A326318 -id:A326318 - cp's OEIS frontend

A308247 a(n) is the least integer not the difference of two prime(n)-smooth numbers.

5, 41, 281, 1849, 9007, 35803

Offset: 1

Esteban Crespi de Valldaura, May 16 2019

The known terms have been found by exhaustive search and then proved not to be the difference of prime(n)-smooth numbers using assertions such as +- a(n) !== (modulo m) meaning that no element of the subgroup of Z/m generated by a,b,... added to a(n) is congruent modulo m with an element of the subgroup generated by . For example: <2> +- 41 !== <3> (mod 91) and the fact that 41+1 is not 3-smooth is enough to prove that 41 is not the difference of 3-smooth numbers; <2> + 281 !== <3,5> (mod 13981), <2> - 281 !== <3,5> (mod 76627) and <3> +- 281 !== <2,5> along with the fact that 281+1 is not 5-smooth is enough to show that 281 is not the difference of 5-smooth numbers. The proofs get exponentially harder as n increases. For example, <2, 11> + 9007 !== <3, 5, 7> (mod 308859288230831), or <2,5,7> + 35803 !== <3,11,13> (mod 2219897250633559197203).

The next few terms are conjectured to be 158857, 681179, 2516509, 10772123, 51292187, 186323681; if they were not, they would provide examples of ABC-triples with quality greater than 2.

			We see that 1 = 2-1, 2 = 4-2, 3 = 4-1, and 4 = 8-4. It is easy to see that 5 is not the difference of two powers of 2, so a(1) = 5. In the same way we can see that all the integers up to 40 are the difference of 3-smooth numbers, but as shown above 41 is not, so a(2)=41.

Esteban Crespi de Valldaura, Proof that a(n) is not prime(n)-smooth for n=2,3,4,5,6
Wikipedia, abc conjecture

P-smooth_numbers: A000079, A003586, A051037, A002473, A051038, ...

a(i) is the first term in each of A101082, A290365, A308456, A326318, A326319, A326320.

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

Original entry on oeis.org

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

Esteban Crespi de Valldaura, May 26 2019

nonn

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.

			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.

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

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.

\\ 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();

A326319 Numbers that cannot be written as a difference of 11-smooth numbers.

Original entry on oeis.org

9007, 10091, 10531, 10831, 11801, 12197, 12431, 12833, 12941, 13393, 13501, 13619, 13679, 13751, 13907, 13939, 14219, 14423, 14737, 14851, 14893, 15217, 15641, 15767, 15773, 15803, 15959, 16019, 16201, 16241, 16393, 16397, 16417, 16441, 16517, 16559

Offset: 1

Esteban Crespi de Valldaura, Jun 26 2019

nonn

Terms were found by generating in sequential order the 11-smooth numbers up to some limit and collecting the differences. The first 100 candidates k were then proved to be correct by showing that each of the following 15 congruences holds:
  <2> +- k !== <3, 5, 7, 11> mod 563213996185633,
  <3> +- k !== <2, 5, 7, 11> mod 194191394486113583,
  <2, 3> +- k !== <5, 7, 11> mod 1762314762258271,
  <5> +- k !== <2, 3, 7, 11> mod 220836983154619,
  <2, 5> +- k !== <3, 7, 11> mod 2128827364461031,
  <3, 5> +- k !== <2, 7, 11> mod 3521575252831519,
  <7, 11> +- k !== <2, 3, 5> mod 497846284658749,
  <7> +- k !== <2, 3, 5, 11> mod 5489574535421899,
  <2, 7> +- k !== <3, 5, 11> mod 6600281111334703,
  <3, 7> +- k !== <2, 5, 11> mod 834486158701066937,
  <5, 11> +- k !== <2, 3, 7> mod 239190476358328703,
  <5, 7> +- k !== <2, 3, 11> mod 3288443009987083,
  <3, 11> +- k !== <2, 5, 7> mod 14071029652900961,
  <2, 11> +- k !== <3, 5, 7> mod 1762314762258271,
  <11> +- k !== <2, 3, 5, 7> mod 411934385702047,
where  represents any element in the subgroup generated by a,b,... of the multiplicative subgroup modulo m. For a discussion iof this method of proof see A308247.

			9007 = A308247(5) cannot be written as the difference of 11-smooth numbers. All smaller numbers can; for example, 1849 = 3^4*5^2 - 2^4*11, 2309 = 2*3^5*5 - 11^2.

Esteban Crespi de Valldaura, Table of n, a(n) for n = 1..101

Cf. A051038 (11-smooth numbers).
Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A308456 (p=5), A326318 (p=7), A326320 (p=13).
Cf. A308247.

A326320 Numbers that cannot be written as a difference of 13-smooth numbers.

Original entry on oeis.org

35803, 36349, 41299, 43591, 45109, 45583, 53821

Offset: 1

Esteban Crespi de Valldaura, Jun 26 2019

Terms were found by generating in ascending order the 13-smooth numbers up to some limit and checking the differences. The first 7 candidates k have been proved true by showing a set of 31 congruences to be impossible (see link below). For a discussion iof this method of proof see A308247.

			35803 = A308247(6) cannot be written as the difference of 13-smooth numbers. All smaller numbers can; for example, 9007 = 3^2*7*11*13 - 2, 10091 = 2^2*3*5*13^2 - 7^2.

Esteban Crespi de Valldaura, Congruences needed in proof of a(1)..a(7)

Cf. A080197 (13-smooth numbers).
Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A308456 (p=5), A326318 (p=7), A326319 (p=11).
Cf. A308247.

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A308247 a(n) is the least integer not the difference of two prime(n)-smooth numbers.

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

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A326319 Numbers that cannot be written as a difference of 11-smooth numbers.

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A326320 Numbers that cannot be written as a difference of 13-smooth numbers.

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