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User: Richard S. Chang

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Richard S. Chang has authored 4 sequences.

A386525 a(n) is the least k such that at least n terms of A063655 starting from index k are strictly decreasing.

Original entry on oeis.org

1, 5, 13, 37, 122, 3004, 26283, 53411, 109453, 4117156, 16831081

Offset: 1

Richard S. Chang, Jul 24 2025

Lai and Reinfeld conjecture that the sequence is infinite.

			The first 40 values of A063655 are: 2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, and 13. Because the 37th term is the first of 4 strictly decreasing values and there is not previous occurrence of four decreasing values, a(3) = 37.

H. Lai and M. Reinfeld, An Introduction to LR Numbers, Girls' Angle Bulletin, Vol. 18, No. 5 (2025), 11-15.

Cf. A063655.

a(11) from Sean A. Irvine, Aug 10 2025

A385827 a(n) is the unique k such that 2^k is closest to 5^n.

Original entry on oeis.org

0, 2, 5, 7, 9, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 35, 37, 39, 42, 44, 46, 49, 51, 53, 56, 58, 60, 63, 65, 67, 70, 72, 74, 77, 79, 81, 84, 86, 88, 90, 93, 95, 97, 100, 102, 104, 107, 109, 111, 114, 116, 118, 121, 123, 125, 128, 130, 132, 135, 137, 139, 142, 144, 146, 149

Offset: 0

Richard S. Chang, Jul 09 2025

a(n) is the unique k such that 2^k/5^n is closest to 1.
For all n, 2/3 < 2^a(n)/5^n < 4/3.
For n > 0, a(n) = A061785(n) if log_2(2*5^n/3) < floor(n*log_2(5)), otherwise a(n) = A061785(n) + 1.

			a(3) = 7 because 2^7 = 128 is the closest power of 2 to 5^3 = 125.

A. Gorman-Huang and E. Marks, Approximating Powers of 2 Using Powers of 3 and Break Point Rounding, Girls' Angle Bulletin, Vol. 18, No. 5 (2025), 8-10.

Cf. A020857, A061785.

a[n_]:=-Floor[-n*Log2[5] + Log2[3/2]]; Array[a,65,0] (* Stefano Spezia, Jul 17 2025 *)

def A385827(n):
    if n == 0: return 0
    k = 5**n
    m = k.bit_length()-2
    return m+1+(k>3<Chai Wah Wu, Jul 22 2025

a(n) = -floor(-n*log_2(5) + log_2(3/2)).

A385812 Numbers k such that A063655(k) > A063655(k+1).

Original entry on oeis.org

5, 7, 11, 13, 14, 17, 19, 23, 26, 27, 29, 31, 34, 37, 38, 39, 41, 43, 44, 47, 51, 53, 55, 59, 61, 62, 65, 67, 69, 71, 73, 74, 76, 79, 83, 86, 87, 89, 94, 95, 97, 98, 101, 103, 107, 109, 111, 113, 116, 118, 119, 122, 123, 124, 125, 127, 129, 131, 134, 137, 139, 142, 146, 149

Offset: 1

Richard S. Chang, Jul 09 2025

Lai and Reinfeld show that:
  Terms include all primes greater than 3.
  Terms include 2p where p is prime and 2p+1 is composite.
  a(n) + 1 is never a perfect square.
  Let b be a real number greater than 1 and let P(n) be the probability of getting n as the product of two independent die rolls where each die comes up k with probability (b-1)/b^k. A number is a term if and only if P(n)Lai and Reinfeld conjecture that:
  Asymptotically half the positive integers are terms.
  For any positive integer L, there exist L consecutive numbers in this sequence.
Also, a(n) is never a perfect square.

			A063655(14) = 9 and A063655(15) = 8, so 14 is a term.
A063655(50) = 15 and A063655(51) = 20, so 50 is not a term.

Robert Israel, Table of n, a(n) for n = 1..10000
Helena Lai and Margot Reinfeld, An Introduction to LR Numbers, Girls' Angle Bulletin, Vol. 18, No. 5 (2025), 11-15.

Cf. A063655.

Res:= NULL: count:= 0:
v:= A063655(1):
for i from 2 while count < 100 do
  w:= A063655(i);
  if w < v then Res:= Res,i-1; count:= count+1 fi;
  v:= w
od:
Res; # Robert Israel, Aug 10 2025

Position[Differences[Array[2*Median[Divisors[#]] &, 150]], ?Negative] // Flatten (* _Amiram Eldar, Jul 10 2025 *)

s(n) = my(md=if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2])); md + n/md; \\A063655
isok(k) = s(k) > s(k+1); \\ Michel Marcus, Jul 09 2025

More terms from Michel Marcus, Jul 09 2025

A333354 Minimum cost of path that starts at 1 and visits integers from 1 to n, inclusive, each at least once, where the cost to travel from a to b is LCM(a, b).

Original entry on oeis.org

0, 2, 7, 12, 21, 28, 40, 51, 65, 79, 100, 114, 138, 158, 182, 205, 238, 259, 295, 324, 358, 390, 435, 463, 511, 549, 593, 634

Offset: 1

Richard S. Chang, May 04 2020

			For n = 3, the optimal path is 1, 2, 1, 3, which has cost 2 + 2 + 3 = 7.
For n = 4, the optimal path is 1, 3, 1, 2, 4, which has cost 3 + 3 + 2 + 4 = 12.
For n = 7, there are multiple optimal paths of which 1, 3, 6, 2, 4, 1, 5, 1, 7 is one and has cost 3 + 6 + 6 + 4 + 4 + 5 + 5 + 7 = 40.
For n = 20, an optimal path is 1, 11, 1, 13, 1, 17, 1, 19, 1, 7, 14, 2, 16, 8, 4, 12, 6, 18, 9, 3, 15, 5, 10, 20.

A. Catanzaro, J. Feldman, M. Higgins, B. Kimball, H. Kirk, A. C Maravelias, and D. Sinha, LCM Optimal Paths, Girls' Angle Bulletin, Vol. 13, No. 4 (2020), 13-19.

a(21)-a(28) from Bert Dobbelaere, Aug 22 2020

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User: Richard S. Chang

A386525 a(n) is the least k such that at least n terms of A063655 starting from index k are strictly decreasing.

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A385827 a(n) is the unique k such that 2^k is closest to 5^n.

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A385812 Numbers k such that A063655(k) > A063655(k+1).

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A333354 Minimum cost of path that starts at 1 and visits integers from 1 to n, inclusive, each at least once, where the cost to travel from a to b is LCM(a, b).

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