A189123 -id:A189123 - cp's OEIS frontend

A189125 Primes p such that A005245(p) < A005245(p-1)+1.

353942783, 1163385647, 1932319583, 2336924879, 4166809919, 4937621453, 5123340683, 5184740299, 5390865059, 5455982879, 5467766947, 6432033887, 6459553799, 6545574839, 6714582263, 6988649399, 7378517519, 7515851039, 7657182539, 7756383347

Offset: 1

Max Alekseyev, Apr 17 2011

Counterexamples to the conjecture from section F26 in UPINT.
a(1) was found by Martin Fuller; a(2) was found by Janis Iraids.
The prime terms of each of the sequences A189123 and A189124.

R. K. Guy, Unsolved Problems in Number Theory, Section F26.

Janis Iraids, Table of n, a(n) for n = 1..1000
Martin N. Fuller, C program for A005245

Cf. A005245, A189123, A189124.

More terms from Janis Iraids, Apr 20 2011.

A189124 Positive integers n such that A005245(n) < min{ A005245(n-1) + 1, A005245(x) + A005245(y) } where x,y range over positive integers with x*y = n.

Original entry on oeis.org

353942783, 516743639, 1163385647, 1542243239, 1932319583, 2336924879, 3113713259, 3444631199, 3878989487, 4103787551, 4166809919, 4937621453, 5123340683, 5170931639, 5184740299, 5200683263, 5390865059, 5455982879, 5467766947, 5570566315

Offset: 1

Max Alekseyev, Apr 17 2011

Counterexamples to the second conjecture of David Wilson on A005245.
a(1), a(2) were found by Martin Fuller; a(3) was found by Janis Iraids.
Prime terms are given by A189125.

Janis Iraids, Table of n, a(n) for n = 1..1000
Martin N. Fuller, C program for A005245

Cf. A005245, A189123, A189125.

More terms from Janis Iraids, Apr 20 2011

A195101 Solid numbers. Numbers m such that A005245(m) < A005245(k) + A005245(m-k) for 1<= k < m.

Original entry on oeis.org

1, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 26, 27, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111

Offset: 1

Juan Arias-de-Reyna, Sep 09 2011

nonn

It is useful for computing  A005245(m). To compute min_k  A005245(k) + A005245(m-k) we only need to check the cases in which k is a solid number.
The solid numbers  <= x appear to be  <= 0.6 * x.
We find many values where the minimum of A005245(k) + A005245(m-k) is not taken for k = 1. This is sequence A189123.
The first value of m needing k = 6 is 21080618, the first k = 8 is 385159320, the first with k = 9 is 3679353584.
Conjecture that for every solid number m > 1 there is some number n such that  A005245(n) = A005245(m)+A005245(n-m) and such that  for every representation as a product n = u*v with u, v >= 2 or every 1 < = k < m, we have A005245(n) < A005245(u)+A005245(v) and A005245(n) < A005245(k) + A005245(n-k).
The solid numbers are infinite. Proof by H. Altman, mentioned in link. For n>1, 3^n is a solid number.  If 3^n=a+b with 3n=||a||+||b||, then 3log_3(a)+3log_3(b)<=3n, and so ab<=3^n=a+b.  So either a=b=2 (impossible), or a=1 or b=1.  So suppose a=1.  Then b=3^n-1.  But since n>1 we have 3^n-1>(3/4)3^n>=E(3n-1) and thus ||3^n-1||>=3n, ||a||+||b||>=3n+1, contradiction. - Juan Arias-de-Reyna, Jan 09 2014

			m = 8 is a term of the sequence because
A005245(8) = 6 < A005245(7) + A005245(1)=6+1; A005245(8) < A005245(6) + A005245(2)=5+2; A005245(8)  < A005245(5) + A005245(3)=5+3;
A005245(8) < A005245(4) + A005245(4)=4+4.
m = 7 is not a term of the sequence because A005245(7) = 6 = A005245(6) + A005245(1) = 5 + 1.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
H. Altman and J. Zelinsky, Numbers with integer complexity close to the lower bound, Integers, 12 (2012) 1093-1125 (article where sequence is first introduced).
Juan Arias de Reyna, Complexity of natural numbers and arithmetic compact sets, arXiv:2302.06224 [math.NT], 2023.
Juan Arias de Reyna, Arithmetical Self-Similar Compact Sets, Integers (2024) Vol. 24, A21. See p. 13.

Cf. A005245, A189123.

nn = 200; a5245[n_] :=  a5245[n] = If[n == 1, 1, Min[Sequence @@ Table[a5245[i] + a5245[n - i], {i, 1, n/2}], Sequence @@ Table[a5245[d] + a5245[n/d], {d, Divisors[n]~Complement~{1, n}}]]]; t = Table[a5245[n], {n, nn}]; Select[Range[nn], And @@ Table[t[[#]] < t[[k]] + t[[# - k]], {k, # - 1}] &] (* T. D. Noe, Apr 09 2014 *)

Name and comments change using "solid numbers" notation by Juan Arias-de-Reyna, Jan 09 2014

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A189125 Primes p such that A005245(p) < A005245(p-1)+1.

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A189124 Positive integers n such that A005245(n) < min{ A005245(n-1) + 1, A005245(x) + A005245(y) } where x,y range over positive integers with x*y = n.

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A195101 Solid numbers. Numbers m such that A005245(m) < A005245(k) + A005245(m-k) for 1<= k < m.

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