A085324 -id:A085324 - cp's OEIS frontend

A085325 a(n) is the least number m such that the minimal exponent for which reverse(m^n) = prime holds is n. Thus reverse(m^k) is composite for k = 1, .., n-1.

2, 4, 52, 61, 43, 49, 29, 8, 223, 53, 83, 59, 25, 568, 47, 221, 229, 1286, 427, 629, 637, 46, 109, 652, 458, 925, 1438, 86, 674, 535, 574, 314, 623, 173, 236, 676, 689, 205, 67, 419, 161, 976, 634, 818, 2104, 304, 26, 2392, 5012, 767, 238, 1769, 185, 3148, 3554

Offset: 1

Labos Elemer, Jul 02 2003

			For n = 10, a(10) = 53: This means that reverse(53^10) = 940315563074788471 is prime, but reverse(53^k) is composite for k=1, ..., 9. Also, reverse(m^10) for m < a(10) = 53 is not prime. However m > 53 is possible like, e.g., reverse(103^10) is prime. 10 as the least exponent belongs to several bases of which a(10) = 53 is the smallest one.

Amiram Eldar, Table of n, a(n) for n = 1..245

Cf. A004086 (reverse), A085324.

q[m_, n_] := AllTrue[Range[n - 1], CompositeQ[IntegerReverse[m^#]] &] && PrimeQ[IntegerReverse[m^n]]; a[n_] := Module[{m = 2}, While[! q[m, n], m++]; m]; Array[a, 30] (* Amiram Eldar, Feb 11 2025 *)

More terms from Amiram Eldar, Feb 11 2025

A058998 Least exponent k for which n^k reversed (leading zeros are not allowed) is a prime, or 0 if impossible.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 8, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 0, 8, 0, 13, 47, 0, 2, 7, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 5, 0, 0, 22, 15, 0, 6, 0, 0, 3, 10, 0, 0, 143, 0, 88, 12, 0, 4, 2, 0, 4, 8, 0, 39, 83, 0, 0, 1, 0, 1, 1, 0

Offset: 1

Robert G. Wilson v, Jan 17 2001

There are two different versions of this sequence: A085324 and this sequence which agrees with A085324 on the first 19 terms, but differs at a(20).

			a(4) is 2, because 4^2 is 16, and 16 reversed is 61 which is prime.

Robert Israel, Table of n, a(n) for n = 1..168

Cf. A085324.

Rev:= proc(n) local L;
L:= convert(n,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f:= proc(n) local k;
  if igcd(n,33) <> 1 or (n/10)::integer then return 0 fi;
  for k from 1 do if isprime(Rev(n^k)) then return k fi od:
end proc:
f(1):= 0: f(3):= 1: f(11):= 1:
map(f, [$1..168]); # Robert Israel, Apr 08 2018

Do[ If[ Mod[ n, 3 ] != 0 && Mod[ n, 10 ] != 0 && Mod[ n, 11 ] != 0, k = 1; While[ !PrimeQ[ ToExpression[ StringReverse[ ToString[ n^k ] ] ] ], k++ ]; Print[ k ], Print[ 0 ] ], {n, 2, 75} ]

a(n*10^k) = 0 for all k > 0 since definition does not allow leading 0's.

A085326 a(n)=p is smallest prime such that rev(p)=n^j with some exponent, or 0 if no such prime exists [when e.g. n=1,n=3k or n=11k, k>1].

Original entry on oeis.org

0, 2, 3, 61, 5, 0, 7, 61277761, 0, 0, 11, 0, 31, 41, 0, 61, 71, 0, 163, 2, 0, 0, 18258901387, 0, 5265674839116110941, 6716872795737314976899264656807717363719079328404119318887571869813, 0

Offset: 1

Labos Elemer, Jul 03 2003

			n=86: a(86)=6505868216024313214870917495263873755562243530151045641,
and rev[86]=86^28.

Cf. A085298, A085299, A085300, A085324, A085325.

cp's OEIS Frontend

A085325 a(n) is the least number m such that the minimal exponent for which reverse(m^n) = prime holds is n. Thus reverse(m^k) is composite for k = 1, .., n-1.

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A058998 Least exponent k for which n^k reversed (leading zeros are not allowed) is a prime, or 0 if impossible.

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A085326 a(n)=p is smallest prime such that rev(p)=n^j with some exponent, or 0 if no such prime exists [when e.g. n=1,n=3k or n=11k, k>1].

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