{"schema":"vela.problem-packet.v0.1","problem":470,"statement":"Call $n$ weird if $\\sigma(n)\\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_5cfde0ba44d6cbac","kind":"dead_end","claim":"Erdős #470 (weird numbers: abundant but not pseudoperfect) — OBSTRUCTION MAP, not a settlement. UNCONDITIONAL: Q3 (weird numbers have positive density) is closed [Benkoski-Erdős 1974]; Melfi's family n=2^k*p*q (p=2^(k+2)-a, q=2^(k+2)+b primes, b+3<a<2^((k-1)/2)) yields PRIMITIVE WEIRD numbers, re-verified on a battery incl. 2^6*251*257=4128448; and the obstruction to making Q2 unconditional is exactly a lacunary square-root short-interval problem — Melfi's window has radius L_k=sqrt(2^(k+2))/(2*sqrt2) centered at the sparse points 2^(k+2), exponent 1/2 with no log, so BHP (x^0.525), bounded gaps (Zhang/Maynard) and RH (sqrt(x)*log x) all OVERSHOOT it by a factor ~X^0.025. A prime in (x,x+c*sqrt x] for any c<1/(4*sqrt2)=0.17677 suffices (sharpening Melfi's stated 0.1). CONDITIONAL (NOT asserted): Q2 (infinitely many primitive weird) holds under g_n=o(sqrt p_n) [Cramer-type]; Q1 (odd weird numbers) reduces to a uniform Omega-bound on odd primitive weird n [unknown; none below 10^28, Fang 2022].","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[{"id":"A002975","name":"Primitive weird numbers: weird numbers with no proper weird divisors.","terms":"70,836,4030,5830,7192,7912,9272,10792,17272,45356,73616,83312,91388,113072,243892,254012,338572,343876,388076,519712,539","url":"https://oeis.org/A002975"},{"id":"A006037","name":"Weird numbers: abundant (A005101) but not pseudoperfect (A005835).","terms":"70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,12110,12530,12670,13370,13510,13790,13930,14770,1561","url":"https://oeis.org/A006037"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}