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.
a(5)=c(a(3))=c(9)=16, because 5=prime(3), and the 9th composite number is c(9)=16. Thus a(10)=prime(a(5))=prime(16)=53 (since 10 is the 5th composite), a(18)=prime(a(10))=prime(53)=241 (since 18 is the 10th composite), a(28)=prime(a(18))=prime(241)=1523. A significant record value is a(198) = prime(a(152)) = prime(563167303) since 198=c(152); a(152)=prime(a(115)) since 152=c(115); a(115)=prime(a(84)); a(84)=prime(a(60)); a(60)=prime(a(42)); a(42)=prime(a(28)).
terms = 150; cc = Select[Range[4, 2 terms^2(*empirical*)], CompositeQ]; compositePi[k_?CompositeQ] := FirstPosition[cc, k][[1]]; a[1] = 1; a[p_?PrimeQ] := a[p] = cc[[a[PrimePi[p]]]]; a[k_] := a[k] = Prime[a[ compositePi[k]]]; Array[a, terms] (* Jean-François Alcover, Mar 02 2016 *)
A236854(n)={if(isprime(n), A002808(A236854(primepi(n))), n==1&&return(1);prime(A236854(n-primepi(n)-1)))} \\ without memoization: not much slower. - M. F. Hasler, Feb 03 2014
a236854=vector(999);a236854[1]=1;A236854(n)={a236854[n]&&return(a236854[n]); a236854[n]=if(isprime(n), A002808(A236854(primepi(n))), prime(A236854(n-primepi(n)-1)))} \\ Version with memoization. - M. F. Hasler, Feb 03 2014
from sympy import primepi, prime, isprime
def a002808(n):
m, k = n, primepi(n) + 1 + n
while m != k: m, k = k, primepi(k) + 1 + n
return m # this function from Chai Wah Wu
def a(n): return n if n<2 else a002808(a(primepi(n))) if isprime(n) else prime(a(n - primepi(n) - 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 07 2017
Start with: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... After the first step: 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, ... After the 2nd step: 2, 1, 6, 3, 4, 7, 8, 5, 12, 9, 10, 13, 14, 11, 18, 15, 16, ... After the 3rd step: 2, 1, 6, 3, 4, 12, 5, 8, 7, 14, 13, 10, 9, 16, 15, 18, 11, ...
a249990 n = a249990_list !! (n-1)
a249990_list = f 2 [1..] where
f k xs = reverse ys ++ f (k + 1) (g zs) where
g us = reverse vs ++ g ws where
(vs, ws) = splitAt k us
(ys, zs) = splitAt k xs
-- Reinhard Zumkeller, Dec 17 2014
TOP = 100
a = list(range(TOP))
for step in range(2,TOP):
numBlocks = (len(a)-1) // step
if numBlocks==0: break
a = a[:(1+numBlocks*step)]
for pos in range(1,len(a),step):
a[pos:pos+step] = a[pos+step-1:pos-1:-1]
for i in range(1, step+1): print(str(a[i]), end=',')
a[1:] = a[step+1:]
p:= 1: c:= 1: R:= 1:
for n from 2 to 100 do
if isprime(n) then
c:= max(c,n)+1;
while isprime(c) do c:=c+1 od:
R:= R,c
else
p:= nextprime(max(p,n));
R:= R,p
fi
od:
R; # Robert Israel, Apr 10 2024
TOP = 100000
a = list(range(TOP))
for step in range(2,TOP):
numBlocks = (len(a)-1) // step
if numBlocks==0: break
a = a[:(1+numBlocks*step)]
for pos in range(1,len(a),step):
a[pos:pos+step] = a[pos+step-1:pos-1:-1]
print(a[1], end=', ')
a[1:] = a[2:]
For n = 2, (prime), a(2) = 9, the smallest composite number not divisible by 2. For n = 6, (composite), a(6) = 5, the smallest novel prime which does not divide 6.
nn = 120; c[_] := False; a[1] = 1; c[1] = True; u = 2; v = 4;
Do[If[PrimeQ[n],
k = v; While[Or[c[k], PrimeQ[k], Divisible[k, n]], k++],
k = u; While[Or[c[k], CompositeQ[k], Divisible[n, k]], k++]];
Set[{a[n], c[k]}, {k, True}];
If[k == u, While[Or[c[u], CompositeQ[u]], u++]];
If[k == v, While[Or[c[v], PrimeQ[v]], v++]], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, May 31 2024 *)
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