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 1-5 of 5 results.

A383268 Numbers k for which sigma(k - x) + sigma(k + x) = 4*k has at least one nonnegative solution.

Original entry on oeis.org

6, 13, 15, 17, 28, 33, 39, 42, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 69, 71, 77, 78, 82, 89, 90, 93, 95, 99, 101, 107, 111, 115, 118, 120, 121, 123, 125, 129, 131, 139, 141, 149, 153, 161, 165, 167, 171, 177, 179, 182, 183, 190, 195, 196, 197, 201, 204, 213, 215
Offset: 1

Views

Author

Felix Huber, Apr 24 2025

Keywords

Comments

Supersequence of A000396 because sigma(A000396(n) - x) + sigma(A000396(n) + x) = 4*A000396(n) has the solution x = 0.

Examples

			15 is in the sequence because sigma(15 - x) + sigma(15 + x) = 4*15 has the solution x = 5: sigma(15 - 5) + sigma(15 + 5) = sigma(10) + sigma(20) = 18 + 42 = 60 = 4*15.

Links

Crossrefs

Cf. A000203, A000396, A186103, A383269.

Programs

  • Maple
    with(NumberTheory):
A383268:=proc(N) # To get the first N terms.
    local k,x,K;
    K:=[];
    for k while nops(K)A383268(59);
  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 4*k, return(1))); \\ Michel Marcus, Apr 26 2025

A227306 Numbers k that divide sigma(k) + sigma(k-1).

Original entry on oeis.org

2, 6, 34, 50, 216, 236, 262, 386, 898, 924, 945, 1456, 2380, 5356, 6468, 6624, 8362, 14100, 23496, 26938, 46594, 80876, 196344, 212796, 1661136, 4070200, 4160920, 4626700, 5244548, 5462384, 17062316, 60464628, 217408416, 248621604, 262792908, 265371336, 323987588
Offset: 1

Views

Author

Alex Ratushnyak, Jul 05 2013

Keywords

Comments

Is 945 the only odd term? - Zak Seidov, Jul 06 2013
945 and 19910536425 are the only odd terms below 2^36. - Alex Ratushnyak, Jul 08 2013
The third odd term is a(58) = 841488503841. - Giovanni Resta, Apr 04 2014

Links

Crossrefs

Cf. A000203, A092403, A007691, A068078, A186103, A072188.

Programs

  • Mathematica
    With[{nn=324*10^6},Select[Thread[{Total/@Partition[DivisorSigma[ 1,Range[ nn]],2,1],Range[ 2,nn]}],Divisible[#[[1]],#[[2]]]&][[All,2]]] (* Harvey P. Dale, May 29 2020 *)

A383269 a(n) is the smallest nonnegative solution to sigma(A383268(n) - x) + sigma(A383268(n) + x) = 4*A383268(n).

Original entry on oeis.org

0, 1, 5, 11, 0, 7, 17, 28, 26, 37, 23, 14, 7, 13, 17, 49, 11, 22, 11, 5, 1, 58, 70, 13, 20, 37, 19, 11, 17, 31, 41, 67, 6, 16, 13, 73, 49, 11, 55, 91, 19, 73, 119, 5, 11, 77, 53, 43, 103, 86, 7, 114, 173, 88, 71, 59, 124, 95, 139, 7, 128, 31, 92, 143, 83, 227, 163
Offset: 1

Views

Author

Felix Huber, Apr 24 2025

Keywords

Examples

			a(2) = 1 because sigma(A383268(2) - 1) + sigma(A383268(2) + 1) =  sigma(13 - 1) + sigma(13 + 1) = sigma(12) + sigma(14) = 28 + 24 = 52 = 4*13 = 4*A383268(2).

Links

Crossrefs

Cf. A000203, A000396, A186103, A383268.

Programs

  • Maple
    with(NumberTheory):
A383269:=proc(N) # To get the first N terms.
    local k,x,X;
    X:=[];
    for k while nops(X)A383269(67);
  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 4*k, return(x))); return(-1);
lista(nn) = for (n=1, nn, my(m=isok(n)); if (m != -1, print1(m, ", "))); \\ Michel Marcus, Apr 26 2025

Formula

a(n) = 0 iff A383268(n) is a perfect number (A000396) and vice versa.

