cp's OEIS Frontend

A278981 a(n) is the first composite number having the same base-n digits as its prime factors (with multiplicity), excluding zero digits (or 0 if no such composite number exists).

%I A278981 #114 Feb 29 2020 17:59:54
%S A278981 15,399,85,318,57,906,85,1670,1111,18193,185,7205205,4119,63791,4369,
%T A278981 1548502,489,258099,451,408166,13315,1012985,679,25841526,26533,
%U A278981 2884373,985,49101338,1057,5362755,1285,2447558,179503,3091422,1387,5830693854,82311,149338,2005
%N A278981 a(n) is the first composite number having the same base-n digits as its prime factors (with multiplicity), excluding zero digits (or 0 if no such composite number exists).
%C A278981 For an alternate program that only checks a single base at a time, use the code from "#the actual function (alternate)" instead of "#the actual function".
%C A278981 The computation of a(n) is exceedingly inefficient, requiring the checking of all natural values less than a(n). A more efficient way to compute a(n) is very desirable. - _Ely Golden_, Dec 25 2016
%C A278981 There is a lower bound on a(n), if not 0, of n^2 + n + 1. As well, a(n) must have 3 or more nonzero digits in base n (if n is odd, this lower bound is n^3 + n^2 + n + 1, and a(n) must have 4 or more nonzero digits in base n). This does not significantly improve the computation of a(n), however. - _Ely Golden_, Dec 30 2016
%C A278981 The pattern in the magnitude of a(n) is unclear. For some values of n, a(n) is much larger than for other values. For example, a(65) is 2460678262, whereas a(64) is only 4369 and a(66) is 4577. It seems as though even values of n typically have smaller values of a(n). - _Ely Golden_, Dec 30 2016
%C A278981 It is known that a(n) > 0 for any nonzero member of this sequence, as well as any n >= 2 of the form A280270(m), A070689(m), A279480(m), 2*A089001(m), 2*A115104(m), and 2*A280273(m)-1. It is likely, but not known, that a(n) > 0 for all n >= 2. - _Ely Golden_, Dec 30 2016
%H A278981 Ely Golden, <a href="/A278981/b278981.txt">Table of n, a(n) for n = 2..72</a> (terms a(67), a(69), and a(71) computed by Chai Wah Wu)
%H A278981 Ely Golden, <a href="/A278981/a278981_18.txt">Table of n, a(n) for n = 2..11584</a> (a-file, contains every value of a(n) <= 2^27)
%H A278981 Ely Golden, <a href="/A278981/a278981_11.txt">Proofs regarding the lower bound of A278981(n)</a>
%e A278981 a(2) = 15, as 15 is the first composite number whose base-2 nonzero digits (1111) are the same as the base-2 nonzero digits of its prime factors (11_2 and 101_2).
%t A278981 g[n_] := g[n] = Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]];
%t A278981 f[b_] := Block[{c = b^2}, While[ PrimeQ@ c || DeleteCases[ Sort[ IntegerDigits[c, b]], 0] != DeleteCases[ Sort[ Flatten[ IntegerDigits[g[c], b]]], 0], c++]; c]; Array[f, 39, 2] (* _Robert G. Wilson v_, Dec 30 2016 *)
%o A278981 (SageMath)
%o A278981 def nonZeroDigits(x,n):
%o A278981     if(x<=0|n<2):
%o A278981         return []
%o A278981     li=[]
%o A278981     while(x>0):
%o A278981         d=divmod(x,n)
%o A278981         if(d[1]!=0):
%o A278981             li.append(d[1])
%o A278981         x=d[0]
%o A278981     li.sort()
%o A278981     return li;
%o A278981 def nonZeroFactorDigits(x,n):
%o A278981     if(x<=0|n<2):
%o A278981         return []
%o A278981     li=[]
%o A278981     f=list(factor(x))
%o A278981     #ensures inequality of nonZeroFactorDigits(x,n) and nonZeroDigits(x,n) if x is prime
%o A278981     if((len(f)==1)&(f[0][1]==1)):
%o A278981         return [];
%o A278981     for c in range(len(f)):
%o A278981         for d in range(f[c][1]):
%o A278981             ld=nonZeroDigits(f[c][0],n)
%o A278981             li+=ld
%o A278981     li.sort()
%o A278981     return li;
%o A278981 #the actual function
%o A278981 def a(n):
%o A278981     c=2
%o A278981     while(nonZeroFactorDigits(c,n)!=nonZeroDigits(c,n)):
%o A278981         c+=1;
%o A278981     return c;
%o A278981 index=2
%o A278981 while(index<=100):
%o A278981     print(str(index)+" "+str(a(index)))
%o A278981     index+=1
%o A278981 print("complete")
%o A278981 #the actual function (alternate)
%o A278981 def a(n):
%o A278981     c=2
%o A278981     while(nonZeroFactorDigits(c,n)!=nonZeroDigits(c,n)):
%o A278981         c+=1;
%o A278981         if(c%1000000==1):
%o A278981             print("checked up to "+str(c-1))
%o A278981     return c;
%o A278981 x=3 # <some base you want to check>
%o A278981 print(str(x)+" "+str(a(x)))
%o A278981 print("complete")
%Y A278981 a(10) = A176670(1); a(2) = A278909(1).
%K A278981 nonn,base
%O A278981 2,1
%A A278981 _Ely Golden_, Dec 02 2016