A052027 -id:A052027 - cp's OEIS frontend

A052026 Composites base 10 that remain composite in all bases b, 2<=b<=10, expansions interpreted as decimal numbers.

14, 18, 20, 24, 30, 32, 36, 40, 42, 44, 51, 54, 60, 62, 69, 70, 72, 74, 76, 78, 80, 86, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 114, 120, 124, 125, 126, 128, 129, 130, 132, 135, 140, 144, 146, 148, 150, 152, 156, 158, 159, 160, 162, 164, 168, 170, 174

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 24. Primes in several bases, The Prime Puzzles & Problems Connection.

Cf. A038537, A052027-A052033, A084482, A236356

Select[Range@ 174, AllTrue[Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], CompositeQ] &] (* Michael De Vlieger, Mar 24 2015, version 10 *)

A038537 Primes base 10 that remain primes in eight bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

Original entry on oeis.org

2, 3, 379081, 59771671, 146752831, 764479423, 1479830551, 3406187401, 5631714889, 7740024337, 8256310441, 8772257161, 9522879913, 10350894331, 12852250993, 14261996563, 16082349433, 16199980009, 17727606151, 18172964503, 18294784903, 19393314433, 19472325391, 20582035993

Offset: 1

Carlos Rivera

Sebastian Petzelberger, Table of n, a(n) for n = 1..54 (terms up to 10^12).
Carlos Rivera, Puzzle 24: Primes in several bases, The Prime Puzzles & Problems Connection.

Cf. A052026, A038537, A052027-A052033, A084482, A236356

Select[Prime[Range@ 50000], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 8 &] (* Michael De Vlieger, Mar 20 2015, after Harvey P. Dale at A052032 *)

a(4)-a(7) found by Jack Brennen (see link) added by Patrick De Geest, Dec 15 1999

Terms beyond a(7) from Sebastian Petzelberger, Mar 21 2015

A084482 Primes base 10 that remain primes in all nine bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

Original entry on oeis.org

50006393431, 727533146383, 2250332130313, 2651541199513, 4437592255351, 4877749016143, 6777899690983, 7417899095713, 7431376081543, 7766799025303, 9078654198463, 10712216924641, 12244626455491, 13562282568103, 14180813918071, 14833027106593, 19479075240913, 19971686697103, 23196986067193, 34431442237963, 36429184518721, 49198998504223

Offset: 1

Jack Brennen, Jun 29 2003

a(1) found by Jack Brennen on Jul 13 2001; remaining terms computed by Jack Brennen, Nov 15 2001.

The number must end with 1, 3, 7, or 9 in each base from 2 to 10; thus must be congruent to: 1 (mod 2), 1 (mod 3), 1 or 3 (mod 4), 1 or 3 (mod 5), 1 (mod 6), 1 or 3 (mod 7), 1 or 3 or 7 (mod 8), 1 or 7 (mod 9), 1 or 3 or 7 or 9 (mod 10).

Carlos Rivera, PP and P Puzzle 24: Primes in several bases

Cf. A052026, A038537, A052027-A052033, A084482, A236356.

isok(n) = sum(b=2, 10, isprime(subst(Pol(digits(n, b)), x, 10))) == 9; \\ Michel Marcus, Mar 22 2015

Thanks to David W. Wilson for proposing the sequence and W. Edwin Clark for verifying the terms using Maple's command isprime.

A256350 Composites in base 10 that remain composite in exactly eight bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

4, 6, 12, 26, 27, 35, 38, 45, 46, 48, 49, 50, 52, 56, 57, 58, 63, 64, 65, 66, 68, 77, 81, 82, 84, 85, 88, 95, 105, 116, 117, 118, 119, 121, 122, 134, 136, 138, 142, 153, 154, 161, 165, 166, 171, 175, 176, 187, 188, 190, 192, 195, 207, 208, 218, 219, 220, 225

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases, The Prime Puzzles & Problems Connection.

Cf. A052026, A256350-A256356, A038537, A052027-A052033, A084482, A236356.

A256356 Composites in base 10 that remain composite in exactly two bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

33247243, 64037779, 104865433, 130237003, 238561081, 550677781, 947051353, 1013991553, 1246382791, 1343122201, 1607697631, 1609062751, 1632753601, 1788658063, 2203645111, 2364166213, 2393866411, 2480419783, 2518589671, 2544177511, 2668538575, 3029334883

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Are there any remaining composites in only one other base?

Sebastian Petzelberger, Table of n, a(n) for n = 1..150
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350, A256351, A256352, A256353, A256354, A256355.

Cf. A038537, A052027-A052033, A084482, A236356.

