cp's OEIS Frontend

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.

Showing 1-2 of 2 results.

A208768 The distinct values of A070198.

Original entry on oeis.org

0, 1, 5, 11, 59, 419, 839, 2519, 27719, 360359, 720719, 12252239, 232792559, 5354228879, 26771144399, 80313433199, 2329089562799, 72201776446799, 144403552893599, 5342931457063199, 219060189739591199, 9419588158802421599, 442720643463713815199
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 01 2012

Keywords

Comments

The terms of A070198, and duplicates removed.
a(n) = A051451(n) - 1 = A051452(n) - 2.
From Daniel Forgues, Apr 27 2014: (Start)
Factorizations:
5, 11, 59, 419, 839 are primes;
2519 = 11*229, 27719 = 53*523, 360359 = 173*2083,
720719 = 31*67*347, 12252239 = 29*647*653;
232792559, 5354228879 are primes;
26771144399 = 47*12907*44131, 80313433199 = 29*61*45400471;
2329089562799 is prime;
72201776446799 = 37*149*239*1091*50227;
144403552893599 is prime;
Very likely contains an infinite number of primes (see A057824). (End)
A more natural (compare with A051452) name for the sequence: lcm(1, ..., k) - 1, where k is the n-th prime power A000961(n). - Daniel Forgues, May 09 2014

Programs

  • Haskell
    import Data.List (nub)
    a208768 n = a208768_list !! (n-1)
    a208768_list = nub a070198_list
    
  • Python
    from math import prod
    from sympy import primepi, integer_nthroot, integer_log, primerange
    def A208768(n):
        def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
        m, k = n, f(n)
        while m != k:
            m, k = k, f(k)
        return prod(p**integer_log(m, p)[0] for p in primerange(m+1))-1 # Chai Wah Wu, Aug 15 2024

A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 29, 2521, 19, 89, 71, 1693, 232792561, 6659, 26771144401, 331, 101, 72201776446801, 1801, 173, 54941, 89, 442720643463713815201, 593, 5171, 239, 1222615931, 103, 7265496855919, 6562349363, 4447, 147099357127, 1931
Offset: 1

Views

Author

Keywords

Examples

			1 + lcm(1..8) = 29^2, so its smallest prime divisor is 29; it occurs as the 7th term in the sequence because 8 is the 7th prime power: A000961(7) = 8.
		

Crossrefs

Programs

  • Magma
    a:=[]; lcm:=1; for k in [1..83] do if (k eq 1) or IsPrimePower(k) then lcm:=Lcm(lcm,k); a:=a cat [Factorization(1+lcm)[1][1]]; end if; end for; a; // Jon E. Schoenfield, May 28 2018
    
  • Mathematica
    Join[{2},With[{ppwr=Select[Range[200],PrimePowerQ]},Table[FactorInteger[LCM@@Take[ ppwr,n]+ 1][[1,1]],{n,40}]]] (* Harvey P. Dale, May 28 2024 *)
  • PARI
    a(n) = {my(nb = 1, lc = 1, k = 2); while (nb != n, if (isprimepower(k), nb++; lc = lcm(lc, k)); k++;); vecmin(factor(lc +1)[,1]);} \\ Michel Marcus, May 29 2018
    
  • Python
    from math import prod
    from sympy import primepi, integer_nthroot, integer_log, primerange, primefactors
    def A051454(n):
        def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
        m, k = n, f(n)
        while m != k:
            m, k = k, f(k)
        return min(primefactors(1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)))) # Chai Wah Wu, Aug 15 2024
Showing 1-2 of 2 results.