A376400 -id:A376400 - cp's OEIS frontend

A376401 a(n) = A276075(A376400(n)); Partial sums of A376400.

0, 1, 3, 9, 39, 1089, 70814494839, 7568077812763134673885891483463343434987134201379042046671543939118568810481776089

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(8) has 2129 (decimal) digits.
From the second term onward also the partial sums of A376400.
By induction, it is easy to see that formula a(n) = A276075(A376400(n)) implies that from the second term onward, this sequence gives the partial sums of A376400, as A276075 is fully additive.

Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A376400, A376403.

Cf. also A143293 (when prepended with 0, an analogous sequence for A276085).

A276075(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*(primepi(f[k, 1])!)); };
A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376400(n) = if(!n,1,my(x=A376400(n-1)); x*A276076(x));
A376401(n) = A276075(A376400(n));
\\ Or alternatively as:
A376401(n) = if(!n,0,A376401(n-1)+A376400(n-1));

a(0) = 0; and for n >= 1, a(n) = a(n-1) + A376400(n-1) = Sum_{i=0..n-1} A376400(i).

A376408 a(0) = 1, and for n > 0, a(n) = a(n-1) * A019565(a(n-1)), where A019565 is the base-2 exp-function.

Original entry on oeis.org

1, 2, 6, 90, 353430, 274407373885179150, 2443417474326613595267894539584266773823049253134356678751627846400290750

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(7) has 407 digits, and a(8) has 2804 digits.

Like A376406, this satisfies A048675(a(n)) = a(n-1) + A048675(a(n-1)), for all n >= 1, that is, applying A048675 to the terms gives the partial sums shifted right once, A376409. However, unlike A376406, this is not a subsequence of A005117: a(3) = 90 is the first term that is not squarefree. Neither can we say that this is the lexicographically largest of such sequences, as there are also infinite sequences that begin as 1, 2, 6, 120, 38, ... or as 1, 2, 6, 120, 2042040, ... that satisfy the same condition.

Antti Karttunen, Table of n, a(n) for n = 0..7
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A376406.
Cf. A376409 (= A048675(a(n)), also partial sums from its second term onward).
Cf. also analogous sequences A002110 (for A276086) and A376400 (for A276076).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376408(n) = if(!n,1,my(x=A376408(n-1)); x*A019565(x));

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.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

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).

			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.

Antti Karttunen, Table of n, a(n) for n = 0..7
Index entries for sequences related to factorial base representation

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).

\\ 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];

\\ 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];

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

cp's OEIS Frontend

A376401 a(n) = A276075(A376400(n)); Partial sums of A376400.

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A376408 a(0) = 1, and for n > 0, a(n) = a(n-1) * A019565(a(n-1)), where A019565 is the base-2 exp-function.

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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.

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