A256351 Composites in base 10 that remain composite in exactly seven bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

8, 9, 15, 16, 21, 22, 25, 28, 34, 75, 87, 91, 93, 94, 106, 111, 123, 141, 143, 145, 147, 155, 172, 201, 205, 214, 217, 237, 255, 298, 304, 305, 363, 371, 376, 377, 385, 388, 395, 403, 411, 423, 428, 442, 458, 466, 471, 473, 483, 495, 501, 505, 507, 531, 533

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350 - A256356.
Cf. A038537, A052027, A052028, A052029, A052030, A052031, A052032, A084482.
Cf. A052033, A236356.

f:= proc(b,x) local L,i;
L:= convert(x,base,b);
isprime(add(10^(i-1)*L[i],i=1..nops(L)))
end proc:
select(t -> not isprime(t) and nops(select(f,[$2..9],t))=2, [$1..1000]); # Robert Israel, Mar 26 2015

fQ[n_] := CompositeQ@ n && Count[ CompositeQ[ FromDigits[ IntegerDigits[n, #]] & /@ Range[2, 9]], True] == 6; Select[ Range@ 500, fQ] (* Robert G. Wilson v, Mar 26 2015 *)

A256352 Composites in base 10 that remain composite in exactly six bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

10, 33, 39, 133, 183, 185, 203, 235, 291, 295, 303, 325, 343, 381, 391, 451, 475, 517, 535, 539, 561, 583, 655, 703, 723, 753, 775, 791, 799, 841, 867, 889, 895, 943, 1003, 1023, 1083, 1099, 1121, 1159, 1165, 1173, 1186, 1198, 1207, 1219, 1263, 1333, 1366

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350-A256356, A038537, A052027-A052033, A084482, A236356.

A256353 Composites in base 10 that remain composite in exactly five bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

55, 169, 247, 253, 323, 493, 529, 556, 671, 1027, 1111, 1243, 1261, 1339, 1375, 1711, 1751, 1803, 2185, 2413, 2431, 2881, 3193, 4381, 4417, 4843, 5029, 5203, 5251, 6631, 7093, 7999, 8515, 8653, 9271, 9307, 9481, 9523, 9593, 9727, 9745, 9937, 9955, 10393, 10555

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Less remaining is not possible for even numbers.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350-A256356, A038537, A052027-A052033, A084482, A236356.

A256354 Composites in base 10 that remain composite in exactly four bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

115, 2563, 3523, 5071, 9193, 10873, 12223, 12811, 13231, 15775, 19111, 20203, 23089, 25831, 27007, 28171, 34189, 39859, 40033, 43361, 55033, 57871, 58813, 74371, 84253, 89377, 93043, 95833, 101683, 117001, 125359, 126673, 128953, 131029, 134527, 137467, 138193

Offset: 1

Sebastian Petzelberger, Mar 25 2015

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350-A256356, A038537, A052027-A052033, A084482, A236356.

A256355 Composites in base 10 that remain composite in exactly three bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

Original entry on oeis.org

11233, 42241, 98281, 131239, 161953, 315151, 358135, 606553, 692263, 785851, 1114081, 1130419, 1525777, 1906363, 3369313, 3403081, 3880873, 5616721, 6036103, 6947611, 7253191, 7516783, 7886593, 8799127, 8811223, 9108289, 9113203, 9195313, 9450361, 9600769

Offset: 1

Sebastian Petzelberger, Mar 25 2015

			11233 = 324413_5 and 324413_10 is composite; 11233 = 44515_7 and 44515_10 is composite; 11233_10 itself is composite. Interpreted in base 2, 3, 4, 6, 8, and 9 the result is prime. Hence 11233 is in this sequence.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A052026, A256350, A256351, A256352, A256353, A256354, A256356, A038537, A052027, A052028, A052029, A052030, A052031, A052032, A052033, A084482, A236356.

is(n)=!isprime(n) && sum(b=2,9,isprime(fromdigits(digits(n,b))))==6 \\ Charles R Greathouse IV, Apr 23 2015

cp's OEIS Frontend

A052026 Composites base 10 that remain composite in all bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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A038537 Primes base 10 that remain primes in eight bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

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A084482 Primes base 10 that remain primes in all nine bases b, 2<=b<=10, when the expansions are interpreted as decimal numbers.

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A256350 Composites in base 10 that remain composite in exactly eight bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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A256356 Composites in base 10 that remain composite in exactly two bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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A256351 Composites in base 10 that remain composite in exactly seven bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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Maple

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A256352 Composites in base 10 that remain composite in exactly six bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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A256353 Composites in base 10 that remain composite in exactly five bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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A256354 Composites in base 10 that remain composite in exactly four bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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A256355 Composites in base 10 that remain composite in exactly three bases b, 2 <= b <= 10, expansions interpreted as decimal numbers.

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PARI