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.

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A131190 Numbers n>=0 such that d(n) = (n^1 + 1) (n^2 + 2) ... (n^25 + 25) / 25! is nonintegral.

Original entry on oeis.org

2, 7, 12, 18, 22, 27, 29, 37, 40, 47, 51, 52, 62, 72, 73, 77, 84, 87, 95, 97, 102, 106, 112, 122, 127, 128, 137, 139, 147, 150, 152, 161, 162, 172, 177, 183, 187, 194, 197, 202, 205, 212, 216, 222, 227, 237, 247, 249, 252, 260, 262, 271, 272, 277, 282, 287, 293, 297, 302, 304, 312, 315, 322, 326, 327, 337
Offset: 1

Views

Author

Alexander R. Povolotsky, Sep 25 2007

Keywords

Comments

If n is in this sequence the so is n+6050. - Max Alekseyev, Feb 02 2015

Crossrefs

Cf. A129995, A131189, A131191, A131192, A131685

Programs

  • PARI
    { is_A131190(n) = setsearch([2,12,22,27,37,47],n%50) || ( (n%11)==7 && (n%121)!=117 ) } /* Max Alekseyev, Feb 02 2015 */

Formula

Notice that 25! = 2^22 * 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23. The value of (n^1+1)(n^2+2)...(n^25+25) is always divisible by all these prime powers, except 5^6 and 11^2. There is no divisibility by 5^6 for n in {50m+2, 50m+12, 50m+22, 50m+27, 50m+37, 50m+47} and by 11^2 for n in {11m+7} \ {121m+117}. Therefore, the sequence is the union {50m+2} U {50m+12} U {50m+22} U {50m+27} U {50m+37} U {50m+47} U ( {11m+7} \ {121m+117} ). - Max Alekseyev, Nov 10 2007

Extensions

Initial terms were calculated by Peter J. C. Moses; see comment in A129995.
More terms from Max Alekseyev, Feb 02 2015

A131192 Numbers n >= 0 such that d(n) = (n^1 + 1)*(n^2 + 2)*...*(n^26 + 26) / 26! is nonintegral.

Original entry on oeis.org

7, 11, 18, 24, 29, 37, 40, 50, 51, 62, 73, 76, 84, 89, 95, 102, 106, 115, 128, 139, 141, 150, 154, 161, 167, 172, 180, 183, 193, 194, 205, 206, 216, 219, 227, 245, 249, 258, 260, 271, 282, 284, 293, 297, 304, 310, 315, 323, 326, 336, 337, 348, 349, 362, 370, 375, 381, 388, 392, 403, 414, 425, 427, 436
Offset: 1

Views

Author

Alexander R. Povolotsky, Sep 25 2007

Keywords

Comments

Notice that 26! = 2^23 * 3^10 * 5^6 * 7^3 * 11^2 * 13^2 * 17 * 19 * 23. There is no divisibility for 11^2 and n in {11m+7} \ {121m+117} and for 13^2 and n in {13m+11} \ {169m+63}. Therefore, this sequence is formed by the union ( {11m+7} \ {121m+117} ) U ( {13m+11} \ {169m+63}). - Max Alekseyev, Nov 10 2007

Crossrefs

Cf. A129995, A131189, A131190, A131191, A131685.

Programs

  • Maple
    d:=proc(n) options operator, arrow: (product(n^j+j,j=1..26))/factorial(26) end proc: a:=proc(n) if type(d(n), integer) = false then n else end if end proc; seq(a(n),n=1..300); # Emeric Deutsch, Oct 24 2007

Extensions

Initial terms were calculated by Peter J. C. Moses; see comment in A129995
More terms from Emeric Deutsch, Oct 24 2007
More terms from Max Alekseyev, Feb 02 2015

A131677 a(n) = (Product_{i=1..7} n^i+i) / 7!.

Original entry on oeis.org

1, 8, 274725, 8903032600, 21521701559085, 9892478959203456, 1527238784041075105, 109733832449349303000, 4483781212288588835625, 118795734924428077310080, 2233888850312257843810061, 31811523551546985038211552, 359951182400070234774044725
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

Links

Programs

Extensions

Definition made explicit by M. F. Hasler, May 02 2015

A131678 a(n) = (Product_{i=1..8} n^i+i) / 8!.

Original entry on oeis.org

1, 9, 9065925, 7310502643675, 176327300873583405, 483041091658815453456, 320648364425775841520065, 79074323113562613259765875, 9403175220694650942397475625, 639220975955961365494757841040, 27923612862792073359883606310061, 852385355738368243011331354210716, 19346552845649626158477975728463925
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

Links

Programs

Extensions

Definition made explicit by M. F. Hasler, May 02 2015

A133154 a(n) is the smallest m<=p-1 such that p-1 is the only value of j in 1<=j<=2p for which m^j+j==0 (mod p), where p is the n-th prime.

Original entry on oeis.org

0, 0, 0, 2, 7, 11, 2, 5, 3, 8, 5, 26, 2, 2, 9, 16, 6, 14, 9, 9, 3, 10, 3, 10, 4, 2, 5, 2, 13, 2, 3, 2, 3, 21, 8, 22, 2, 3, 2, 5, 5, 2, 3, 2, 4, 2, 2, 7, 44, 7, 16, 3, 4, 3, 2, 19, 22, 3, 3, 26, 7, 16, 12, 2, 9, 6, 2, 14, 3, 4, 9, 6, 4, 19, 15, 6, 4, 6, 16, 5, 11, 9, 5, 4, 2, 3, 18, 3, 7, 9, 18, 16, 3, 8
Offset: 1

Views

Author

Alexander R. Povolotsky, Oct 08 2007

Keywords

Comments

Andrew Granville, based on submitter's analysis of the data in A131685, made the following conjecture: "For some n with 1<=n<=p-1, there does not exist a value of j, with 1 <= j <= 2p, other than j=p-1, for which n^j+j == 0 (mod p)." Max Alekseyev's calculations confirm that the conjecture is true for the primes between 5 and 10^5. The sequence consists of the first such "n" (referred to as "m" in this sequence's definition) for each prime. a(n)=0 means that there is no corresponding m; this occurs at n=1 (p=2), n=2 (p=3), and n=3 (p=5), and at no other primes p<10^5.

Crossrefs

Cf. A131685.

Programs

  • PARI
    { a(p) = for(n=1,p-1, local(j=1); while(j<=2*p, if( j!=p-1 && Mod(n,p)^j==-j, break); j++); if(j>2*p,return(n)); ); 0 }
vector(100,n,a(prime(n))) /* Max Alekseyev */

Extensions

Definition simplified and comments edited by Jon E. Schoenfield, Nov 29 2013
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