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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A295299 Numbers k such that the sum of the divisors (A000203) and the binary weight of k (A000120) have different parity.

Original entry on oeis.org

7, 9, 11, 13, 14, 18, 19, 21, 22, 26, 28, 31, 35, 36, 37, 38, 41, 42, 44, 47, 52, 55, 56, 59, 61, 62, 67, 69, 70, 72, 73, 74, 76, 79, 82, 84, 87, 88, 91, 93, 94, 97, 103, 104, 107, 109, 110, 112, 115, 117, 118, 122, 124, 127, 131, 133, 134, 137, 138, 140, 143, 144, 145, 146, 148, 151, 152, 155, 157, 158, 161, 164, 167, 168, 169
Offset: 1

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Author

Antti Karttunen, Nov 26 2017

Keywords

Comments

Numbers k for which A010059(k) = A053866(k).
This sequence is the union of all terms of A028982 (squares and twice squares) that are evil (A001969), and all odious numbers (A000069) that are neither a square or twice a square. See also A231431, A235001.
This is a subsequence of A004169, numbers k such that phi(k) is not a power of 2. See comment in A295298 for the reason. - Antti Karttunen, Nov 27 2017

Links

Crossrefs

Complement of A295298.
Subsequence of A004169.
Cf. A000069, A001969, A000120, A000203, A010059, A028982, A053866, A231431, A235001, A295297 (characteristic function).

Programs

  • Mathematica
    Select[Range@ 169, UnsameQ @@ Map[EvenQ, {DivisorSigma[1, #], DigitCount[#, 2, 1]}] &] (* Michael De Vlieger, Nov 26 2017 *)

A343722 a(n) is the number of starting residues r modulo n from which repeated iterations of the mapping r -> r^2 mod n never reach a fixed point.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 8, 0, 8, 8, 0, 0, 0, 8, 16, 0, 12, 16, 20, 0, 16, 16, 16, 16, 24, 0, 28, 0, 24, 0, 20, 16, 32, 32, 24, 0, 32, 24, 40, 32, 20, 40, 44, 0, 40, 32, 0, 32, 48, 32, 40, 32, 48, 48, 56, 0, 56, 56, 48, 0, 40, 48, 64, 0, 60, 40, 68, 32
Offset: 1

Views

Author

Jon E. Schoenfield, Apr 27 2021

Keywords

Comments

a(n) = 0 iff n is a term of A003401, that is, A000010(n) is a power of 2.

Examples

			For every n >= 1, the residue r = 0 is a fixed point under the mapping r -> r^2 mod n, since we have 0 -> 0^2 mod n = 0. Also, for every n >= 2, the residue r = 1 is a fixed point, since we have 1 -> 1^2 mod n = 1.
For n=1, the only residue mod n is 0 (a fixed point), so a(1) = 0.
For n=2, the only residues are 0 and 1 (each a fixed point), so a(2) = 0.
For n=3, the only residue other than 0 and 1 is 2; 2 -> 2^2 mod 3 = 4 mod 3 = 1, a fixed point, so a(3) = 0.
For n=4, we have 0 -> 0, 1 -> 1, 2 -> 2^2 mod 4 = 4 mod 4 = 0, and 3 -> 3^2 mod 4 = 9 mod 4 = 1, each trajectory ending at a fixed point, so a(4) = 0.
For n=5, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 1 -> 1
  3 -> 4 -> 1 -> 1
  4 -> 1 -> 1
(each ending at a fixed point), so a(5) = 0.
For n=6, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 4
  3 -> 3
  4 -> 4
  5 -> 1 -> 1
(each ending at a fixed point), so a(6) = 0.
For n=7, however, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 2 -> ...       (a loop)
  3 -> 2 -> 4 -> 2 -> ...  (a loop)
  4 -> 2 -> 4 -> ...       (a loop)
  5 -> 4 -> 2 -> 4 -> ...  (a loop)
  6 -> 1 -> 1
so 4 of the 7 trajectories never reach a fixed point, so a(7)=4.

Links

Crossrefs

Cf. A003401, A004169, A279185, A343720, A343721.

Programs

  • PARI
    pos(list, r) = forstep (k=#list, 1, -1, if (list[k] == r, return (#list - k + 1)););
isok(r, n) = {my(list = List()); listput(list, r); for (k=1, oo, r = lift(Mod(r, n)^2); my(i = pos(list, r)); if (i==1, return (1)); if (i>1, return(0)); listput(list, r); );} \\ reaches a fixed point
a(n) = sum(r=0, n-1, 1 - isok(r, n)); \\ Michel Marcus, May 02 2021

Formula

a(n) is the number of terms of n-th row of A279185 that are greater than 1. - Pontus von Brömssen, Apr 27 2021
a(n) + A343721(n) = n. - Michel Marcus, May 02 2021

A272370 Number of geometrically inscriptible regular polygons with fewer than 2^n + 1 sides.

Original entry on oeis.org

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 559, 591, 623, 655, 687, 719, 751, 783, 815, 847, 879, 911, 943, 975, 1007, 1039, 1071, 1103, 1135, 1167, 1199, 1231, 1263, 1295
Offset: 1

Views

Author

Michel Marcus, Apr 28 2016

Keywords

Comments

a(n) is the number of terms of A003401, except its first two degenerate case terms, that are less than 2^n + 1.

Examples

			For n=2, there are 2 such polygons, those with 3 and 4 sides, below 2^2+1 = 5.

Links

Crossrefs

Cf. A000051, A003401, A004169, A019434, A045544.

Programs

  • PARI
    a(n) = if(n < 32, (n-1)*(n+2)/2, if(n < 2^33, 32*n-497));
  • PARI
    a(n) = {i = 2^n; j = 2*n - 2; k = 1; while(i > A045544(k) && k < 31, k++; j+=floor(log(i/A045544(k))/log(2))+1); j; } \\ Jinyuan Wang, Jul 29 2019

Formula

a(n) = (n-1)*(n+2)/2 if n < 32, otherwise 32*n-497 if n < 2^33.
Previous Showing 11-13 of 13 results.

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