cp's OEIS Frontend

A068916 Smallest positive integer that is equal to the sum of the n-th powers of its prime factors (counted with multiplicity).

%I A068916 #63 Apr 29 2021 01:08:23
%S A068916 2,16,1096744,3125,256,823543,19683
%N A068916 Smallest positive integer that is equal to the sum of the n-th powers of its prime factors (counted with multiplicity).
%C A068916 Does a(n) exist for all n?
%C A068916 a(12)=65536, a(27)=4294967296. a(n) exists for all n of the form n=p^i-i, where p is prime and i > 0, since p^p^i is an example (see A067688 and A081177). - _Jud McCranie_, Mar 16 2003
%C A068916 a(23) <= 298023223876953125. a(24) <= 7625597484987. - _Jud McCranie_, Jan 18 2016
%C A068916 a(10) = 285311670611. - _Jud McCranie_, Jan 25 2016
%C A068916 a(24) = 7625597484987. - _Jud McCranie_, Jan 30 2016
%H A068916 S. P. Hurd and J. S. McCranie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Hurd/hurd1.html">Integers that are Sums of Uniform Powers of all their Prime Factors: the sequence A068916</a>, J. of Int. Seq., vol 22, article 19.3.4.
%e A068916 a(3) = 1096744 = 2^3*11^3*103; the sum of the cubes of the prime factors is 3*2^3 + 3*11^3 + 103^3 = 1096744.
%t A068916 a[n_] := For[x=1, True, x++, If[x==Plus@@(#[[2]]#[[1]]^n&/@FactorInteger[x]), Return[x]]]
%o A068916 (PARI) isok(k, n) = {my(f=factor(k)); sum(j=1, #f~, f[j,2]*f[j,1]^n) == k;}
%o A068916 a(n) = {my(k = 1); while(! isok(k,n), k++); k;} \\ _Michel Marcus_, Jan 25 2016
%o A068916 (Python)
%o A068916 from sympy import factorint
%o A068916 def a(n):
%o A068916   k = 1
%o A068916   while True:
%o A068916     f = factorint(k)
%o A068916     if k == sum(f[d]*d**n for d in f): return k
%o A068916     k += 1
%o A068916 for n in range(1, 8):
%o A068916   print(a(n), end=", ") # _Michael S. Branicky_, Feb 16 2021
%Y A068916 Cf. A067688, A268036.
%Y A068916 Cf. A081177, A000325, A024024, A024050.
%K A068916 nonn,hard,more
%O A068916 1,1
%A A068916 _Dean Hickerson_, Mar 07 2002