A376399 - OEIS

A376399

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.

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

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OFFSET

0,2

COMMENTS

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

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

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

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.

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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..7

Index entries for sequences related to factorial base representation

FORMULA

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

EXAMPLE

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

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

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

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

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

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

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

etc.

PROG

(PARI)

\\ Do it hard way, by searching:

up_to = 12;

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

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); };

v376399 = A376399list(1+up_to);

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

(PARI)

\\ Compute, do not search, much faster:

up_to = 8;

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

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); };

v376399 = A376399list(1+up_to);

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

CROSSREFS

Cf. A276075, A276076, A376400 (variant).

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

Subsequence of A276078.

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

Sequence in context: A007280 A280260 A102927 * A376400 A227105 A127295

Adjacent sequences: A376396 A376397 A376398 * A376400 A376401 A376402

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 02 2024

STATUS

approved