A052033 -id:A052033 - cp's OEIS frontend

A236437 Primes which occur in their proper place in A236174.

2, 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, 1733, 1759, 1811, 1823, 1949, 1973, 1997

Offset: 1

N. J. A. Sloane, Jan 25 2014

Primes p such that A236174(k) = prime(k) for some k. The values of k are (essentially) given in A235377.

Same as A052033 if the initial 2 is omitted.

			263 is the 56th prime and is also the 56th term in A236174.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A052033, A236174, A235377, A235354.

from sympy import prime, isprime
def A236174(n):
    p = prime(n)
    for b in range(2,11):
        x, y, z = p, 0, 1
        while x >= b:
            x, r = divmod(x,b)
            y += r*z
            z *= 10
        y += x*z
        if isprime(y):
            return y
A236437_list = [prime(n) for n in range(1,10**6) if A236174(n) == prime(n)]
# Chai Wah Wu, Jan 03 2015

A235377 Positions of 10s in A235354.

Original entry on oeis.org

56, 57, 69, 78, 83, 89, 92, 95, 104, 109, 111, 123, 124, 128, 136, 139, 142, 158, 160, 171, 173, 176, 177, 178, 180, 185, 194, 200, 201, 203, 211, 214, 219, 222, 223, 228, 231, 236, 237, 241, 245, 246, 248, 256, 259, 270, 274, 280, 281, 296, 298, 302, 307, 314

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2014

If prime(a(n)) is written in base k>=2, and the k-representation is read in decimal, then all such numbers, for k = 2,3,...,9, are composite.

Cf. A052033, A235354, A235376, A236174, A236437.

isok(n) = {my(p = prime(n)); for (b = 2, 9, if (isprime(subst(Pol(digits(p, b)), x, 10)), return(0));); return (1);} \\ Michel Marcus, Jan 18 2014

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A236437 Primes which occur in their proper place in A236174.

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A235377 Positions of 10s in A235354.

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PARI

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

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Mathematica

PARI

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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Mathematica

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

Mathematica

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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