A094487
Primes p such that 2^j+p^j are primes for j=0,1,2,4.
Original entry on oeis.org
3, 5, 17, 4517, 5477, 5867, 7457, 8537, 13877, 16067, 22697, 27917, 56477, 59357, 90437, 97577, 101747, 118247, 122207, 124247, 135467, 139457, 140417, 153947, 208697, 247067, 267677, 306947, 419927, 470087, 489407, 520547, 529577, 540347
Offset: 1
For j=0 1+1=2 is prime; also terms should be lesser-twin-primes
because of p^1+2^1=p+2=prime; 3rd and 4th conditions are as
follows: prime=p^2+4 and prime=16+p^4.
-
{ta=Table[0, {100}], u=1}; Do[s0=2;s1=Prime[j]+2;s2=4+Prime[j]^2;s4=16+Prime[j]^4; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s4], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[45000]],AllTrue[{2+#,4+#^2,16+#^4},PrimeQ]&] (* Harvey P. Dale, Sep 18 2022 *)
A094489
Primes p such that 2^j+p^j are primes for j=0,1,4,32.
Original entry on oeis.org
59, 5417, 19079, 33827, 136949, 181871, 242519, 284897, 421607, 452537, 552401, 598187, 962681, 1068251, 1081979, 1163231, 1317761, 1760279, 1801361, 1891499, 1895081, 1919459, 2056907, 2131601, 2427461, 2557601, 2579177, 2826737
Offset: 1
For j=0 1+1=2 is prime; also terms should be lesser-twin-primes
because of p^1+2^1=p+2=prime; 3rd and 4th conditions are as
follows: prime=p^4+16 and prime=2^32+p^32.
-
{ta=Table[0, {100}], u=1}; Do[s0=2;s1=Prime[j]+2;s2=4+Prime[j]^2;s8=2^32+Prime[j]^32; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s8], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[210000]],AllTrue[{2+#,16+#^4,2^32+#^32},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 13 2015 *)
A094490
Primes p such that 2^j+p^j are primes for j=0,2,4,64.
Original entry on oeis.org
37, 1423, 8537, 61333, 397963, 419927, 699217, 1151603, 1156823, 1210793, 1746923, 1809163, 1915477, 2012113, 2713127, 3617683, 4001567, 4192033, 4760117, 4768133, 5099623, 5432153, 5801737, 5909737, 5924833, 6118157
Offset: 1
For j=0 1+1=2 is prime; other conditions are:
because of p^2+4==prime; 3rd and 4th conditions are as
follows: prime=p^4+16 and prime=2^64+p^64.
-
{ta=Table[0, {100}], u=1}; Do[s0=2;s2=4+Prime[j]^2;s4=16+Prime[j]^4;s64=2^64+Prime[j]^64 If[PrimeQ[s0]&&PrimeQ[s2]&&PrimeQ[s4]&&PrimeQ[s64], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[500000]],AllTrue[Table[2^j+#^j,{j,{0,2,4,64}}], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 29 2015 *)
A094492
Primes p such that 2^j+p^j are primes for j=0,1,4,16.
Original entry on oeis.org
179, 461, 521, 1877, 4259, 9767, 30389, 33071, 33329, 93701, 120077, 124247, 145547, 163481, 181871, 245627, 344171, 345731, 487427, 492671, 522281, 598187, 700199, 709739, 736061, 769259, 833717, 955709, 966869, 1009649, 1030739
Offset: 1
For j=0 1+1=2 is prime; other conditions are:
because of p^1+2=prime; 3rd and 4th conditions are as
follows: prime=p^4+16 and prime=65536+p^16.
-
{ta=Table[0, {100}], u=1}; Do[s0=2;s1=2+Prime[j]^1;s8=16+Prime[j]^4;s16=65536+Prime[j]^16 If[PrimeQ[s0]&&PrimeQ[s4]&&PrimeQ[s8]&&PrimeQ[s128], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
With[{j={0,1,4,16}},Select[Prime[Range[81000]],And@@PrimeQ[2^j+#^j]&]] (* Harvey P. Dale, Oct 17 2011 *)
A094316
Primes p for which 2^j+p^j is also prime for j in {0,2,8,512}.
Original entry on oeis.org
13, 4133, 1831343, 2320583, 3828673, 9173893, 23658377, 24037537, 42489677, 56253203, 78222863, 96325093, 99846337, 110453773, 110468653, 117748427, 122173187, 130937467, 138072163, 146981537, 174978913, 184050553, 186927817
Offset: 1
Smallest such prime is 13 and the relevant four primes are
2, 173, 815730977 and a 571-digit prime.
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{ta=Table[0, {100}], u=1}; {exponents, {a, b, c, d}={0, 2, 8, 512}} Do[s0=Prime[j]^a+2^a;s1=Prime[j]^b+2^b;s2=Prime[j]^c+2^c;s3=Prime[j]^d+2^d; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}] ta
A094478
Primes of form 2^j + 59^j.
Original entry on oeis.org
2, 61, 12117377, 464798130469793589516643498190087912509935907401081390977
Offset: 1
j=0: p=1+1=2;
j=1: p=2+59=61;
j=4: p=16+12117361=12117377;
j=32: p=2^32+59^32=464798130469793589516643498190087912509935907401081390977;
the j exponents are powers of 2.
A094483
Primes of form 2^j + 179^j.
Original entry on oeis.org
2, 181, 1026625697, 1110832290554380967776058484990830657
Offset: 1
-
Select[Table[2^j+179^j, {j,0,30}], PrimeQ] (* Harvey P. Dale, Apr 24 2013 *)
A094484
Primes of form 2^j + 461^j.
Original entry on oeis.org
2, 463, 45165175457, 4161163747708008324368372925882377717624897
Offset: 1
The relevant exponents are powers of 2: 0, 1, 4, 16; a(4) = 65536+461^16 = 4161163747708008324368372925882377717624897.
A094493
Primes p such that 2^j+p^j are primes for j=0,1,2,16.
Original entry on oeis.org
43577, 84317, 93887, 108377, 124247, 346667, 379997, 431867, 461297, 579197, 681257, 819317, 863867, 889037, 1143047, 1146797, 1271027, 1306817, 1518707, 1775867, 1926647, 1948517, 2119937, 2177447, 2348807, 2491607, 2604557
Offset: 1
For j=0: 1+1=2 is prime; other conditions are:
because of p^1+2=prime; 3rd and 4th conditions are as
follows: prime=p^2+4 and prime=65536+p^16.
-
{ta=Table[0, {100}], u=1}; Do[s0=2;s1=2+Prime[j]^1;s2=4+Prime[j]^2;s16=65536+Prime[j]^16 If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s16], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[2*10^5]],AllTrue[Table[2^k+#^k,{k,{0,1,2,16}}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 05 2021 *)
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