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A182532
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Carmichael numbers divisible by 7 and 17.
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1
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825265, 6840001, 16778881, 47006785, 413631505, 490503601, 547652161, 1180398961, 1529544961, 1597009393, 2265650401, 2313774001, 2523947041, 2560104001, 2586927553, 3180632833, 3754483201, 4477793761, 5106068065, 5394826801, 6218177329, 6453043345, 8053562881, 10152380161
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OFFSET
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1,1
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COMMENTS
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Conjecture (1): Any Carmichael number C divisible by 7 and 17 of the form 10k+1 can be written as C = 7*17*(120n+119). We got for the tested numbers the following values for n: 478, 1174, 34348, 38350, 82660, 107110, 158658, 162028, 176746, 179278, 262918, 313570, 377788, 563974, 710950.
Conjecture (2): Any Carmichael number C divisible by 7 and 17 of the form 10k+5 can be written as C = 7*17*(120n+95). We got for the tested numbers the following values for n: 57, 3291, 28965, 357567, 451893.
Conjecture (3): Any Carmichael number C divisible by 7 and 17 of the form 10k+3 can be written as C = 7*17*(120n+47). We got for the tested numbers the following values for n: 111835, 181157, 222733.
Conjecture (4): Any Carmichael number C divisible by 7 and 17 of the form 10k+9 can be written as C = 7*17*(120n+71). We got for the tested number the following value for n: 435446.
Note: the property of being factorizable as C = p*q*(n*(p*q+1) + p*q), where p and q are prime, is not derived from Korselt's criterion, and is not shared by all Carmichael numbers; e.g., for the Carmichael number 340561 = 13*17*23*67, even though it is of the form 10k+1, we have (23*67) mod (13*17+1) = 209 != 221 = 13*17.
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LINKS
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PROG
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(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
list(lim)=my(v=List()); forstep(n=825265, lim, 5712, if(Korselt(n), listput(v, n))); Vec(v)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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