cp's OEIS Frontend

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.

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.

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

Views

Author

Esteban Crespi de Valldaura, Jun 26 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

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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