A052026 -id:A052026 - cp's OEIS frontend

A052030 Primes base 10 that remain primes in four bases b, 2<=b<=10, expansions interpreted as decimal numbers.

19, 23, 37, 67, 79, 103, 127, 191, 193, 211, 229, 277, 311, 313, 337, 379, 409, 433, 443, 577, 613, 619, 631, 643, 647, 653, 787, 857, 883, 907, 919, 947, 997, 1021, 1039, 1087, 1097, 1123, 1171, 1279, 1423, 1429, 1447, 1459, 1471, 1567, 1597, 1669, 1693

Offset: 1

Patrick De Geest, Dec 15 1999

			19 is 103_4, 31_6, 23_8 and 19_10.

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@ 265], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 4 &] (* Michael De Vlieger, Mar 20 2015 after Harvey P. Dale at A052032 *)

A052031 Primes base 10 that remain primes in three bases b, 2<=b<=10, expansions interpreted as decimal numbers.

Original entry on oeis.org

11, 13, 17, 29, 31, 47, 59, 61, 83, 89, 97, 101, 109, 149, 151, 179, 181, 197, 227, 241, 251, 281, 331, 349, 353, 359, 373, 383, 419, 421, 439, 449, 457, 487, 503, 541, 547, 563, 587, 601, 617, 659, 673, 709, 727, 733, 743, 751, 773, 811, 823, 877, 953, 967

Offset: 1

Patrick De Geest, Dec 15 1999

			11 is 23_4, 13_8 and 11_10.

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@ 165], Count[PrimeQ /@ Table[FromDigits[IntegerDigits[#, i]], {i, 2, 10}], True] == 3 &] (* Michael De Vlieger, Mar 20 2015 after Harvey P. Dale at A052032 *)

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

A319858 a(n) is the number of values of m in the interval [2,10] such that the base-m expansion of n, interpreted as a base-10 number, yields a prime.

Original entry on oeis.org

0, 8, 8, 1, 7, 1, 5, 2, 2, 3, 3, 1, 3, 0, 2, 2, 3, 0, 4, 0, 2, 2, 4, 0, 2, 1, 1, 2, 3, 0, 3, 0, 3, 2, 1, 0, 4, 1, 3, 0, 2, 0, 5, 0, 1, 1, 3, 1, 1, 1, 0, 1, 2, 0, 4, 1, 1, 1, 3, 0, 3, 0, 1, 1, 1, 1, 4, 1, 0, 0, 5, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 1, 3, 1, 1, 0, 2, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 0, 0, 0

Offset: 1

Anton Deynega, Sep 29 2018

			a(31)=3 because 31 yields primes for 3 bases in [2,10]:  31 = 11111_2 = 1011_3 = 133_4 = 111_5 = 51_6 = 43_7 = 37_8 = 34_9 = 31_10, and of the decimal numbers 11111, 1011, 133, 111, 51, 43, 37, 34, and 31, the 3 primes are 43, 37, and 31.

Cf. A052026 (where a(n) is 0).

Array[Count[FromDigits /@ IntegerDigits[#, Range[2, 10]], ?PrimeQ] &, 105] (* _Michael De Vlieger, Oct 11 2018 *)

a(n) = sum(b=2, 10, isprime(fromdigits(digits(n, b), 10))); \\ Michel Marcus, Sep 30 2018

A386929 a(n) is the least base b in {2,...,10} such that the base-b expansion of n, when read as a decimal integer, is prime; a(n) = 0 if no such base exists.

Original entry on oeis.org

0, 3, 2, 3, 2, 5, 4, 5, 6, 3, 4, 9, 4, 0, 6, 5, 4, 0, 4, 0, 5, 3, 2, 0, 6, 5, 6, 5, 4, 0, 7, 0, 5, 3, 8, 0, 4, 7, 6, 0, 5, 0, 4, 0, 6, 3, 2, 9, 8, 7, 0, 7, 4, 0, 4, 5, 8, 3, 7, 0, 4, 0, 5, 9, 8, 9, 3, 5, 0, 0, 4, 0, 4, 0, 8, 0, 4, 0, 3, 0, 5, 9, 4, 9, 7, 0, 6, 9, 2, 0, 4, 0, 6, 3, 8

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Aug 08 2025

There are infinitely many zeros since if n is a multiple of 2520, then each base-b expansion ends with digit 0.

			a(10) = 3 since 10 in base 3 is "101" and 101 is prime; base 2 is "1010" -> 1010 composite.
a(11) = 4 since base 4 gives "23" -> 23 is prime; base 2 "1011" -> 1011 composite; base 3 "102" -> 102 composite.
a(23) = 2 since base 2 gives "10111" -> 10111 is prime.

Cf. A038537, A052026 (the zeros), A052033 (the tens).

a[n_] := Block[{m}, Do[m = FromDigits[IntegerDigits[n, b], 10]; If[PrimeQ[m], Return[b]], {b, 2, 10}]; 0]

a(n) = for(b=2, 10, if (isprime(fromdigits(digits(n, b))), return(b))); \\ Michel Marcus, Aug 09 2025

a(2520*n) = 0.

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A052030 Primes base 10 that remain primes in four bases b, 2<=b<=10, expansions interpreted as decimal numbers.

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A052031 Primes base 10 that remain primes in three 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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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

A319858 a(n) is the number of values of m in the interval [2,10] such that the base-m expansion of n, interpreted as a base-10 number, yields a prime.

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A386929 a(n) is the least base b in {2,...,10} such that the base-b expansion of n, when read as a decimal integer, is prime; a(n) = 0 if no such base exists.

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