cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A175420 Sequence of numbers after 1st step of iteration defined in A175419.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 0, 1, 64, 729, 4096, 15625, 46656
Offset: 0

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Examples

			For n = 33: a(33) = 27 because for the number 33 there are 4 steps of defined iteration: {3^3 = 27}, {7^2 = 49}, {9^4 = 6561}, {((1^6)^5)^6 = 1} and the 1st step of the iteration ends with 27.

Links

Crossrefs

Cf. A175419, A175421, A175422, A175423, A175424, A175425, A175426, A175427.
Cf. A055642.

Programs

  • Maple
    A175420 := proc(n) local dgs,a,i ; if n = 0 then return 0 ; end if; dgs := convert(n,base,10) ; a := op(1,dgs) ; for i from 2 to nops(dgs) do a := a^ op(i,dgs) ; end do: a ; end proc: seq(A175420(n),n=0..120) ; # R. J. Mathar, May 12 2010
  • Mathematica
    Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; a[0]=0; a[n_]:=If[(len=IntegerLength[n])==1, n, Last[list=IntegerDigits[n]]^Product[Part[Drop[list, -1], i], {i, len-1}]]; Array[a, 67, 0] (* Stefano Spezia, Feb 25 2024 *)
  • PARI
    a(n) = if (n, my(d=digits(n), r=d[#d]); forstep (k=#d-1, 1, -1, r = r^d[k];); r); \\ Michel Marcus, Jan 20 2022

Formula

a(n) = (((D_k^D_(k-1))^D_(k-2))^...)^D_1, where D_k = k-th digit D of number n and k = the number of digits of number n in decimal expansion of n (A055642).

Extensions

More terms from R. J. Mathar, May 12 2010

A175425 The smallest number m which needs n iterations of the map x->A175420(x) to reach a single-digit number.

Original entry on oeis.org

0, 10, 24, 27, 33
Offset: 0

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Comments

Conjecture: sequence a(n) is finite for n = natural number, a(n) = 25 for n = infinity. See A175424.

Examples

			a(4) = 33 because 33 is the smallest numbers with 4 steps of defined iteration: 3^3 = 27, 7^2 = 49, 9^4 = 6561, ((1^6)^5)^6 = 1.

Crossrefs

Cf. A175419 - A175427.

Formula

a(n) = min {j : A175425(j) = n}.

A175419 The single-digit number obtained by iterated mapping of r (starting with n) to a power-tower of its digits, or -1 if such a single-digit number is never reached.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, -1, -1, 1, 1, 1, 0, 1, 8, 1, 1, -1, -1, 1, 8, 1, 0, 1, 6, 1, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Comments

Define a map r->A175420(r) which takes the base-10 digits of r = Sum_{i>=0} d_i*10^i and assigns the power-tower ((d_0^d_1)^d_2)^d^3... to the result. There are A055642(r)-1 exponentiations in this expression. Single-digit numbers are fixed points of the map.
Starting with n, this map is iterated as often as needed to result in a single-digit number, which becomes a(n). In case the iteration does not reach a single-digit number (i.e., enters cycles with only multi-digit numbers), a(n)= -1.
The entries 1 to 9 appear infinitely often in the sequence.
The entry -1 appears infinitely often in the sequence, see A175426.
After 1 to 4 iterations we reach sequences A175420 to A175423.

Examples

			For n = 33: a(33) = 1 because starting with 33 we reach a single-digit 1 after 4 iterations: 3^3 = 27, 7^2 = 49, 9^4 = 6561, ((1^6)^5)^6 = 1.
For n = 25: a(25) = -1 because starting with 25 the iteration enters a loop of 2-digit numbers: 5^2 = 25, 5^2 = 25, ...

Crossrefs

Cf. A175420, A175421, A175422, A175423, A175424, A175425, A175426, A175427.

A175422 Sequence of numbers after 3rd step of iteration defined in A175419.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, 25, 36, 1, 1, 1, 0, 1, 8, 6561, 1, 25, 216, 1, 8, 1, 0, 1, 6, 1, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 1, 867361737988403547205962240695953369140625, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Examples

			For n = 33: a(33) = 6561 because for the number 33 there are 4 steps of defined iteration: {3^3 = 27}, {7^2 = 49}, {9^4 = 6561}, {((1^6)^5)^6 = 1} and the 3rd step of the iteration ends with 6561.

Crossrefs

Cf. A175419, A175420, A175421, A175423, A175424, A175425, A175426, A175427.

