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.

User: Esteban Crespi de Valldaura

Esteban Crespi de Valldaura's wiki page.

Esteban Crespi de Valldaura has authored 7 sequences.

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.

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.

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.

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();

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.

A181667 Least integer m > 0 such that none of the first n primes divides any value of the polynomial x^2 + x + m.

Original entry on oeis.org

1, 5, 11, 11, 17, 17, 41, 41, 41, 41, 41, 41, 19421, 19421, 333491, 601037, 601037, 5237651, 9063641, 12899891, 24073871, 24073871, 28537121, 67374467, 67374467, 67374467, 67374467, 146452961, 13236860171, 13236860171, 17959429571, 57391479317, 57391479317
Offset: 1

Views

Author

Esteban Crespi de Valldaura, Feb 04 2011

Keywords

Comments

All the elements of this sequence with n > 2 are congruent mod 30 to one of the polynomials x^2 + x + 11 or x^2 + x + 17.
The elements of the sequence have been taken from A060392, see below.

Examples

			x^2 + x + 11 takes the values 11, 13, 17, 23, 31, 41, 53, 67, 83, ... never divisible by any of the primes 2, 3, or 5.

Links

Crossrefs

a(n) equals min_{k > n} A060392(k).

Extensions

a(29) corrected and more terms added by William P. Orrick, Mar 17 2017

A181437 Size of the longest increasing sequence of primes starting with 2, 3 and with second-order differences bounded by n.

Original entry on oeis.org

4, 57, 57, 421, 421, 1860, 1860, 24661, 24661, 380028, 380028, 2964603, 2964603
Offset: 1

Views

Author

Esteban Crespi de Valldaura, Jan 31 2011, Feb 03 2011

Keywords

Comments

For n > 1, a(2n + 1) = a(2n) because after the primes (2,3,5), all second differences are even numbers. It is not known if the sequence remains finite for all n.
To give an idea of the size of the sequences, the largest prime in the sequence corresponding to a(12) is 1280522207.
For a given n, the Mathematica program uses recursion to find the longest list of primes with second differences bounded by n. This is not the best language to use for this problem. Even n=4 takes quite a while. The program prints longer lists as they are found. - T. D. Noe, Feb 03 2011

Examples

			For n = 1, the sequence is (2, 3, 5, 7) which has 4 members, so a(1) = 4.
For n = 2, a sequence is (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1597, 1669, 1741, 1811, 1879, 1949, 2017, 2083) which has 57 members, so a(2) = 57.

Links

Programs

  • Mathematica
    Valid[lst_List]:=Module[{r1,r2}, r1=-n+2*lst[[-1]]-lst[[-2]]; r2=r1+2*n; r1=Max[lst[[-1]]+1,r1]; Select[Range[r1,r2],PrimeQ]]; FindPath[lst_List]:=Module[{p,ln}, p=Valid[lst]; If[p=={}, ln=Length[lst]; If[ln>len, len=ln; Print[{len,lst}]], Do[FindPath[Append[lst,i]], {i,p}]]]; n=2; len=0; t={2,3}; FindPath[t]; len (* T. D. Noe, Feb 03 2011 *)

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