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.

Previous Showing 21-24 of 24 results.

A097021 Initial terms of chains consisting of five consecutive integers, for neither of which the value of sigma-function is divisible by six.

Original entry on oeis.org

211248, 1146096, 3948048, 6090048, 14590800, 54100800, 61051248, 315758448, 567777600, 1715222448, 1912711248, 2408874048, 2636106048, 2744664048, 2811450096, 3304032048, 4647444048, 4832821296, 6020336448, 6028239600, 6739372800, 7824754800, 10110704400
Offset: 1

Views

Author

Labos Elemer, Aug 23 2004

Keywords

Comments

In A084301, that is among remainders of sigma(n) modulo 6, chains of 0's can be large. On the contrary, the length of non-0-remainder-chains is believed to be limited or occurrence of longer chains is rare. Consider remainders of sigma(5x) modulo 6.
The first 1000 terms are all divisible by 144. - Donovan Johnson, Aug 07 2013

Examples

			n = 14590800: sigma values for {n, n+1, n+2, n+3, n+4} = {59658880, 15110144, 22806063, 20958080, 25533914} have remainders modulo 6 as follows {4,2,3,2,2}.

Links

Crossrefs

Cf. A084301.

Programs

  • PARI
    forstep(m=25, 10110704400, 25, if(sigma(m)%6<>0, n=m; c=1; forstep(j=m-1, m-4, -1, if(sigma(j)%6<>0, c++; n=j, j=m-4)); for(j=m+1, m+4, if(sigma(j)%6<>0, c++, j=m+4)); if(c>=5, print1(n ", ")))) /* Donovan Johnson, Aug 06 2013 */

Extensions

a(6)-a(20) from Donovan Johnson, Sep 03 2008

A097503 Numbers k such that A000203(k) = sigma(k) is not divisible by 6.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 12, 13, 16, 18, 19, 21, 25, 27, 28, 31, 32, 36, 37, 39, 43, 48, 49, 50, 52, 57, 61, 63, 64, 67, 72, 73, 75, 76, 79, 81, 84, 91, 93, 97, 98, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 128, 129, 133, 139, 144, 148, 151, 156, 157, 162, 163, 169
Offset: 1

Views

Author

Labos Elemer, Aug 23 2004

Keywords

Comments

Positions of nonzero terms in A084301. - Ivan Neretin, Jan 26 2017

Links

Crossrefs

Cf. A000203, A084301.
Complement of A074627.

Programs

  • Mathematica
    Select[Range[170], ! Divisible[DivisorSigma[1, #], 6] &] (* Ivan Neretin, Jan 26 2017 *)
  • PARI
    isok(n) = (sigma(n) % 6) != 0; \\ Michel Marcus, Jan 26 2017

Extensions

Definition corrected by Ivan Neretin, Jan 26 2017

A097016 a(n)=x is the first term in chain of consecutive integers, for all of which the value of sigma[x] is divisible by 6.

Original entry on oeis.org

5, 5, 22, 44, 85, 85, 230, 260, 352, 950, 950, 1172, 1172, 1172, 1172, 1172, 1172, 1172, 1172, 1901, 1901, 7249, 7249, 7249, 12932, 12932, 12932, 12932, 12932, 38852, 38852, 226324, 226324, 235372, 235372, 235372, 413974, 413974, 423485, 423485
Offset: 1

Views

Author

Labos Elemer, Aug 23 2004

Keywords

Comments

It appears that arbitrarily long chains exist.

Examples

			n=52: a(52)=1270685 means that all entries in {sigma[a(52)],...,sigma[1270685+52-1]} ={1666224,...,2520672} are divisible by 6.

Crossrefs

Cf. A000203, A084301, A074627.

Programs

  • Mathematica
    With[{ds=Table[If[Divisible[DivisorSigma[1,n],6],1,0],{n,450000}]},Flatten[ Table[SequencePosition[ds,PadRight[{},n,1],1],{n,40}],1][[All,1]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 06 2018 *)

A097020 Initial terms of chains consisting of four consecutive integers, for none of which is the value of sigma-function divisible by six.

Original entry on oeis.org

1, 241, 576, 721, 1249, 2449, 12481, 23761, 40897, 65521, 120049, 126001, 128016, 167041, 186049, 211248, 211249, 344449, 374976, 406801, 417697, 451249, 607201, 647521, 665856, 708049, 746641, 766321, 1039681, 1068721, 1092241, 1146096
Offset: 0

Views

Author

Labos Elemer, Aug 23 2004

Keywords

Comments

It is believed that in all of such quadruples remainder 3 always arises.

Examples

			n=241: sigma-s for {241,242,243,244} are {242,399,364,434}
whose remainders modulo 6 are {2,3,4,2}.

Crossrefs

Cf. A084301.

Programs

  • Mathematica
    SequencePosition[Table[If[Divisible[DivisorSigma[1,n],6],0,1],{n,12*10^5}],{1,1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 21 2017 *)

Extensions

Definition rephrased by Harvey P. Dale, Oct 21 2017
Previous Showing 21-24 of 24 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.