A393128 Positive integers not of the form floor(k + log(k) + log(Pi)).

1, 4, 9, 21, 52, 134, 356, 956, 2588, 7021, 19069, 51818, 140837, 382814, 1040575, 2828552, 7688777, 20900235, 56812699, 154432898, 419792107, 1141113222, 3101867298, 8431749474, 22919871337, 62302669726, 169356214937, 460357921555, 1251382572703, 3401610507781, 9246536030745

Offset

1,2

Comments

It was conjectured that for every prime p there is exactly one integer k such that p = k + floor(log(Pi*k)), but this is not true for the primes in this sequence, the first of which are 19069, 140837, 56812699, ...

(This sequence lists all positive integers that do not occur as values of the function on the right hand side.)

Links

Table of n, a(n) for n=1..31.

Sebastian Martin Ruiz, Pi Conjecture, PrimeNumbersTheory on Groups.io, March 14, 2026.

Formula

a(n+1) / a(n) -> e = 2.718281828...

Prog

(PARI)
A393128_first(N)=vector(N, n, if(n>1, N=N*exp(1)\/1-n; n=N+log(N*Pi)\1; until(n+1<n=N+++log(N*Pi)\1, ); n-1, N=1))

Crossrefs

Cf. A053510 (log Pi).

Sequence in context: A243640 A243634 A076861 * A374943 A048391 A343322

Adjacent sequences: A393125 A393126 A393127 * A393129 A393130 A393131

Keyword

nonn

Author

M. F. Hasler, Mar 17 2026

Status

approved