A376401 -id:A376401 - cp's OEIS frontend

A376403 a(0) = 0, and for n > 0, a(n) = a(n-1) + A276076(a(n-1)), where A276076 is the factorial base exp-function.

0, 1, 3, 9, 39, 1089, 520179, 1466909163669354042297

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(8) has 212 digits, a(9) has 10654 digits.

By induction, it is easy to see that formula a(n) = A276075(A376399(n)) implies that from the second term onward, this sequence gives the partial sums of A376399. See more comments in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..8
Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A376399, A376401.

Cf. also A143293 (when prepended with 0, an analogous sequence for A276086).

A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376403(n) = if(!n,0,A376403(n-1)+A276076(A376403(n-1)));

a(n) = A276075(A376399(n)).

a(0) = 0; and for n > 0, a(n) = a(n-1) + A376399(n-1) = Sum_{i=0..n-1} A376399(i).

A376400 a(0) = 1, and for n > 0, a(n) = a(n-1) * A276076(a(n-1)), where A276076 is the factorial base exp-function.

Original entry on oeis.org

1, 2, 6, 30, 1050, 70814493750, 7568077812763134673885891483463343434987134201379042046671543939118568739667281250

Offset: 0

Antti Karttunen, Nov 02 2024

nonn

a(7) has 2129 (decimal) digits.
Like A376399, this satisfies A276075(a(n)) = a(n-1) + A276075(a(n-1)), for all n >= 1, so also here, applying A276075 to the terms gives the partial sums shifted right once, A376401.
However, unlike A376399, this is not a subsequence of A276078: a(5) = 70814493750 is the first term that is in A276079.

Index entries for sequences related to factorial base representation

Cf. A276075, A276076, A276078, A276079, A376399, A376403.
Cf. A376401 (= A276075(a(n)), also gives the partial sums from its second term onward).
Cf. also analogous sequences A002110 (for A276086) and A376408 (for A019565).

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));

A376409 a(n) = A048675(A376408(n)); Partial sums of A376408.

Original entry on oeis.org

0, 1, 3, 9, 99, 353529, 274407373885532679, 2443417474326613595267894539584266773823049253134356679026035220285823429

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(8) has 407 digits, a(9) has 2804 digits.

By induction, it is easy to see that formula a(n) = A048675(A376408(n)) implies that from the second term onward, this sequence gives the partial sums of A376408, as A048675 is fully additive.

Antti Karttunen, Table of n, a(n) for n = 0..8
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A376407, A376408.

Cf. also A376401 (an analogous sequence for A276075).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A376408(n) = if(!n,1,my(x=A376408(n-1)); x*A019565(x));
A376409(n) = A048675(A376408(n));

a(0) = 0; and for n >= 1, a(n) = a(n-1) + A376408(n-1) = Sum_{i=0..n-1} A376408(i).

cp's OEIS Frontend

A376403 a(0) = 0, and for n > 0, a(n) = a(n-1) + A276076(a(n-1)), where A276076 is the factorial base exp-function.

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A376400 a(0) = 1, and for n > 0, a(n) = a(n-1) * A276076(a(n-1)), where A276076 is the factorial base exp-function.

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A376409 a(n) = A048675(A376408(n)); Partial sums of A376408.

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