A376924 a(n) is the largest number such that both a(n) and a(n) - n are 7-smooth numbers, or 0 if no such number exists.

4375, 8750, 13125, 17500, 21875, 26250, 30625, 35000, 39375, 43750, 3136, 52500, 1728, 61250, 65625, 70000, 5120, 78750, 6144, 87500, 91875, 6272, 2048, 105000, 109375, 3456, 118125, 122500, 2430, 131250, 4000, 140000, 9408, 10240, 153125, 157500, 32805, 12288, 5184, 175000

Offset

1,1

Comments

From Zhicheng Wei, May 26 2026: (Start)

All terms n>1 are currently conjectural.

a(1) is just A002072(4) + 1, which is proven by Størmer's theorem. This provides a(n) is not zero where n is 7-smooth. All other terms are conjectured, but the abc-conjecture can help.

a(n) can be less than n and this first occurs at a(1849) = 1800 because there is no answer to >1849. 1800 = 2^3 * 3^2 * 5^2 and 1800-1849 = -49 = -(7^2). Here we also consider negative numbers as smooth numbers. (End)

Links

Table of n, a(n) for n=1..40.

Formula

If a(n) or a(p*n) exists then a(p*n) >= p * a(n) for p in {2, 3, 5, 7}. - Zhicheng Wei, May 26 2026

Example

a(31) = 4000 since both 4000 and 4000 - 31 = 3969 are 7-smooth numbers, and it is conjectured that no number k larger than 4000 such that both k and k - 31 are 7-smooth numbers.

Prog

(PARI)
M(v, u, lim)={vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))}
Gen(lim, k)={my(v=[1]); forprime(p=2, k, v=M(v, vector(logint(lim, p)+1, e, p^(e-1)), lim)); v}
lista(n, lim=10^30)={my(v=vector(n), G=Gen(lim, 7)); 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} \\ Andrew Howroyd, Oct 11 2024
(PARI)
A006530(n)=if(n>1, vecmax(factor(n)[, 1]), 1)
b(n)=for(t=1, n, A006530(t*(n-t))<=7&return(t))
A376924(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)) \\ Zhicheng Wei, May 26 2026

Crossrefs

Cf. A002072, A002473, A106600.

Sequence in context: A206148 A230523 A375622 * A111345 A203874 A205317

Adjacent sequences: A376921 A376922 A376923 * A376925 A376926 A376927

Keyword

nonn,changed

Author

Zhicheng Wei, Oct 10 2024

Extensions

a(23) onwards from Andrew Howroyd, Oct 11 2024

Running Andrew Howroyd's PARI program with limit=10^100 does not change the terms in DATA. - Hugo Pfoertner, Oct 13 2024

Status

approved