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.
p=2, e=7: 1+(1+A000040(1))*A000040(1)^7 = 1+(1+2)*2^7 = 1+3*128 = 385 = a(17); p=7, e=2: 1+(1+A000040(4))*A000040(4)^2 = 1+(1+7)*7^2 = 1+8*49 = 393 = a(18).
A000961 := proc(n) option remember ; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if nops( ifactors(a)[2] ) = 1 then RETURN(a); fi; od: fi; end: A025473 := proc(n) option remember ; if n <= 2 then n; else ifactors( A000961(n))[2] ; op(1,op(1,%)) ; fi; end: A137152 := proc(n) option remember ; local a,m,i; if n = 1 then RETURN([1]) ; else a := procname(n-1) ; m := A025473(n) ; for i from 1 to nops(a) do if gcd(op(i,a),m) <> 1 then m := m*op(i,a) ; a := subsop(i=1,a) ; fi; od; a := [op(a),m] ; fi; RETURN(a) ; end: A135524 := proc(n) add(k,k=A137152(n)) ; end: for n from 1 to 80 do printf("%d,",A135524(n)) ; od: # R. J. Mathar, Dec 17 2008
A000961 := proc(n) option remember ; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if nops( ifactors(a)[2] ) = 1 then RETURN(a); fi; od: fi; end: A025473 := proc(n) option remember ; if n <= 2 then n; else ifactors( A000961(n))[2] ; op(1,op(1,%)) ; fi; end: A137152 := proc(n) option remember ; local a,m,i; if n = 1 then RETURN([1]) ; else a := procname(n-1) ; m := A025473(n) ; for i from 1 to nops(a) do if gcd(op(i,a),m) <> 1 then m := m*op(i,a) ; a := subsop(i=1,a) ; fi; od; a := [op(a),m] ; fi; RETURN(a) ; end: A135525 := proc(n) local a,k,i ; a := 0 ; k :=A137152(n) ; for i in k do if i <> 1 then a := a+i; fi; od; a ; end: for n from 1 to 80 do printf("%d,",A135525(n)) ; od: # R. J. Mathar, Dec 17 2008
A number's longest string of consecutive divisors is a(5)=6 iff the integer is a multiple of 60 but not of 7. As n increases, the asymptotic fraction of positive integers satisfying those conditions equals 1/60 * 6/7 = 1/70. Therefore a(5) = 70.
forstep(n=3, 839, 2, p=isprimepower(n); if(p>0, m=Mod(round(n^(1/p)), 9); if(m==2||m==5, print1(n, ", "))));
a(16) is 2 since A000961(16) is 27 which is 3^3 = (p_2)^3, i.e., the prime index of 3 is 2.
b:= proc(n) option remember; local k; for k from
1+b(n-1) while nops(ifactors(k)[2])>1 do od; k
end: b(1):=1:
a:= n-> `if`(n=1, 0, numtheory[pi](ifactors(b(n))[2, 1$2])):
seq(a(n), n=1..100); # Alois P. Heinz, Feb 20 2020
mxn = 500; Join[{0}, Transpose[ Sort@ Flatten[ Table[ {Prime@n^ex, n}, {n, PrimePi@ mxn}, {ex, Log[Prime@n, mxn]}], 1]][[2]]]
lista(nn) = {print1(0); for(n=2, nn, if(isprimepower(n, &p), print1(", ", primepi(p)))); } \\ Jinyuan Wang, Feb 19 2020
from sympy import primepi, integer_nthroot, primefactors def A330669(n): if n == 1: return 0 def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))) kmin, kmax = 1,2 while f(kmax) >= kmax: kmax <<= 1 while True: kmid = kmax+kmin>>1 if f(kmid) < kmid: kmax = kmid else: kmin = kmid if kmax-kmin <= 1: break return int(primepi(primefactors(kmax)[0])) # Chai Wah Wu, Aug 20 2024
s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 2]]^f[[1, 1]], Nothing]]; s[1] = 0; Array[s, 250] (* Amiram Eldar, May 16 2025 *)
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