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: Igor Schein

Igor Schein's wiki page.

Igor Schein has authored 4 sequences.

A219113 Sequence of integers which are simultaneously a sum of consecutive squares and a difference of consecutive cubes.

Original entry on oeis.org

1, 61, 5941, 582121, 57041881, 5589522181, 547716131821, 53670591396241, 5259170240699761, 515345012997180301, 50498552103482969701, 4948342761128333850361, 484887092038473234365641, 47513986677009248633982421, 4655885807254867892895911581
Offset: 1

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Author

Igor Schein, Nov 12 2012

Keywords

Examples

			1 = 0^2+1^2 = 1^3-0^3.
61 = 5^2+6^2 = 5^3-4^3.

Links

Crossrefs

Cf. A105038.

Programs

  • PARI
    A(n)=my(v=[0,4],k,a);a=if(n<3,v[n],for(k=3,n,v=concat(v,v[#v]*10-v[#v-1]+4));v[#v]);3*a*(a+1)+1

Formula

a(n) = 3*A105038(n)*(A105038(n)+1)+1.
G.f.: -x*(x^2-38*x+1) / ((x-1)*(x^2-98*x+1)). [Colin Barker, Nov 21 2012]
a(n) = (6+(5-2*sqrt(6))^(2*n-1)+(5+2*sqrt(6))^(2*n-1))/16. - Gerry Martens, Jun 06 2015
a(n) = 98*a(n-1)-a(n-2)-36, n>=3. - R. J. Mathar, Nov 07 2015

A218343 a(n) is the largest n-digit palindromic integer surrounded by twin primes, if one exists, or 0 otherwise.

Original entry on oeis.org

6, 0, 858, 2112, 89898, 276672, 8916198, 89522598, 898373898, 8998558998, 89984248998, 899931139998, 8999867689998, 89999400499998, 899998808899998, 8999950220599998, 89999961316999998, 899999985589999998, 8999999741479999998, 89999993611639999998
Offset: 1

Views

Author

Igor Schein, Oct 26 2012

Keywords

Links

Programs

  • PARI
    a(n)=my(p,t,v,k,j,d,N=n-1);if(n==1,6,if(n==2,0,t=9*10^(N\2);d=6*11^(N%2);forstep(k=t,0,-1,v=digits(k);v=concat(v,vecextract(Vecrev(v),concat(Str((N+1)%2+1),"..")));p=subst(Pol(v),x,10);if(p%d==0&&ispseudoprime(p+1)&&ispseudoprime(p-1),return(p)))))

A110585 Smallest number k of consecutive primes > p_n such that p_n^2 * p_(n+1) * p_(n+2) * ... * p_(n+k) is an abundant number.

Original entry on oeis.org

1, 3, 7, 16, 29, 44, 65, 89, 120, 155, 192, 236, 282, 332, 390, 453, 520, 589, 666, 746, 832, 927, 1026, 1131, 1239, 1350, 1467, 1592, 1725, 1867, 2017, 2161, 2313, 2469, 2634, 2800, 2975, 3155, 3339, 3532, 3729, 3931, 4143, 4356, 4579, 4809, 5051, 5291
Offset: 1

Views

Author

Igor Schein, Sep 13 2005

Keywords

Comments

The sequence arose while solving puzzle 329 from Carlos Rivera's Prime Puzzles & Problems Connection site.

Examples

			a(2)=3 because the second prime being 3, then 3^2 * 5 * 7 * 11 = 3465 and sigma(3465) - 2*3465 = 558, a positive number (i.e., 3465 is abundant), but 3^2 * 5 * 7 = 315 and sigma(315) - 2*315 = -6, a nonpositive number (i.e., 315 is not abundant).

Links

Crossrefs

Cf. A005101.
Cf. A005231, A047802.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2n; f[0] = 0; f[n_] := f[n] = Block[{k = f[n - 1]}, p = Fold[Times, Prime[n], Prime[ Range[n, n + k]]]; While[ !abQ[p], k++; p = p*Prime[n + k]]; k]; Table[ f[n], {n, 48}] (* Robert G. Wilson v *)
  • PARI
    forprime(p=2,100,k=0;while(k++,if(sigma(n=p^2*prod(j=1,k,prime(j+primepi(p))))-n>n,print(k);break)))

Extensions

Edited and extended by Robert G. Wilson v, Sep 15 2005

A097149 Numbers n such that Galois group of polynomial x^8+n*x^4+1 is E(8) (the group C_2 X C_2 X C_2).

Original entry on oeis.org

7, 23, 47, 62, 79, 119, 142, 167, 194, 223, 287, 322, 359, 439, 482, 527, 574, 623, 674, 727, 782, 839, 959, 1022, 1087, 1223, 1294, 1367, 1442, 1519, 1598, 1679, 1762, 1847, 1934, 2023, 2114, 2207, 2399, 2498, 2599, 2807, 2914, 3023
Offset: 1

Views

Author

Igor Schein, Jul 27 2004

Keywords

Links

Formula

n^2+8*n+14 with some gaps.

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