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
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.
Cf.
A376403 (=
A276075(a(n)), also gives the partial sums from its second term onward).
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\\ 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];
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\\ 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];
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
Cf.
A376401 (=
A276075(a(n)), also gives the partial sums from its second term onward).
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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));
Original entry on oeis.org
0, 1, 3, 9, 39, 1089, 70814494839, 7568077812763134673885891483463343434987134201379042046671543939118568810481776089
Offset: 0
Cf. also
A143293 (when prepended with 0, an analogous sequence for
A276085).
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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));
A376407
a(0) = 0, 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
0, 1, 3, 9, 23, 353, 10519, 12086209, 1174153011340170531, 73582975079922326904310062621361286634299329277087298285
Offset: 0
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A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376407(n) = if(!n,0,my(x=A376407(n-1)); x+A019565(x));
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