A378472 Position of start of first run of exactly n zeros in the base-2 representation of Pi, or -1 if no such run exists.

17, 1, 26, 7, 109, 135, 96, 189, 2610, 902, 4267, 36139, 17317, 8375, 479166, 11791, 112954, 436893, 1286743, 726844, 5572140, 27456324, 2005750, 42248747, 200643872, 547151636, 171498580, 469458286, 1222711767, 2151391703, 1407238214

Offset

1,1

Comments

In base-2, Pi is: 11.00100100001111110110101010001... For this sequence, the integer part of Pi is ignored, and the first fractional bit is numbered one.

No further terms <= 4*10^9. - Michael S. Branicky, Dec 04 2024

See A178708 for the start of the first string of n zeros, not necessarily followed by a digit 1. - M. F. Hasler, Feb 04 2026

Links

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

Formula

a(n) >= A178708(n). - Michael S. Branicky, Dec 13 2024

Example

The first run of a single "0" bit is at position 17, so a(1) = 17.
The first run of exactly 2 zeros is at position 1, so a(2) = 1.

Prog

(Python)
import gmpy2
gmpy2.get_context().precision = 2000000
pi = gmpy2.const_pi()
# Convert Pi to binary representation
binary_pi = gmpy2.digits(pi, 2)[0] # zero-th element is the string of bits
outVec = []
for lenRun in range(1, 20):
  l1 = binary_pi.find("1" + "0"*lenRun + "1")
  outVec.append(l1) # edited by M. F. Hasler, Feb 04 2026
print(outVec)

Crossrefs

Cf. A004601, A175945, A178708 (start of run of exactly n '0's), A233836.

Sequence in context: A040304 A347142 A347143 * A124517 A241121 A040305

Adjacent sequences: A378469 A378470 A378471 * A378473 A378474 A378475

Keyword

nonn,base,more

Author

James S. DeArmon, Nov 27 2024

Extensions

a(21)-a(31) from Michael S. Branicky, Dec 04 2024

Clarified definition, added escape clause - N. J. A. Sloane, Dec 23 2024

Status

approved