A113122 -id:A113122 - cp's OEIS frontend

A113173 Ascending descending base exponent transform of semiprimes (A001358).

256, 5392, 315361, 11667713, 717360537, 83932270482, 27775696582531, 22260761742531649, 109563850113131234720, 2013390472722146301196, 1899501614194512059559835, 85600281199526209989968735

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. a(7) is itself semiprime. The smallest primes in this sequence are a(3) = 315361 and a(4) = 11667713. What is the next prime?

			a(1) = 256 because semiprime(1)^semiprime(1) = 4^4 = 256.
a(2) = 5392 because prime(1)^prime(2) + prime(2)^prime(1) = 4^6 + 6^4 = 5392.
a(3) = 315361 because 4^9 + 6^6 + 9^4 = 315361.
a(4) = 11667713 = 4^10 + 6^9 + 9^6 + 10^4.
a(5) = 717360537 = 4^14 + 6^10 + 9^9 + 10^6 + 14^4.
a(6) = 83932270482 = 4^15 + 6^14 + 9^10 + 10^9 + 14^6 + 15^4.
a(7) = 27775696582531 = 4^21 + 6^15 + 9^14 + 10^10 + 14^9 + 15^6 + 21^4.
a(8) = 22260761742531649 = 4^22 + 6^21 + 9^15 + 10^14 + 14^10 + 15^9 + 21^6 + 22^4.
a(9) = 109563850113131234720 = 4^25 + 6^22 + 9^21 + 10^15 + 14^14 + 15^10 + 21^9 + 22^6 + 25^4.

G. C. Greubel, Table of n, a(n) for n = 1..100

Cf. A001358, A005408, A113122, A113153, A113154.

A001358[A001358%5Bk%5D%5B%5Bk%5D%5D)%5E((A001358%5Bn%20-%20k%20+%201%5D%5B%5Bn%20-%20k%20+%201%5D%5D)),%20%7Bk,%201,%20n%7D%5D,%20%7Bn,%201,%2010%7D%5D%20(*%20_G.%20C.%20Greubel">] := Select[Range[100], PrimeOmega[#] == 2 &]; Table[Sum[(A001358[k][[k]])^((A001358[n - k + 1][[n - k + 1]])), {k, 1, n}], {n, 1, 10}] (* _G. C. Greubel, May 19 2017 *)

a(n) = Sum_{i=1..n} (semiprime(i))^(semiprime(n-i+1)).

a(n) = Sum_{i=1..n} (A001358(i))^(A001358(n-i+1)).

A113170 Ascending descending base exponent transform of odd numbers A005408.

Original entry on oeis.org

1, 4, 33, 376, 5665, 115356, 3014209, 95722288, 3619661121, 161338248820, 8349617508961, 493959321484584, 33041900704133473, 2479933070973253516, 207343189445230918785, 19175058576632809926496, 1949302342535131018462849, 216707770770991401785821668

Offset: 1

Jonathan Vos Post, Jan 06 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The parity of this sequence cycles odd, even, odd, even, ... There is no nontrivial integer fixed point of the transform.

			a(2) = 4 because 1^3 + 3^1 = 1 + 3 = 4.
a(3) = 33 because 1^5 + 3^3 + 5^1 = 1 + 27 + 5 = 33.
a(4) = 406 because 1^7 + 3^5 + 5^3 + 7^1 = 1 + 243 + 125 + 7 = 376.
a(5) = 5665 because 1^9 + 3^7 + 5^5 + 7^3 + 9^1 = 5665.
a(6) = 115356 = 1^11 + 3^9 + 5^7 + 7^5 + 9^3 + 11^1.
a(7) = 3014209 = 1^13 + 3^11 + 5^9 + 7^7 + 9^5 + 11^3 + 13^1.
a(8) = 95722288 = 1^15 + 3^13 + 5^11 + 7^9 + 9^7 + 11^5 + 13^3 + 15^1.
a(9) = 3619661121 = 1^17 + 3^15 + 5^13 + 7^11 + 9^9 + 11^7 + 13^5 + 15^3 + 17^1.
a(10) = 161338248820 = 1^19 + 3^17 + 5^15 + 7^13 + 9^11 + 11^9 + 13^7 + 15^5 + 17^3 + 19^1.

