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

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

Author

Richard S. Chang, Jul 09 2025

Keywords

• easy
• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A063655.

Programs

Maple

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

Mathematica

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

PARI

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

Extensions

More terms from Michel Marcus, Jul 09 2025

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