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

%I A308456 #27 Aug 23 2019 13:31:49
%S A308456 281,289,353,413,421,439,443,457,469,493,541,562,563,578,581,583,641,
%T A308456 653,661,677,683,691,701,706,707,731,733,737,751,761,769,779,787,793,
%U A308456 803,811,817,823,826,827,829,841,842,843,853,857,867,877,878,881,883,886
%N A308456 Numbers that cannot be written as a difference of 5-smooth numbers (A051037).
%C A308456 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:
%C A308456   {5} +- k !== {2,3} mod 205910575871,
%C A308456   {3} +- k !== {2,5} mod 220411358713,
%C A308456   {2} +- k !== {3,5} mod 3019333681,
%C A308456 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.
%H A308456 Esteban Crespi de Valldaura, <a href="/A308456/b308456.txt">Table of n, a(n) for n = 1..1000</a>
%e A308456 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.
%o A308456 (PARI)
%o A308456 \\ Computes the first N elements in the sequence.
%o A308456 \\ At least the first 10000 are correct.
%o A308456 N=100;
%o A308456 \\computes the multiplicative subgroup generated
%o A308456 \\by the elements of the vector L modulo m.
%o A308456 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}
%o A308456 m1=205910575871; L1= SGR([2,3],m1); M1 = SGR([5],m1);
%o A308456 m2=220411358713; L2= SGR([2,5],m2); M2 = SGR([3],m2);
%o A308456 m3=  3019333681; L3= SGR([3,5],m3); M3 = SGR([2],m3);
%o A308456 chkdif(k)={r=1;
%o A308456    D=1;while(gcd(k/D,30)>1,D*=gcd(k/D,30));
%o A308456    fordiv(D,d,
%o A308456      if(vecmax(factor(k/d+1)[,1])<= 5 ,r=0);
%o A308456      if(r,for(t=1,length(M1),
%o A308456        if(setsearch(L1,(M1[t]+k/d)%m1),r=0;break)));
%o A308456      if(r,for(t=1,length(M2),
%o A308456        if(setsearch(L2,(M2[t]+k/d)%m2),r=0;break)));
%o A308456      if(r,for(t=1,length(M3),
%o A308456        if(setsearch(L3,(M3[t]+k/d)%m3),r=0;break)));
%o A308456      if(r,for(t=1,length(M1),
%o A308456        if(setsearch(L1,(M1[t]-k/d)%m1),r=0;break)));
%o A308456      if(r,for(t=1,length(M2),
%o A308456        if(setsearch(L2,(M2[t]-k/d)%m2),r=0;break)));
%o A308456      if(r,for(t=1,length(M3),
%o A308456        if(setsearch(L3,(M3[t]-k/d)%m3),r=0;break)));
%o A308456      if(r==0, break)
%o A308456    );
%o A308456    r
%o A308456 }
%o A308456 for(k=1,m3,if(chkdif(k),print1(k,", ");if(N--==0, break))); print();
%Y A308456 Cf. A051037 (5-smooth numbers).
%Y A308456 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).
%Y A308456 Cf. A308247.
%K A308456 nonn
%O A308456 1,1
%A A308456 _Esteban Crespi de Valldaura_, May 26 2019