Jason Holt - cp's OEIS frontend

User: Jason Holt

Jason Holt has authored 3 sequences.

A191592 Numbers in A191311 but not in A129521.

4, 11305, 13981, 23001, 30889, 39865, 68101, 88561, 91001, 93961, 107185, 137149, 152551, 157641, 176149, 204001, 228241, 251251, 276013, 401401, 464185, 493697, 493885, 534061, 563473, 574561, 622909, 631351, 683761, 786961, 915981, 950797, 1106785, 1141141

Offset: 1

Jason Holt, Jun 04 2011

nonn

David W. Wilson, Table of n, a(n) for n = 1..697

Cf. A191311, A129521.

More terms from David W. Wilson, Aug 16 2011

A191311 Numbers n such that exactly half of the a such that 0

Original entry on oeis.org

4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, 23001, 30889, 38503, 39865, 49141, 68101, 79003, 88561, 88831, 91001, 93961, 104653, 107185, 137149, 146611, 152551, 157641, 176149, 188191, 204001, 218791, 226801, 228241

Offset: 1

Jason Holt, Jun 04 2011

Values of n for which half the witnesses in the Fermat primality test are false.
When n=pq with p,q=2p-1 prime, a^(n-1) = 1 (mod p) iff a is a quadratic residue mod q. So A129521 is a subsequence. - Gareth McCaughan, Jun 05 2011
From Robert G. Wilson v, Aug 13 2011: (Start)
Number of terms less than 10^n: 2, 4, 5, 7, 22, 60, 129, 303, 690, 1785, …, .
In reference to the numbers in the b-file: (1) number of terms which have k>0 prime factors: 1, 1058, 139, 512, 339, 102, 6; (2) about half of the terms, 1058, are members of A129521, those which have just two prime factors; (3) except for the first term, all terms are squarefree, except for the first two terms, all terms are odd; and (4) most terms, more than 98.5%, are congruent to 1 modulo 6. (End)

David W. Wilson and Robert G. Wilson v, Table of n, a(n) for n = 1..2157
While exploring Carmichael numbers, I noticed a few values on the chart on this page for which exactly half of the relatively prime witnesses to the Fermat primality test were false witnesses.

A063994 gives the number of false witnesses for every n.

A129521 is a subsequence. See also A191592.

fQ[n_] := Block[{pf = First /@ FactorInteger@ n}, 2Times @@ GCD[n - 1, pf - 1] == n*Times @@ (1 - 1/pf)]; Select[ Range@ 250000, fQ] (* Robert G. Wilson v, Aug 08 2011 *)

import math
for x in range(2, 1000):
  false_witnesses = 0
  relatively_prime_values = 0
  for y in range(x):
    if math.gcd(y, x) == 1:
      relatively_prime_values += 1
    if (pow(y, x-1, x) == 1):
      false_witnesses += 1
  if false_witnesses * 2 == relatively_prime_values:
    print(x, "is a Fermat Half-Prime")

Integers, n, such that A063994(n) = 2*A000010(n). - Robert G. Wilson v, Aug 13 2011

Edited by N. J. A. Sloane, Jun 07 2011. I made use of a more explicit definition due to Gareth McCaughan, Jun 05 2011.

A124484 Maximum number of unit squares aligned with unit-spaced horizontal lines that can be enclosed by a circle of radius n.

Original entry on oeis.org

0, 1, 8, 21, 40, 65, 97, 135, 180, 229, 286, 350, 419, 495, 575, 664, 761, 860, 966, 1079, 1200, 1326, 1458, 1595, 1741, 1892, 2050, 2213, 2383, 2558, 2741, 2930, 3124, 3328, 3534, 3746, 3967, 4194, 4428, 4666, 4910, 5162, 5420, 5682, 5952, 6231, 6517, 6802, 7097

Offset: 0

Jason Holt, Nov 10 2006

nonn

I don't know how many of these entries have been proved to be optimal. The Erdős-Graham paper shows how subtle such problems can be. - N. J. A. Sloane, Dec 19 2006 [This comment was written before the July 2024 clarifications to the name and definition. - Editors]

In the Erdős-Graham paper and on Erich Friedman's website, the orientation of the squares is not restricted to a position parallel to the axes. - Hugo Pfoertner, Jul 14 2024

P. Erdős and R. L. Graham, On packing squares with equal squares, J. Comb. Theory (A), 19 (1975), 119-123.
Erich Friedman, Squares in Circles.
Jason Holt, Packing squares into circles.
Hugo Pfoertner, Illustration of a(2)-a(12), showing results of Jason Holt's C program.

Cf. A374505.

a(n) = A374505(2*n). - David Dewan, Jul 10 2024

a(1) corrected by David Dewan, Jun 13 2024
Name and definition amended to be consistent with the author's program by Hugo Pfoertner, Jul 14 2024
a(20) onwards from David Dewan, Jul 14 2024

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User: Jason Holt

A191592 Numbers in A191311 but not in A129521.

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A191311 Numbers n such that exactly half of the a such that 0

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A124484 Maximum number of unit squares aligned with unit-spaced horizontal lines that can be enclosed by a circle of radius n.

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