A320022 Numbers equal to the sum of the aliquot parts of the following k numbers, for some k.

Original entry on oeis.org

1, 3, 7, 9, 15, 31, 33, 56, 63, 127, 135, 168, 255, 511, 1023, 2047, 2401, 4095, 5328, 8191, 16383, 17360, 21003, 32767, 41163, 54721, 65535, 131071, 262143, 524287, 557280, 1048575, 1060801, 2097151, 4194303, 5026561, 8388607, 10800111, 11108163, 14366401, 16777215
Offset: 1

Views

Author

Paolo P. Lava, Oct 03 2018

Keywords

Comments

Any number of the form 2^j-1, with j > 0, is part of the sequence (with k=1).
So far 1 <= k <= 3 (k = 2 for 9, 33, 135, 168, 2401, 5328, 21003, 41163, 54721, 1060801, 5026561, ...; k = 3 for 56, 17360, ...). Are there terms with k = 4, 5, 6, ...? No k=4 up to 10^9.
If we were looking at numbers equal to the sum of the aliquot parts of the previous k numbers and of the following k, for some k, the first terms would be 2263024 and 128508838576, as confirmed by Giovanni Resta.
Up to n = 6*10^12 there are no terms with k>3. - Giovanni Resta, Oct 11 2018

Examples

			1 is in the sequence because aliquot part of 2 is 1.
9 is in the sequence because aliquot parts of 10 are 1, 2, 5 and of 11 is 1: 1 + 2 + 5 + 1 = 9.
56 is in the sequence because aliquot parts of 57 are 1, 3, 19, of 58 are 1, 2, 29, of 59 is 1: 1 + 3 + 19 + 1 + 2 + 29 + 1 = 56.

Crossrefs

Cf. A000225, A001065, A186103, A320021.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,j,k,n; for n from 1 to q do
a:=0; k:=0; while a

Formula

a(n) = Sum_{i = 1..k} A001065(a(n)+i), for some k.

Extensions

a(38)-a(41) from Giovanni Resta, Oct 09 2018

A296027 Numbers k such that k | (sigma(k-2) + sigma(k-1) + sigma(k+1) + sigma(k+2)).

Original entry on oeis.org

6, 57, 443, 1407, 1410, 12242, 15051, 30952, 44277, 65190, 68697, 609531, 921774, 951092, 2012670, 2820460, 11961680, 32886944, 3450005970
Offset: 1

Views

Author

Paolo P. Lava, Dec 04 2017

Keywords

Comments

Values of the ratio ( sigma(k-2)+sigma(k-1)+sigma(k+1)+sigma(k+2) ) / k: 6, 6, 7, 6, 6, 8, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, ...

Examples

			6 is in the sequence because (sigma(4) + sigma(5) + sigma(7) + sigma(8))/6 = (7 + 6 + 8 + 15)/6 = 6;
443 is in the sequence because (sigma(441) + sigma(442) + sigma(444) + sigma(445))/443 = (741 + 756 + 1064 + 540)/443 = 7.

Crossrefs

Cf. A186103, A227306.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,d,f,k,n;
a:=sigma(0); b:=sigma(1); c:=sigma(2); d:=sigma(3); f:=sigma(4);
for n from 2 to q do if type((a+b+d+f)/n,integer) then print(n,(a+b+d+f)/n); fi; a:=b; b:=c; c:=d; d:=f; f:=sigma(n+3); od; end: P(10^9);
  • PARI
    lista(nn) = {my(v = vector(nn, k, sigma(k))); for (k=3, nn-3, if (!((v[k-2]+v[k-1]+v[k+1]+v[k+2]) % k), print1(k, ", ")););} \\ Michel Marcus, Sep 10 2019
  • PARI
    upto(n) = my(v=List(vector(5,i,sigma(i))), res=List()); for(i=6, n, if((v[1] + v[2] + v[4] + v[5]) % (i-3) == 0, listput(res, i-3)); listpop(v,1); listput(v,sigma(i))); res \\ David A. Corneth, Sep 10 2019

Extensions

Term 2 removed and a(17)-a(18) added by Michel Marcus, Sep 10 2019
a(19) from Daniel Suteu, Sep 10 2019
Showing 1-5 of 5 results.

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