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Old 2021-01-23, 00:02   #16
kriesel
 
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Mar 2017
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Default Mersenne semiprimes

Semiprimes are natural numbers that are products of exactly two factors other than one and themselves (and they may be squares). https://prime-numbers.info/article/semiprimes

http://oeis.org/A092561 lists Mersenne numbers that are semiprimes without regard to whether n is composite.
n Mn
4 15,
9 511,
11 2047,
23 8388607,
37 137438953471,
41 2199023255551,
49 562949953421311,
59 576460752303423487,
67 147573952589676412927,
83 9671406556917033397649407,
97 158456325028528675187087900671,
101 2535301200456458802993406410751

Subsetting to prime exponent < 100 gives
p for Mp and factors
11 2047 f=23; 89
23 8,388,607 f=47; 178481
37 137438953471 f=223; 616318177
41 2,199,023,255,551 f=13367, 164,511,353
59 576,460,752,303,423,487 f= 179951, 3 203431 780337
67 147,573,952,589,676,412,927 = 193 707721 × 761838 257287
83 9,671,406,556,917,033,397,649,407 = 167 × 57912 614113 275649 087721
97 158,456,325,028,528,675,187,087,900,671 = 11447 × 13 842607 235828 485645 766393
below p=100, 8 Mersenne semiprimes of prime exponent vs. 10 Mersenne primes.

in p= 100 to 200, 8 semiprimes vs. only 2 primes of prime exponent.
in p=200 to 500, 9 vs 0.
https://www.mersenne.org/report_expo..._hi=500&full=1

Last fiddled with by kriesel on 2021-01-29 at 17:22 Reason: add a semiprime definition and link
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