Smarandache semiprimes
 

Sequence A046461 Sloane's OEIS, about finding semiprime Smarandache numbers (concatenation the positive integers, Sm(11)=1234567891011, etc.). The known values this sequence are 3, 4, 7, 34, 97. For example, Sm(34) = 2 * 6172839455055606570758085909601061116212631364146515661667.

Further searching indicates the only other possible members below 2300, are:

[CODE]Sm(631)
Sm(691)
Sm(859)
Sm(1033)
Sm(1051)
Sm(1291)
Sm(1651)
Sm(1657)
Sm(1831)
Sm(1951)
Sm(2041)
Sm(2047)
Sm(2107)[/CODE]

Finding a factor any these numbers sufficient (after determing cofactor primality etc.) determine they are the sequence.

I've run 100 ecm curves with b1=10000 each candidate, but nothing with higher limits.

Hmm, for some reason the forum i.s losing two letter words like i.s from m.y posts.

[quote=sean;210253]Hmm, for some reason the forum i.s losing two letter words like i.s from m.y posts.[/quote]
I suspect (and hope) this (and other anomalies) will only annoy today...

Ugh, ,base,-sequences. I'll take Sm(631) t.o 250,000.

Some weeks ago i begun to make a summary-page for Smarandache-type sequences but not yet time to complete everything and the FactorDB-issue for small numbers is annoying,too.

See [url=www.rieselprime.de/Others/Smarandache.htm]here[/url] for Smarandache- and Reverse-Smarandache-sequences.

PrimeForm/GW has found Sm(5053) = 133283 * prp19099.

I'm not attempting to prove the prp.

[QUOTE=CRGreathouse;210305]Ugh, ,base,-sequences. I'll take Sm(631) t.o 250,000.[/QUOTE]

If nothing goes wrong, I'll have it done through 35 digits (B1 = 1,000,000) tomorrow.

Sm(9706) = 2 * prp37716, found by PrimeForm/GW.

Sm(631) = 414941628631826493984534937401473 * C1752

Working on Sm(691) now.

Many thanks for the Sm(631) and the results for higher n. In a few days I will submit some extra comments on the OEIS entry.

[QUOTE=sean;210671]Many thanks for the Sm(631) and the results for higher n. In a few days I will submit some extra comments on the OEIS entry.[/QUOTE]

It would be really nice to find the next term, but that's going to take more than "a few days" unless you get a lot more people.

Another one eliminated:

[CODE]Sm(1033) = 30915171019253321813 * C3005[/CODE]

I have trial factored the first 25000 Smarandache numbers to 2^32 with a C program. The factors are attached.
PrimeForm/GW has prp tested the cofactor of the first 10000, and those from 10000 to 25000 which have exactly one prime factor below 2^32.

No probable semiprimes were found beyond the two earlier reported: Sm(5053) and Sm(9706).
39 complete factorizations up to Sm(200) were rediscovered.
25 complete factorizations above Sm(200) were found:

[CODE]Sm(203) = 3 * 521 * p497
Sm(265) = 5 * 17 * 887 * 1787 * p678
Sm(279) = 3^2 * 458891 * p722
Sm(327) = 3 * 23 * 132861611 * p863
Sm(343) = 19 * 271 * 1601 * p914
Sm(345) = 3 * 5 * 37 * 47 * 67 * 157 * 441121 * 43455173 * 1503598049 * 2913393619 * p886
Sm(452) = 2^2 * 3 * 67 * 163 * 3023 * 83177 * p1234
Sm(461) = 3 * 23^2 * 1033 * p1268
Sm(572) = 2^2 * 3 * 2399 * p1603
Sm(601) = 2749 * 3403013 * 33226397 * 3222045163 * p1668
Sm(780) = 2^2 * 3 * 5 * 17^2 * 73 * 167 * 229 * p2221
Sm(872) = 2^6 * 3^2 * p2505
Sm(1386) = 2 * 3^2 * 19 * 269 * 12301 * prp4428
Sm(1927) = 7^2 * 13 * prp6598
Sm(3231) = 3^2 * 50411 * prp11811
Sm(3418) = 2 * 7 * 752933 * prp12558
Sm(3506) = 2 * 3 * 34693 * 186227 * 204749 * prp12901
Sm(5053) = 133283 * prp19099
Sm(5417) = 3^2 * 19 * prp20558
Sm(5677) = 17^2 * 7307 * 19507 * prp21590
Sm(6182) = 2 * 3^3 * 29 * 277 * prp23615
Sm(6419) = 3 * 61 * 6827 * 2368097 * 113970541 * prp24548
Sm(7582) = 2 * 13 * 17 * 31 * prp29216
Sm(8550) = 2 * 3^2 * 5^2 * 7 * 3499 * 307019 * 1551997 * prp33074
Sm(9706) = 2 * prp37716[/CODE]

Marcel Martin's Primo proved the first 12 cofactors. I am not planning to prove or prp test more cofactors.

Starting from Jens' list I can eliminate the following candidates:

[CODE]133 8223519074965787731
391 269534025881.1944200997131
463 95762360557419696305637209522797
631 414941628631826493984534937401473
649 877026198670416511
787 46110336443
859 77278117165680372333434870566009
1027 35488519258785497579
1033 30915171019253321813
1039 1479264257587679378617
1207 6176528903
1561 43008171535331
1609 129787493421373
1783 55786004333
1807 738921181
1861 1471795943419
1939 10981653341127770593
1951 891704872753975021087
1963 15654967526938910563
1999 11687073721669454909
2029 42457568152889999
2107 5135394736173493133
2191 434290438959877
2539 4840263101
2659 17678395741
3121 4745966107
3289 1787896747099
3547 222029309596931
3619 1685013448673
3673 465890314573
3883 47784961208459
3907 7663417791922241
4261 32322505871
4393 7446261457
4429 264146732728106877929671
4477 145122725496587
4549 9457516676479
4663 279168988896034784147
4711 3059843006852143
4717 7436263183
4897 37089928868738827
[/CODE]

This leaves the following as possible semiprimes between 97 and the next confirmed member Sm(5053):

[CODE]691
1051
1291
1651
1657
1831
2041
2047
2377
2383
2491
2761
3007
3163
3391
3493
3571
3937
3991
4057
4087
4141
4213
4399
4819
4831
4873
[/CODE]

In each case I have run at least 100 curves with b1=10000.

828297113610116869 | Sm(1291)

I've eliminated 691 and 1051:

[CODE]
691 1304238680165623831238651513722972177904593843651
1051 94182284179957932178843944956212141
[/CODE]