A075241 -id:A075241 - cp's OEIS frontend

A075242 Least base for which the n-th composite number whose reversal in that base is a prime, or zero if impossible.

0, 2, 4, 6, 2, 2, 2, 3, 8, 3, 2, 3, 2, 2, 2, 2, 9, 2, 6, 4, 3, 2, 3, 12, 6, 3, 2, 6, 2, 3, 2, 2, 3, 2, 9, 2, 3, 2, 2, 3, 2, 12, 2, 3, 12, 3, 6, 2, 3, 10, 6, 2, 3, 10, 2, 26, 2, 27, 2, 12, 3, 2, 9, 2, 12, 2, 2, 3, 2, 3, 2, 4, 3, 2, 34, 2, 3, 2, 6, 2, 3, 2, 38, 2, 2, 3, 4, 7, 24, 2, 2, 3, 2, 3, 18, 4, 18

Offset: 1

Robert G. Wilson v, Sep 09 2002

Question: Other than 4, is there a composite that cannot be made a prime by base reversal? I have found none < (10^5)-th composite.

			a(1) = 0 because 4 (2) = 1 and 4 (3) = 4 and any base greater than 3 always gives the composite 4 as its base reversal. a(3) = 4 because 8 (2) = 1, 8 (3) = 8 but 8 (4) = 2 a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A075241, A075241.

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n]; f[n_] := Block[{b = 2}, While[b < n && !PrimeQ[ FromDigits[ Reverse[ IntegerDigits[n, b]], b]], b++ ]; If[b != n, b, 0]]; Table[ f[ Composite[n]], {n, 1, 105}]

A075240 Smallest prime whose base-n reversal is not a prime.

Original entry on oeis.org

2, 3, 11, 5, 13, 7, 11, 29, 19, 11, 19, 13, 29, 23, 19, 17, 23, 19, 31, 29, 29, 23, 29, 37, 37, 29, 31, 29, 43, 31, 37, 37, 59, 59, 43, 37, 41, 59, 43, 41, 59, 43, 47, 47, 53, 47, 59, 61, 59, 61, 59, 53, 67, 59, 59, 59, 61, 59, 73, 61, 71, 67, 79, 71, 89, 67, 71, 101, 89

Offset: 2

Robert G. Wilson v, Sep 09 2002

All primes show up in the sequence eventually.

			11_10 = 23_4, reverse(23) = 32, 32_4 = 14_10, so a(4) = 11 since this does not work for smaller primes.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A075241, A075242.

a = {}; Do[k = 1; While[ PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[k], n]], n]], k++ ]; a = Append[a, Prime[k]], {n, 2, 70}]; a

a(n) = my(p=2); while (isprime(fromdigits(Vecrev(digits(p, n)), n)), p = nextprime(p+1)); p; \\ Michel Marcus, Mar 28 2021

A075244 Least number requiring the base n to produce a prime by base reversal.

Original entry on oeis.org

2, 3, 15, 8, 109, 9, 119, 16, 27, 70, 2197, 36, 1265, 158, 213, 178, 4205, 126, 14189, 260, 273, 304, 4865, 120, 1295, 78, 81, 532, 44323, 150, 47317, 952, 771, 102, 16705, 492, 6209, 114, 1209, 2020, 132743, 294, 22945, 2834, 2721, 2276, 66455, 144

Offset: 1

Robert G. Wilson v, Sep 09 2002

Question, Is every base necessary to convert the natural numbers into primes?

			a(1) = 2 because two = 11 in unary (A000042) and its reversal 11 = 2. a(2) = 3 because three = 11 in base 2 (A007088) and its reversal 11 in base 2 = 3. a(3) = 15 because fifteen = 120 in base 3 (A007089) and its reversal 21 in base 3 = 7. a(4) = 8 -> 2. a(7) = 119 because 119 base 7 = 230 in base 7 (A007093) and its reversal 32 base 7 = 161.

Cf. A075241 & A075242.

f[n_] := Block[{b = 2}, While[b < n && !PrimeQ[ FromDigits[ Reverse[ IntegerDigits[n, b]], b]], b++ ]; If[b != n, b, 0]]; a = Table[0, {70}]; Do[b = f[n]; If[b < 76 && a[[b]] == 0, a[[b]] = n], {n, 2, 133000}]

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A075242 Least base for which the n-th composite number whose reversal in that base is a prime, or zero if impossible.

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A075240 Smallest prime whose base-n reversal is not a prime.

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A075244 Least number requiring the base n to produce a prime by base reversal.

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