A308456 Numbers that cannot be written as a difference of 5-smooth numbers (A051037).

281, 289, 353, 413, 421, 439, 443, 457, 469, 493, 541, 562, 563, 578, 581, 583, 641, 653, 661, 677, 683, 691, 701, 706, 707, 731, 733, 737, 751, 761, 769, 779, 787, 793, 803, 811, 817, 823, 826, 827, 829, 841, 842, 843, 853, 857, 867, 877, 878, 881, 883, 886

Offset

1,1

Comments

Terms were found by generating in sequential order the 5-smooth numbers up to some limit and collecting the differences. The first 1000 candidates k were then proved to be correct by showing that each of the following congruences holds:

{5} +- k !== {2,3} mod 205910575871,

{3} +- k !== {2,5} mod 220411358713,

{2} +- k !== {3,5} mod 3019333681,

where {a,b,...} represents the subgroup generated by a,b,... of the multiplicative subgroup modulo m. For a discussion iof this method of proof see A308247.

Links

Esteban Crespi de Valldaura, Table of n, a(n) for n = 1..1000

Example

281 = A308247(3) cannot be written as the difference of 5-smooth numbers. All smaller numbers can; for example, 277 = 3^4*5 - 2^7, 271 = 2^3*5^3 - 3^6.

Prog

(PARI)
\\ Computes the first N elements in the sequence.
\\ At least the first 10000 are correct.
N=100;
\\computes the multiplicative subgroup generated
\\by the elements of the vector L modulo m.
SGR(L, m)={S=[1]; for(l=1, length(L), z=znorder(Mod(L[l], m)); T=[1]; for(t=1, z, s=lift(Mod(L[l], m)^t); if(setsearch(S, s), break, T=concat(T, s); )); for(t=1, length(T), S=Set(concat(S, lift(S*Mod(T[t], m)))))); S}
m1=205910575871; L1= SGR([2, 3], m1); M1 = SGR([5], m1);
m2=220411358713; L2= SGR([2, 5], m2); M2 = SGR([3], m2);
m3=  3019333681; L3= SGR([3, 5], m3); M3 = SGR([2], m3);
chkdif(k)={r=1;
   D=1; while(gcd(k/D, 30)>1, D*=gcd(k/D, 30));
   fordiv(D, d,
     if(vecmax(factor(k/d+1)[, 1])<= 5 , r=0);
     if(r, for(t=1, length(M1),
       if(setsearch(L1, (M1[t]+k/d)%m1), r=0; break)));
     if(r, for(t=1, length(M2),
       if(setsearch(L2, (M2[t]+k/d)%m2), r=0; break)));
     if(r, for(t=1, length(M3),
       if(setsearch(L3, (M3[t]+k/d)%m3), r=0; break)));
     if(r, for(t=1, length(M1),
       if(setsearch(L1, (M1[t]-k/d)%m1), r=0; break)));
     if(r, for(t=1, length(M2),
       if(setsearch(L2, (M2[t]-k/d)%m2), r=0; break)));
     if(r, for(t=1, length(M3),
       if(setsearch(L3, (M3[t]-k/d)%m3), r=0; break)));
     if(r==0, break)
   );
   r
}
for(k=1, m3, if(chkdif(k), print1(k, ", "); if(N--==0, break))); print();

Crossrefs

Cf. A051037 (5-smooth numbers).

Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A326318 (p=7), A326319 (p=11), A326320 (p=13).

Cf. A308247.

Sequence in context: A294165 A259079 A296506 * A175145 A142444 A139655

Adjacent sequences: A308453 A308454 A308455 * A308457 A308458 A308459

Keyword

nonn

Author

Esteban Crespi de Valldaura, May 26 2019

Status

approved