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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

5, 41, 281, 1849, 9007, 35803
Offset: 1

Views

Author

Esteban Crespi de Valldaura, May 16 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Views

Author

Esteban Crespi de Valldaura, May 26 2019

Keywords

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.

Examples

			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.

Links

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.

Programs

  • 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();

A326318 Numbers that cannot be written as a difference of 7-smooth numbers (A002473).

Original entry on oeis.org

1849, 2309, 2411, 2483, 2507, 2531, 2629, 2711, 2753, 2843, 2851, 2921, 2941, 3139, 3161, 3167, 3181, 3217, 3229, 3251, 3287, 3289, 3293, 3323, 3379, 3481, 3487, 3541, 3601, 3623, 3653, 3697, 3698, 3709, 3737, 3739, 3803, 3827, 3859, 3877, 3901, 3923, 3947
Offset: 1

Views

Author

Esteban Crespi de Valldaura, Jun 26 2019

Keywords

Comments

Terms were found by generating in sequential order the 7-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 congruences holds:
<2> +- k !== <3, 5, 7> mod 31487336959,
<3> +- k !== <2, 5, 7> mod 121328339431,
<2, 3> +- k !== <5, 7> mod 5699207989579,
<5> +- k !== <2, 3, 7> mod 1206047658673,
<2, 5> +- k !== <3, 7> mod 11174958041,
<3, 5> +- k !== <2, 7> mod 31487336959,
<7> +- k !== <2, 3, 5> mod 1116870318707,
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.

Examples

			1849 = A308247(4) cannot be written as the difference of 7-smooth numbers. All smaller numbers can; for example, 281 = 2^5*3^2 - 7, 289 = 2*3*7^2 - 5, ..., 1847 = 3*5^4 - 2^2*7.

Links

Crossrefs

Cf. A002473 (7-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), A326319 (p=11), 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

Views

Author

Esteban Crespi de Valldaura, Jun 26 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.
Showing 1-4 of 4 results.

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