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Concatenations of pairs of primes that differ by 1000000.
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31000003, 371000037, 1511000151, 1931000193, 1991000199, 2111000211, 3131000313, 3671000367, 3971000397, 4091000409, 4571000457, 5411000541, 5471000547, 5771000577, 6191000619, 6911000691, 8291000829, 8591000859
COMMENTS
After the first element, 31000003, which is prime, integers in this sequence can never be prime, as they are all multiples of 3. They can be semiprimes, as is the case for 3671000367 = 3 x 1223666789, 4571000457 = 3 x 1523666819, 5411000541 = 3 x 1803666847, 9071000907 = 3 x 3023666969.
FORMULA
a(n) = Concatenate(P, P+1000000) iff P prime and P+1000000 prime.
EXAMPLE
Prime(47) = 211 and 211 + 1000000 = Prime(78515) = 1000211. Concatenating these two primes gives 2111000211 = 3^4 * 17^2 * 31 * 2909.
Concatenations of pairs of primes that differ by 10^12.
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611000000000061, 1631000000000163, 1931000000000193, 2111000000000211, 2711000000000271, 3311000000000331, 5471000000000547, 6611000000000661, 7511000000000751, 7871000000000787, 9971000000000997, 10511000000001051
COMMENTS
Integers in this sequence can never be prime, as they are all multiples of 3. They can be semiprimes, as is the case for Prime(177) concatenated with Prime(37607912056) = 10511000000001051 = 3 * 3503666666667017.
FORMULA
a(n) = Concatenate(P, P+10^12) iff P prime and P+10^12 prime.
EXAMPLE
61 is prime, specifically prime(18) and 61 + 10^12 is prime, specifically prime(7607912020), so their concatenation is in this sequence: 611000000000061. The concatenation is not itself prime, as it equals 3 * 7 * 23 * 1265010351967.
MATHEMATICA
#*10^13+10^12+#&/@Select[Prime[Range[200]], PrimeQ[#+10^12]&] (* Harvey P. Dale, Jan 18 2021 *)
CROSSREFS
Cf. A000040, A001358, A023201, A100750, A103195, A103206, A104718, A104719, A103523, A103534, A103576, A103617.
Concatenations of pairs of primes that differ by 10^9.
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71000000007, 971000000097, 1031000000103, 1811000000181, 2231000000223, 2411000000241, 2711000000271, 3491000000349, 4091000000409, 4331000000433, 4391000000439, 6071000000607, 6131000000613, 7871000000787, 8291000000829
COMMENTS
Integers in this sequence can never be prime, as they are all multiples of 3. They can be semiprimes, as is the case for Prime(42) concatenated with Prime(50847544) = 1811000000181 = 3 x 603666666727.
FORMULA
a(n) = Concatenate(P, P+1000000000) iff P prime and P+1000000000 prime.
EXAMPLE
181 is prime, 181+10^9 = 1000000181 is prime, so their concatenation is an element of this sequence: 1811000000181. Coincidentally, prime(181)+10^9 = 1000001087 is also prime.
MATHEMATICA
FromDigits[Join[IntegerDigits[#], IntegerDigits[#+10^9]]]&/@Select[Prime[ Range[ 200]], PrimeQ[ #+ 10^9]&] (* Harvey P. Dale, May 14 2022 *)
CROSSREFS
Cf. A000040, A001358, A023201, A100750, A103195, A103206, A104718, A104719, A103523, A103534, A103576.
Concatenations of pairs of primes that differ by 1000.
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131013, 191019, 311031, 611061, 971097, 1031103, 1091109, 1511151, 1631163, 1811181, 1931193, 2231223, 2291229, 2771277, 2831283, 3071307, 3671367, 3731373, 4091409, 4331433, 4391439, 4871487, 4991499, 5231523, 5711571
COMMENTS
All terms are multiples of 3.
FORMULA
a(n) = Concatenate(P, P+1000) iff P prime and P+1000 prime.
EXAMPLE
1811181 is in this sequence because 181 is prime, 181+1000 = 1181 is prime and those two primes are concatenated.
MATHEMATICA
10001#+1000&/@Select[Prime[Range[150]], PrimeQ[#+1000]&] (* Harvey P. Dale, Sep 01 2017 *)
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