A391077 - OEIS

%I #87 Jan 27 2026 05:30:06

%S 81,162,243,324,405,486,250,648,729,810,75,972,256,500,1215,1296,125,

%T 1458,144,1620,750,150,2048,1944,2025,512,2187,1000,125,2430,256,2592,

%U 225,250,1250,2916,32805,288,768,3240,81,1500,243,300,3645,4096,128,3888,625

%N a(n) is the conjectured largest number such that both a(n) and a(n) - n are 5-smooth numbers, or 0 if no such number exists.

%C a(1) is just A002072(3) + 1, which is proven by Størmer's theorem. This provides a(n) is not zero where n is 5-smooth. All other terms are conjectured, but the abc-conjecture can help.

%C a(n) can be less than n and this first occurs at a(281) = 256 because there is no answer to >281. 256 = 2^8 and 256-281 = -25 = -(5^2). Here we also consider negative numbers as smooth numbers.

%C There are no larger possible a(n) under 10^30 for every n.

%F If a(n) or a(p*n) exists then a(p*n) >= p * a(n) for p in {2, 3, 5}. - _David A. Corneth_, Jan 13 2026

%e a(31) = 256 since both 256 and 256 - 31 = 225 are 5-smooth numbers, and it is conjectured that no number k larger than 256 such that both k and k - 31 are 5-smooth numbers.

%o (PARI)

%o M(v, u, lim)={vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))}

%o Gen(lim, k)={my(v=[1]); forprime(p=2, k, v=M(v, vector(logint(lim, p)+1, e, p^(e-1)), lim)); v}

%o lista(n, lim=10^30)={my(v=vector(n), G=Gen(lim, 5)); for(i=1, #G, my(t=G[i], k=i+1); while(k<=#G && G[k]-t<=n, v[G[k]-t]=G[k]; k++)); v}

%o A006530(n)=if(n>1, vecmax(factor(n)[, 1]), 1)

%o b(n)=for(t=1,n,A006530(t*(n-t))<=5&return(t))

%o A391077(n)=if(lista(n)[n]!=0,return(lista(n)[n]);lista(n)[n]==0&b(n)<n,return(n-b(n));lista(n)[n]==0&b(n)==n,return(0))

%Y Cf. A051037, A219794, A392256.

%K nonn

%O 1,1

%A _Zhicheng Wei_, Jan 04 2026