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A308456 Numbers that cannot be written as a difference of 5-smooth numbers (A051037). 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 28 22:29 EST 2020. Contains 331322 sequences. (Running on oeis4.)