A093502 -id:A093502 - cp's OEIS frontend

A074271 Previous term + prime(previous term).

1, 3, 8, 27, 130, 863, 7554, 84385, 1165814, 19408513, 381328184, 8690396907, 226500350374, 6671888768963, 219904832427160, 8040978338951561, 323784549401655250, 14264968083112608087

Offset: 1

Zak Seidov, Sep 21 2002

nonn

			a(4)=27 because a(3)=8, prime(8)=19 and 8+19=27.

Cf. A000040, A000720, A093502.

Cf. also A007097, A179274.

se={1}; a=1; Do[b=a+Prime[a]; se=Append[se, b]; a=b, {i, 13}]; se
NestList[ Prime@# + # &, 1, 14] (* Robert G. Wilson v, Jul 17 2010 *)

a(1) = 1; a(n) = a(n-1) + prime(a(n-1)).

a(n) = A000720(A093502(n)). - Antti Karttunen, Nov 02 2024

a(15) from Robert G. Wilson v, Jul 17 2010

a(16)-a(18) from Charles R Greathouse IV, Mar 15 2011

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)).

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)).

A363655 a(0) = 1; for n > 0, a(n) = prime( Sum_{k=0..n-1} a(k) * a(n-k-1) ).

Original entry on oeis.org

1, 2, 7, 61, 863, 17569, 472741, 16007419, 659408567, 32231133931, 1833425773489, 119498316410171, 8810846732918257, 727089137774221667

Offset: 0

Ilya Gutkovskiy, Jun 13 2023

Cf. A000040, A000108, A014342, A007097, A093502, A338816.

a[0] = 1; a[n_] := a[n] = Prime[Sum[a[k] a[n - k - 1], {k, 0, n - 1}]]; Table[a[n], {n, 0, 11}]

G.f.: sqrt( Sum_{n>=0} index of prime a(n+1) * x^n ).

a(12)-a(13) from Amiram Eldar, Jun 13 2023

cp's OEIS Frontend

A074271 Previous term + prime(previous term).

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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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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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A363655 a(0) = 1; for n > 0, a(n) = prime( Sum_{k=0..n-1} a(k) * a(n-k-1) ).

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