A052026 -id:A052026 - cp's OEIS frontend

A052033 Primes base 10 that are never primes in any smaller base b, 2<=b<10, expansions interpreted as decimal numbers.

263, 269, 347, 397, 431, 461, 479, 499, 569, 599, 607, 677, 683, 719, 769, 797, 821, 929, 941, 1019, 1031, 1049, 1051, 1061, 1069, 1103, 1181, 1223, 1229, 1237, 1297, 1307, 1367, 1399, 1409, 1439, 1453, 1487, 1489, 1523, 1553, 1559, 1571, 1619, 1637

Offset: 1

Patrick De Geest, Dec 15 1999

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

Cf. A038537, A052026-A052033, A236174, A236437, A235354, A235377.

Select[Prime[Range@ 265], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 9}], True] == 0 &] (* Michael De Vlieger, Mar 20 2015 after Harvey P. Dale at A052032 *)

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

A236356 a(n) is the concatenation of the numbers k, 2 <= k <= 9, such that the base-k representation of n, read as a decimal number, is prime; a(n) = 0 if there is no such base.

Original entry on oeis.org

0, 3456789, 2456789, 3, 246789, 5, 4689, 57, 68, 379, 48, 9, 45, 0, 68, 59, 47, 0, 468, 0, 59, 37, 245, 0, 68, 5, 6, 59, 47, 0, 78, 0, 568, 39, 8, 0, 469, 7, 689, 0, 5, 0, 4789, 0, 6, 3, 24, 9, 8, 7, 0, 7, 4, 0, 4689, 5, 8, 3, 78, 0, 49, 0, 5, 9, 8, 9, 368, 5

Offset: 1

Vladimir Shevelev, Jan 23 2014

Composite numbers n for which a(n)=0 we call absolute composite numbers.
Almost evidently that almost all integers are absolute composite numbers. Moreover, since the number of primes<=x containing no at least one digit is o(pi(x)), then, for almost all positions of prime n, a(n)=0. It is interesting to obtain an upper estimate of number of nonzero positions in the sequence, more exactly, than o(x/log(x)).
Only O(sqrt x) numbers up to x have nonzero values in this sequence. - Charles R Greathouse IV, Jan 24 2014

			Let n = 29. In bases 2, 3, ..., 9 the representations of 29 are 11101_2, 1002_3, 131_4, 104_5, 45_6, 41_7, 35_8, 32_9. In this list only 131_4 and 41_7 are primes, so a(29) = 47.
The sequence of numbers whose representations in bases 4 and 7, read in decimal, are primes are the numbers n such that a(n) contains the digits 4 and 7: 2, 3, 5, 17, 29, 43, ....

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A052026.

from sympy import isprime
from sympy.ntheory import digits
def c(n, b): return isprime(int("".join(map(str, digits(n, b)[1:]))))
def a(n): return int("0"+"".join(k for k in "23456789" if c(n, int(k))))
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Sep 22 2022

Name clarified by Jon E. Schoenfield, Sep 21 2022

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.

A052027 Primes in base 10 that remain primes in seven bases b, 2<=b<=10, expansions interpreted as decimal numbers.

Original entry on oeis.org

5, 9241, 17791, 330289, 391231, 1005481, 1210483, 2378143, 2469241, 2779939, 2840041, 6817501, 8320831, 9865711, 10871407, 11087191, 12259603, 13645393, 15665833, 16707883, 17694463, 25751863, 27794287, 31488481

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..1000
Carlos Rivera, PP&P Puzzle 24: Primes in several bases

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

Select[Prime[Range@ 100000], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 7 &] (* Michael De Vlieger, Mar 21 2015, after Harvey P. Dale at A052032 *)

Missing terms 2378143 and 2469241 added by Sebastian Petzelberger, Mar 21 2015

A052032 Primes base 10 that remain prime in one (and only one) other base b, 2<=b<10, expansions interpreted as decimal numbers.

Original entry on oeis.org

41, 53, 73, 107, 113, 131, 137, 139, 167, 173, 223, 233, 239, 257, 271, 293, 317, 389, 401, 467, 491, 509, 521, 557, 593, 641, 661, 691, 701, 739, 761, 809, 827, 829, 839, 853, 859, 863, 881, 887, 911, 937, 971, 977, 991, 1013, 1063, 1109, 1129, 1151, 1153

Offset: 1

Patrick De Geest, Dec 15 1999

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000 [This replaces an earlier b-file computed by Harvey P. Dale]
C. Rivera, PP&P Puzzle 24: Primes in several bases

Cf. A038537, A052026-A052033.

Select[Prime[Range[200]],Count[PrimeQ/@Table[FromDigits[ IntegerDigits[ #,i]],{i,2,9}],True]==1&] (* Harvey P. Dale, Oct 13 2012 *)

Definition clarified by Harvey P. Dale, Oct 13 2012

A052028 Primes base 10 that remain primes in six bases b, 2<=b<=10, expansions interpreted as decimal numbers.

Original entry on oeis.org

157, 523, 1249, 1483, 1753, 4051, 9187, 10531, 22921, 25981, 29599, 35899, 51031, 57751, 67579, 79939, 98323, 103561, 110581, 148471, 150193, 150343, 249703, 259183, 277063, 278623, 331081, 335833, 353401, 391903, 424819, 435553, 504547

Offset: 1

Patrick De Geest, Dec 15 1999

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

Cf. A038537, A052026-A052033.

Select[Prime[Range@ 42000], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 6 &] (* Michael De Vlieger, Mar 20 2015, after Harvey P. Dale at A052032 *)

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.

A052029 Primes base 10 that remain primes in five bases b, 2<=b<=10, expansions interpreted as decimal numbers.

Original entry on oeis.org

7, 43, 71, 163, 199, 283, 307, 367, 463, 571, 757, 1033, 1163, 1627, 1873, 2683, 3041, 3691, 3967, 4483, 4651, 4729, 4951, 4973, 5407, 6073, 6961, 7351, 7537, 8053, 8599, 9103, 9817, 10321, 10831, 11251, 11383, 11743, 12433, 12853, 13219, 14419, 14479

Offset: 1

Patrick De Geest, Dec 15 1999

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

Cf. A038537, A052026-A052033.

Select[Prime[Range@ 1800], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 5 &] (* Michael De Vlieger, Mar 20 2015, after Harvey P. Dale at A052032 *)

lista(nn, nb=5) = {forprime(p=2, nn, if (sum(b=2, 10, isprime(subst(Pol(digits(p, b)), x, 10))) == nb, print1(p, ", ")););} \\ Michel Marcus, Mar 21 2015

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A052033 Primes base 10 that are never primes in any smaller base 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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A236356 a(n) is the concatenation of the numbers k, 2 <= k <= 9, such that the base-k representation of n, read as a decimal number, is prime; a(n) = 0 if there is no such base.

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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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A052027 Primes in base 10 that remain primes in seven bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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A052032 Primes base 10 that remain prime in one (and only one) other base b, 2<=b<10, expansions interpreted as decimal numbers.

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Mathematica

Extensions

A052028 Primes base 10 that remain primes in six bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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Mathematica

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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A052029 Primes base 10 that remain primes in five bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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