Programs

  • Maple
    A175420 := proc(n) option remember; local dgs,a,i ; if n = 0 then return 0 ; end if; dgs := convert(n,base,10) ; a := op(1,dgs) ; for i from 2 to nops(dgs) do a := a^ op(i,dgs) ; end do: a ; end proc:
A175421 := proc(n) A175420(A175420(n)) ; end proc:
A175422 := proc(n) A175420(A175421(n)) ; end proc:
for n from 0 do printf("%d,\n",A175422(n)) ; end do; # R. J. Mathar, May 12 2010
  • Mathematica
    Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; A175420[0]=0; A175420[n_]:=If[(len=IntegerLength[n])==1, n, Last[list=IntegerDigits[n]]^Product[Part[Drop[list, -1], i], {i, len-1}]]; a[n_]:=Nest[A175420[#]&,n,3]; Array[a, 70, 0] (* Stefano Spezia, Feb 25 2024 *)

Extensions

7 more terms from R. J. Mathar, May 12 2010

A175423 Sequence of numbers after 4th step of iteration defined in A175419.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, 25, 216, 1, 1, 1, 0, 1, 8, 1, 1, 25, 36, 1, 8, 1, 0, 1, 6, 1, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Examples

			For n = 33: a(33) = 1 because for the number 33 there are 4 steps of defined iteration: {3^3 = 27}, {7^2 = 49}, {9^4 = 6561}, {((1^6)^5)^6 = 1} and the 4th step of iteration ends with 1. - _Jaroslav Krizek_, May 09 2010

Crossrefs

Cf. A175419, A175420, A175421, A175422, A175424, A175425, A175426, A175427.

Programs

  • Maple
    A175420 := proc(n) option remember; local dgs,a,i ; if n = 0 then return 0 ; end if; dgs := convert(n,base,10) ; if op(1,dgs)= 0 then return 0 ; end if; for i from 1 to nops(dgs) do if op(-i,dgs) = 0 then dgs := [op(-i..nops(dgs),dgs)] ; dgs := subsop(1=1,dgs) ; break ; end if; end do: a := op(1,dgs) ; for i from 2 to nops(dgs) do a := a^ dgs[i] ; end do: a ; end proc:
A175421 := proc(n) A175420(A175420(n)) ; end proc:
A175422 := proc(n) A175420(A175421(n)) ; end proc:
A175423 := proc(n) A175420(A175422(n)) ; end proc:
seq(A175423(n),n=0..100) ; # R. J. Mathar, May 12 2010
  • Mathematica
    Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; A175420[0]=0; A175420[n_]:=If[(len=IntegerLength[n])==1, n, Last[list=IntegerDigits[n]]^Product[Part[Drop[list, -1], i], {i, len-1}]]; a[n_]:=Nest[A175420[#]&, n, 4]; Array[a, 101, 0]

Extensions

More terms from R. J. Mathar, May 12 2010
a(100) corrected by Stefano Spezia, Feb 25 2024

A175427 Starting values which reach a single-digit number after a finite number of iterations of the map x->A175429(x).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Comments

Arguments m such that A175424(m) >= 0. Also arguments m such that A175419(m) >= 0.
Complement of A175426.

Examples

			27 is in the sequence because a single-digit number is reached in 3 iterations: 7^2 = 49, 9^4 = 6561, ((1^6)^5)^6 = 1, so A175424(27) = 3.

Crossrefs

Cf. A175419, A175420, A175421, A175422, A175423, A175424, A175425, A175426.

A175421 Sequence of numbers after 2nd step of iteration defined in A175419.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, 25, 216, 6561, 4096, 1, 0, 1, 8, 49, 4096, 25, 36, 531441, 32, 22876792454961, 0, 1, 6, 1, 60466176, 244140625, 101559956668416, 1, 1, 1, 0
Offset: 0

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Examples

			For n = 33: a(33) = 49 because for the number 33 there are 4 steps of defined iteration: {3^3 = 27}, {7^2 = 49}, {9^4 = 6561}, {((1^6)^5)^6 = 1} and the 2nd step of the iteration ends with 49.

Crossrefs

Cf. A175419, A175420, A175422, A175423, A175424, A175425, A175426, A175427.

Programs

  • Mathematica
    Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; A175420[0]=0; A175420[n_]:=If[(len=IntegerLength[n])==1, n, Last[list=IntegerDigits[n]]^Product[Part[Drop[list, -1], i], {i, len-1}]]; a[n_]:=A175420[A175420[n]]; Array[a, 51, 0] (* Stefano Spezia, Feb 25 2024 *)

A175426 Starting values for which iteration of the map x->A175420(x) never reaches a single-digit value.

Original entry on oeis.org

25, 26, 35, 36, 125, 126, 135, 136, 215, 216, 315, 316, 1125, 1126, 1135, 1136, 1215, 1216, 1315, 1316
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Comments

Arguments m such that A175424(m) = -1. Complement of A175427.

Examples

			26 is in the sequence because starting with 26 the iterations enter a cycle of 2- and 3-digit numbers: 6^2 = 36, 6^3 = 216, (6^1)^2 = 36, 6^3 = 216, (6^1)^2 = 36, ...

Crossrefs

Cf. A175419, A175420, A175421, A175422, A175423, A175424, A175425, A175427.
Showing 1-8 of 8 results.

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