This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
CoefficientList[ Series[ 1/(1 + Sum[ NestList[ Prime, 1, n ] [ [ -1 ] ]*x^n, {n, 1, 17} ] ), {x, 0, 17} ], x ]
2.1659914031722783019082583521101...
CoefficientList[ Series[1/Product[1 - x^Nest[Prime, 1, i], {i, 1, 6}], {x, 0, 75}], x]
A007097(17) = 75063692618249; Primex(75063692618249) = 2586559741900744; A007097(18) = 2586559730396077; Primex(2586559730396077) = 98552043877145945; A007097(19) ~ 98552043800000000.
RiemannGram[x_] := Module[{n = 1, L, s = 1, r}, L = r = Log[x];
While[s < 10^30 r, s = s + r/(Zeta[n + 1] n); n++; r = r L/n]; s];
Primex[n_] := Module[{r1, r2, r, est}, If[n == 1, r = 2, r1 = n Log[n]; r2 = 2 r1; For[i = 1, i < 50, i++, r = (r1 + r2)/2; est = RiemannGram[r]; If[est < n, r1 = r, r2 = r]]]; Round@r];
Primex /@ NestList[Prime, 1, 15] (* Birkas Gyorgy, Apr 04 2011 *)
xeqprimex(n) = {
my(a,x); a = [1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077];
for(x=1,n, print1(round(primex(a[x]))",") ) }
\\ Approximates the n-th prime number to an accuracy of log10(n)/2 places.
primex(n) = {
my(x,px,r1,r2,r,p10,b,e,est);
if(n==1,return(2)); \\ force to 2
b=10; \\ Select base
p10=log(n)/log(10); \\ Determine p10 = power of 10 of n to adjust b^p10
if(Rg(b^p10*log(b^(p10+1)))< b^p10,m=p10+1,m=p10);
r1 = 0; r2 = 7.718281828; \\ Real kicker. if r2=1, it fails at 1e117
for(x=1, 100,
r=(r1+r2)/2;
est = (b^p10*log(b^(m+r)));
px = Rg(est);
if(px <= b^p10,r1=r,r2=r); r=(r1+r2)/2; );
est;
}
Rg(x) = \\ Gram's Riemann Approx of Pi(x)
{ my(n=1,L,s=1,r);
L=r=log(x);
while(s<10^40*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n);
}
a(0) = 0 since there are no primes up to 1 which is A007097(0); a(1) = 2 since the first Primeth recurrence, A007097(1) is 2; a(2) = 5 since the second Primeth recurrence, A007097(2) is 3, and 2+3 = 5; a(3) = 10 since the third Primeth recurrence, A007097(3) is 5, and 2+3+5 = 10; a(4) = 28 since the fourth Primeth recurrence, A007097(4) is 11, 2+3+5+7+11 = 28; a(5) = 160 since the fifth Primeth recurrence, A007097(5) is 31, and 2+3+5+7+11+13+17+19+23+29+31 = 160; etc.
f[n_] := Block[{lmt = Nest[ Prime@# &, 1, n], p = 2, s = 0}, While[p <= lmt, s += p; p = NextPrime@ p]; s]; Array[f, 14, 0]
a(10) is 3 because the tenth prime is 29 -> 10 -> 4 -> 2 and 2 is A007097(1).
f[n_] := Length@ NestWhileList[PrimePi, n, ! MemberQ[{1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041}, #] &] - 1; Array[f, 105]
This sequence is proposed to be a canonical representation for rooted identity trees. The first thirty terms are the following. 1 () 26 (()(()(()))) 62 (()((((()))))) 2 (()) 29 ((()((())))) 65 (((()))(()(()))) 3 ((())) 30 (()(())((()))) 66 (()(())(((())))) 5 (((()))) 31 (((((()))))) 78 (()(())(()(()))) 6 (()(())) 33 ((())(((())))) 79 ((()(((()))))) 10 (()((()))) 39 ((())(()(()))) 82 (()((()(())))) 11 ((((())))) 41 (((()(())))) 87 ((())(()((())))) 13 ((()(()))) 47 (((())((())))) 93 ((())((((()))))) 15 ((())((()))) 55 (((()))(((())))) 94 (()((())((())))) 22 (()(((())))) 58 (()(()((())))) 101 ((()(()(())))) We build the sequence as follows: The empty product is 1, so by (3) 1 is finitary. So is prime(1) = 2 by (2), so is prime(2) = 3 by (2), so is prime(3) = 5 by (2), so is 2*3 = 6 by (3), and so on. - _N. J. A. Sloane_, Oct 03 2016
primeMS[n_Integer?Positive]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_Integer?Positive]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
fin[n_Integer?Positive]:=If[n===1,1,Block[{x=fin[n-1]+1},While[Not[finitaryQ[x]],x++];x]];
Array[fin,200]
a(7) = 3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y. a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.
import Data.List (genericIndex)
a196050 n = genericIndex a196050_list (n - 1)
a196050_list = 0 : g 2 where
g x = y : g (x + 1) where
y = if t > 0 then a196050 t + 1 else a196050 r + a196050 s
where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then 1+a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
a[1] = 0; a[n_?PrimeQ] := a[n] = 1 + a[PrimePi[n]]; a[n_] := Total[#[[2]] * a[#[[1]] ]& /@ FactorInteger[n]]; Array[a, 110] (* Jean-François Alcover, Nov 16 2017 *) difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]]; Table[Length[difac[n]],{n,100}] (* Gus Wiseman, Mar 23 2019 *)
a(n) = my(f=factor(n)); [self()(primepi(p))+1 |p<-f[,1]]*f[,2]; \\ Kevin Ryde, May 28 2021
from functools import lru_cache from sympy import isprime, primepi, factorint @lru_cache(maxsize=None) def A196050(n): if n == 1 : return 0 if isprime(n): return 1+A196050(primepi(n)) return sum(e*A196050(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022
a(7)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y. a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
import Data.List (genericIndex)
a109129 n = genericIndex a109129_list (n - 1)
a109129_list = 0 : 1 : g 3 where
g x = y : g (x + 1) where
y = if t > 0 then a109129 t else a109129 r + a109129 s
where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
Nest[Function[{a, n}, Append[a, If[PrimeQ@ n, a[[PrimePi@ n]], Total@ Map[#2 a[[#1]] & @@ # &, FactorInteger[n]] ]]] @@ {#, Length@ # + 1} &, {0, 1}, 105] (* Michael De Vlieger, Mar 24 2019 *)
ML(n) = if( n==1, 1, my(f=factor(n)); sum(k=1,matsize(f)[1],ML(primepi(f[k,1]))*f[k,2])) ; A109129(n) = if( n==1, 0, ML(n) ); \\ François Marques, Mar 16 2021
from functools import lru_cache from sympy import primepi, isprime, factorint @lru_cache(maxsize=None) def A109129(n): if n <= 2: return n-1 if isprime(n): return A109129(primepi(n)) return sum(e*A109129(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022
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