A051285 -id:A051285 - cp's OEIS frontend

A072651 Table by antidiagonals used in calculating integer solutions to b^c=c^d with b,c,d>0.

1, 1, 4, 1, 16, 27, 1, 0, 0, 16, 1, 65536, 7625597484987, 256, 3125, 1, 0, 0, 0, 0, 46656, 1, 0, 0, 4294967296, 0, 10314424798490535546171949056, 823543, 1, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jun 28 2002

There are also trivial values involving 0 to give 1; for any b and c: b^0=0^0 and 1^c=c^0. For b=p prime, the solutions occur when k is a power of p, say p^x and so c=p^(p^x), d=p^(p^(p^x)-k) and the value is p^(p^(p^x))). For p=2 this seems to give all but the first two terms of A051285.

			Rows start: 1, 1, 1, ...; 4, 16, 0, 65536, 0, 0, 0, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 0, ...; 27, 0, 7625597484987, 0, 0, ...; 16, 256, 0, 4294967296, 0, ... etc. with the nonzero values corresponding to 1^1=1^1, 1^1=1^2, 1^1=1^3; 2^2=2^2, 2^4=4^2, 2^16=16^4, 2^256=256^32; 3^3=3^3, 3^27=27^9; 4^2=2^4, 4^4=4^4, 4^16=16^8; etc. For b=10^9: r=10 and m=9 so there solutions with c=10^k and d=9*10^k/k providing k is in A070023, i.e. if 1/k has period 0 or 1 in base 10.

Cf. A072652, A072653.

T(n, k)=n^(A052410(n)^k) if A052409(n)*A052410(n)^k/k is an integer [and with T(1, k)=1] but otherwise T(n, k)=0 if A052409(n)*A052410(n)^k/k is not an integer. b=n=r^m where r=A052410(n) is the smallest root of n and m=A052409(n) is the power of r that n is; c=r^k; and d=m*r^k/k providing this is an integer [while if n=1 then b=1, c=1 and d=k]. In effect, there is a solution if the largest divisor of k which is coprime to all powers of r is also a divisor of m.

A264933 a(0)=0; a(1)=1; a(2)=2; a(n) = a(n-1)^a(n-2)^a(n-3).

Original entry on oeis.org

0, 1, 2, 2, 4, 256, 340282366920938463463374607431768211456

Offset: 0

Natan Arie Consigli, Dec 17 2015

nonn

			a(3) = a(2)^a(1)^a(0) = 2;
a(4) = a(3)^a(2)^a(1) = 2^2^1 = 4;
a(5) = a(4)^a(3)^a(2) = 4^2^2 = 256;
a(6) = a(5)^a(4)^a(3) = 256^4^2 = 340282366920938463463374607431768211456.

Cf. A051285.

I:=[0, 1, 2]; [n le 2 select I[n] else Self(n-1)^Self(n-2)^Self(n-3): n in [0..6]];

RecurrenceTable[{a[0]==0, a[1]==1, a[2]==2, a[n]==a[n-1]^a[n-2]^a[n-3]}, a, {n, 6}]

a(n) = if(n<3, n, a(n-1)^a(n-2)^a(n-3));

A267218 a(n) is the a(n-1)-st a(n-2)-dimensional orthoplex number, starting with the terms 1, 2.

Original entry on oeis.org

1, 2, 2, 4, 16, 22016

Offset: 1

Robin Powell, Jan 18 2016

nonn

			16 is the 4th 2-orthoplex number = A000290(4).
22016 is the 16th 4-orthoplex number = A014820(16).
The next term will be the 22016th 16-orthoplex number.

OEIS Wiki, Orthoplex numbers

Cf. A000290, A014820, A260161, A051285.

A271938 a(1) = 1, a(2) = 3, a(n) = a(n-1)^a(n-2).

Original entry on oeis.org

1, 3, 3, 27, 19683, 87189642485960958202911070585860771696964072404731750085525219437990967093723439943475549906831683116791055225665627

Offset: 1

Natan Arie Consigli, Apr 17 2016

nonn

Cf. A264932 with a(n-1)^^a(n-2) or a(n-1)^a(n-2)^a(n-2);

Cf. A051285 (with a(2) = 2).

a[1] = 1; a[2] = 3; a[n_] := a[n] = a[n - 1]^a[n - 2]; Array[a, {6}] (* Michael De Vlieger, Apr 17 2016 *)
nxt[{a_,b_}]:={b,b^a}; NestList[nxt,{1,3},5][[;;,1]] (* Harvey P. Dale, Aug 02 2025 *)

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A072651 Table by antidiagonals used in calculating integer solutions to b^c=c^d with b,c,d>0.

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A264933 a(0)=0; a(1)=1; a(2)=2; a(n) = a(n-1)^a(n-2)^a(n-3).

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A267218 a(n) is the a(n-1)-st a(n-2)-dimensional orthoplex number, starting with the terms 1, 2.

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A271938 a(1) = 1, a(2) = 3, a(n) = a(n-1)^a(n-2).

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