A376408 -id:A376408 - cp's OEIS frontend

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

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).

A109162 a(1) = 1; for n > 1, a(n) = A019565(a(n-1)).

Original entry on oeis.org

1, 2, 3, 6, 15, 210, 10659, 54230826, 249853434654335387610276087

Offset: 1

Leroy Quet, Aug 18 2005

nonn

After the initial 1, even-indexed terms are of the form 4k+2 (members of A016825) and odd-indexed terms are of the form 6k+3 (members of A016945). However, not all even terms after 2 are multiples of three, because not all odd-indexed terms are of the form 4k+3. For example, because a(11) is of the form 4k+1, a(12) cannot be a multiple of three. - Antti Karttunen, Jun 18 2017

			a(4) = 6, which is 110 in binary. So a(5) is the product of the primes corresponding to the 1's of 110, 3*5 = 15.

Frank Adams-Watters, Table of n, a(n) for n = 1..11

Cf. A019565, A285320 (a left inverse).
The left edge of A285332 and A285333.
Cf. A153013, A328316 for similar iteration sequences, and also A376406, A376407, A376408.

NestList[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, 1, 11] (* Michael De Vlieger, Aug 20 2017 *)

More terms from Franklin T. Adams-Watters, Aug 29 2006

A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.

Original entry on oeis.org

1, 2, 6, 14, 330, 10166, 12075690, 1174153011328084322, 73582975079922326904310062621361286633125176265747127754

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(9) has 272 digits and a(10) has 1523 digits.
The lexicographically earliest infinite sequence x for which A048675(x(n)) gives the partial sums of x (shifted right once). This follows because the "least k" condition in the alternative formula also ensures that each k is squarefree, as we have A097248(n) = A019565(A048675(n)) <= n for all n, with equivalence only when n is squarefree.
Compare also to A376408.

			Starting with a(0) = 1, we take partial sums of previous terms, and apply A019565 to get the next term, and in the rightmost column, we "unbox" that term by applying A048675 to get A376407(n), which thus gives the partial sums of a(0)..a(n-1):
a(0)                               = 1          -> 0
a(1) = A019565(1)                  = 2,         -> 1     = 1
a(2) = A019565(1+2)                = 6,         -> 3     = 1+2
a(3) = A019565(1+2+6)              = 14,        -> 9     = 1+2+6
a(4) = A019565(1+2+6+14)           = 330,       -> 23    = 1+2+6+14
a(5) = A019565(1+2+6+14+330)       = 10166,     -> 353   = 1+2+6+14+330
a(6) = A019565(1+2+6+14+330+10166) = 12075690,  -> 10519 = 1+2+6+14+330+10166
etc.

Antti Karttunen, Table of n, a(n) for n = 0..9
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A097248, A376408.
Cf. A376407 (= A048675(a(n)), also gives the partial sums from its second term onward).
Subsequence of A005117.
Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376399 (for A276075).

up_to = 12;
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376406list(up_to) = { my(v=vector(up_to), s=1); v[1]=1; for(n=2,up_to,v[n] = A019565(s); s += v[n]); (v); };
v376406 = A376406list(1+up_to);
A376406(n) = v376406[1+n];

a(n) = A019565(A376407(n)) = A019565(Sum_{i=0..n-1} a(i)).
a(0) = 1, and for n > 0, a(n) is the least k such that A048675(k) = a(n-1) + A048675(a(n-1)), where A048675 is the base-2 log-function.
For n > 0, a(n) <= a(n-1) * A019565(a(n-1)).

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));

cp's OEIS Frontend

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

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A109162 a(1) = 1; for n > 1, a(n) = A019565(a(n-1)).

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A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 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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