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 1-4 of 4 results.

A332226 Numbers k such that sigma(k) is congruent to 2 modulo 8.

Original entry on oeis.org

10, 17, 20, 26, 40, 41, 52, 58, 73, 74, 80, 89, 90, 97, 104, 106, 113, 116, 122, 137, 148, 153, 160, 180, 193, 202, 208, 212, 218, 232, 233, 234, 241, 244, 257, 281, 296, 298, 313, 314, 320, 325, 337, 346, 353, 360, 362, 369, 394, 401, 404, 409, 416, 424, 433, 436, 449, 457, 458, 464, 468, 488, 490, 521, 522, 538, 554
Offset: 1

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Author

Antti Karttunen, Feb 13 2020

Keywords

Links

Crossrefs

Cf. A000203.
Subsequence of A191217.
Subsequences: A332227 (odd terms), A332228.

Programs

  • Mathematica
    Select[Range[600],Mod[DivisorSigma[1,#],8]==2&] (* Harvey P. Dale, Mar 30 2025 *)
  • PARI
    isA332226(n) = (2==(sigma(n)%8));

A332228 Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.

Original entry on oeis.org

153, 325, 369, 657, 725, 801, 833, 845, 873, 925, 1017, 1233, 1325, 1377, 1445, 1525, 1737, 2009, 2057, 2097, 2169, 2313, 2525, 2529, 2725, 2817, 2925, 3033, 3177, 3321, 3577, 3609, 3681, 3725, 3757, 3897, 3925, 4041, 4113, 4205, 4325, 4361, 4525, 4689, 4753, 4901, 4925, 4961, 5121, 5193, 5337, 5409, 5537, 5553, 5725
Offset: 1

Views

Author

Antti Karttunen, Feb 13 2020

Keywords

Comments

Proof that any odd perfect number, if such numbers exist at all, has to reside in this sequence: As all terms in A228058 are = 1 modulo 4 (their binary expansion ends as "01"), and taking sigma of an odd perfect number would multiply it by two (shift one bit-position left), the base-2 expansion of that result would end as "010", i.e., sigma(k) modulo 8 should be 2 (not 6) for such numbers k.

Links

Crossrefs

Cf. A000203, A332229.
Subsequence of A228058, of A332226 and of A332227.

Programs

  • PARI
    isA332228(n) = ((n%2)&&!isprimepower(n)&&2==(sigma(n)%8));

A332445 Numbers k of the form 4m+1 for which A087808(sigma(k)) is equal to 2*A087808(k).

Original entry on oeis.org

2009, 19377, 37809, 59373, 74673, 115677, 270041, 310329, 354609, 357309, 720425, 732321, 841437, 2071737, 2612269, 3131149, 3866461, 3930929, 5172093, 5593981, 7118753, 7903961, 8224173, 9327393, 9438129, 11452321, 12708025, 18857209, 18861889, 18875313, 19110321, 20278269, 20709225, 20950061, 23963597, 24895153
Offset: 1

Views

Author

Antti Karttunen, Feb 14 2020

Keywords

Comments

Numbers k such that A332224(k) is equal to A087808(2*k) and k == 1 mod 4.
Notably, the only square among the first 299 terms is a(248) = 5808421369 = 76213^2. sigma(5808421369) = 5808497583 == 3 (mod 4) == 7 (mod 8). Of the remaining 298 terms < 2^33, 92 are such that sigma(k) == 6 (mod 8) and 206 are such that sigma(k) == 2 (mod 8), that is, are terms of A332227.
Question: Why the terms come in clusters? Compare also the scatterplots of A087808 and A332224, and a similar sequence A332465.

Links

Crossrefs

Intersection of A016813 and A332446.
Cf. also A228058, A332227, A332465.
Cf. A000203, A087808, A286357, A332224, A332225, A332458.

Programs

A332465 Numbers n for which A269174(sigma(n)) is equal to 2*A269174(n).

Original entry on oeis.org

6, 28, 348, 496, 732, 886, 2924, 3573, 4972, 5448, 7544, 8128, 23388, 54842, 66928, 89200, 92296, 109786, 118064, 121552, 349512, 356488, 367472, 550432, 634784, 839984, 842452, 1234048, 1561408, 1797496, 2154584, 2364832, 2788808, 2927992, 3451456, 3585328, 5952364, 5991852, 6687136, 8238752, 10594336, 11210712, 11261020
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2020

Keywords

Comments

Numbers n such that A332464(n) is equal to A269174(2*n).
There are only three odd terms <= 2^32 among the first 113 terms of this sequence: 3573, 29255157, 936109557. Because A269174 preserves the 2-adic valuation of its argument, all such odd terms are of the form 4m+1, and must be present in A191218. Incidentally, these three terms are also present in A228058, but not in A332227.
See from the graph how unevenly the terms appear. Compare also the scatter plots of A269174 and A332464, also of a similar sequence A332445.

Examples

			          k   factorization        sigma(k)     A269174(sigma(k)) = A269174(2*k)
        348 = 2^2 * 3 * 29         840          2008,
       3573 = 3^2 * 397            5174         15486,
   29255157 = 3^2 * 3250573        42257462     126737534,
  936109557 = 3^2 * 104012173      1352158262   4055424126.

Links

Crossrefs

Cf. A000203, A269174, A332464.
Cf. A000396 (a subsequence).
Cf. also A191218, A228058, A332227, A332445, A332446.

Programs

  • Mathematica
    b[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];
okQ[n_] := b[DivisorSigma[1, n]] == 2 b[n];
Reap[For[n = 1, n <= 12*10^6, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 23 2020 *)
  • PARI
    A269174(n) = bitand(bitor(n,n<<1),bitor(bitxor(n,n<<1),bitxor(n,n<<2)));
isA332465(n) = (A269174(sigma(n))==2*A269174(n));
Showing 1-4 of 4 results.

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