:32:54

%S 1,2,6,30,1050,519090,1466909163669353522118

%N a(0) = 1, and for n > 0, a(n) is the least k such that A276075(k) = a(n-1) + A276075(a(n-1)), where A276075 is the factorial base log-function.

%C a(7) has 212 digits, a(8) has 10654 digits.

%C The lexicographically earliest infinite sequence x for which A276075(x(n)) gives the partial sums of x (shifted right once).

%C For any a(n), the next term a(n+1) <= a(n) * A276076(a(n)).

%C Conjecture: there are infinitely many variants b of this sequence, such that A276075(b(n)) = partial sums of b (shifted once right). One way to construct them: set i for some value >= 4, construct b first as here, but at point i, set b(i+1) = b(i) * A276076(b(i)), and after that, proceed as before, always finding a minimal k satisfying the condition. Unless b(i+1) = a(i+1), then b differs from this sequence but satisfies the same general condition, except that it is not the lexicographically earliest one. See also A376400.

%C The n-th term can be computed by applying A276076 to A376403(n), i.e., to the partial sums of the preceding terms a(0) .. a(n-1) (see the examples). This follows because all terms are in A276078 by the "least k" condition of the definition (see comment in A376417).

%H Antti Karttunen, <a href="/A376399/b376399.txt">Table of n, a(n) for n = 0..7</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F a(n) = A276076(A376403(n)) = A276076(Sum_{i=0..n-1} a(i)).

%e Starting with a(0) = 1, we take partial sums of previous terms, and apply A276076 to get the next term as:

%e a(1) = A276076(1) = 2,

%e a(2) = A276076(1+2) = 6,

%e a(3) = A276076(1+2+6) = 30,

%e a(4) = A276076(1+2+6+30) = 1050,

%e a(5) = A276076(1+2+6+30+1050) = 519090,

%e a(6) = A276076(1+2+6+30+1050+519090) = 1466909163669353522118,

%e etc.

%o (PARI)

%o \\ Do it hard way, by searching:

%o up_to = 12;

%o A276075(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*(primepi(f[k, 1])!)); };

%o A376399list(up_to) = { my(v=vector(up_to), x); v[1]=1; for(n=2,up_to,x=v[n-1]+A276075(v[n-1]); for(k=1,oo,if(A276075(k)==x,v[n]=k;break)); print1(v[n], ", ")); (v); };

%o v376399 = A376399list(1+up_to);

%o A376399(n) = v376399[1+n];

%o (PARI)

%o \\ Compute, do not search, much faster:

%o up_to = 8;

%o A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };

%o A376399list(up_to) = { my(v=vector(up_to), s=1); v[1]=1; for(n=2,up_to,v[n] = A276076(s); s += v[n]); (v); };

%o v376399 = A376399list(1+up_to);

%o A376399(n) = v376399[1+n];

%Y Cf. A276075, A276076, A376400 (variant).

%Y Cf. A376403 (= A276075(a(n)), also gives the partial sums from its second term onward).

%Y Subsequence of A276078.

%Y Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376406 (for A048675).

%K nonn

%O 0,2

%A _Antti Karttunen_, Nov 02 2024