G. C. Greubel, Table of n, a(n) for n = 1..295

Cf. A005408, A113122, A113153, A113154.

Table[Sum[(2 k + 1)^(2 n - 2 k + 1), {k, 1, n}], {n, 0, 10}] + 1 (* G. C. Greubel, May 18 2017 *)

for(n=0,25, print1(1 + sum(k=1,n, (2*k+1)^(2*n-2*k+1)), ", ")) \\ G. C. Greubel, May 18 2017

a(1) = 1. For n>1: a(n) = Sum_{i=1..n} (2n+1)^(2n-i).

A113492 Least integers, starting with 1, so ascending descending base exponent transforms all triprimes.

Original entry on oeis.org

1, 7, 11, 3, 3, 4, 3, 5, 11, 4, 1, 2, 1, 1, 4, 8, 8, 2, 2, 6, 6, 7, 7, 3, 1, 3, 4, 2, 7, 2, 2, 3, 2, 2, 4, 1, 3, 12, 5, 2, 2, 1, 3, 5, 3, 4, 4, 4, 14, 2, 1, 2, 11, 4, 6, 2, 1, 2, 7, 8, 4, 6, 1, 3, 1, 8, 1, 2, 4, 3, 12, 8, 1, 2, 11, 1, 2, 10, 2, 3, 3, 9, 1, 1

Offset: 1

Jonathan Vos Post, Jan 10 2006

This is the triprime analogy to A113320.

			a(1) = 1 by definition.
a(2) = 7 because 1^7 + 7^1 = 8 = 2^3 is a triprime (A014612).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A014612, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208.

p3[n_] := PrimeOmega[n] == 3; inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! p3[ inve@ Append[ Array[a, n - 1], ++k]]]; k]; Array[a, 75] (* Giovanni Resta, Jun 13 2016 *)

a(1) = 1. For n > 1: a(n) = min {n > 0: Sum_{i=1..n} a(i)^a(n-i+1) is a triprime}. a(n) = min {n > 0: Sum_{i=1..n} a(i)^a(n-i+1) in A014612}.

Corrected and extended by Giovanni Resta, Jun 13 2016

A113497 Ascending descending base exponent transform of sequence A000034(n) = 1 + n mod 2.

Original entry on oeis.org

1, 3, 6, 6, 11, 9, 16, 12, 21, 15, 26, 18, 31, 21, 36, 24, 41, 27, 46, 30, 51, 33, 56, 36, 61, 39, 66, 42, 71, 45, 76, 48, 81, 51, 86, 54, 91, 57, 96, 60, 101, 63, 106, 66, 111, 69, 116, 72, 121, 75, 126, 78, 131, 81, 136, 84, 141, 87, 146, 90, 151, 93, 156, 96, 161, 99, 166, 102, 171

Offset: 1

Jonathan Vos Post, Jan 10 2006

A000034 = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ... = continued fraction for (sqrt(3)+1)/2 (cf. A040001) = base 3 digital root of n+1. In general, the ascending descending base exponent transform of any simple periodic sequence can be written as a periodic set of interleaved sequences.

			a(1) = 1^1 = 1.
a(2) = 1^2 + 2^1 = 3.
a(3) = 1^1 + 2^2 + 1^1 = 6.
a(4) = 1^2 + 2^1 + 1^2 + 2^1 = 6.
a(5) = 1^1 + 2^2 + 1^1 + 2^2 + 1^1 = 11.
a(6) = 1^2 + 2^1 + 1^2 + 2^1 + 1^2 + 2^1 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000034, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208.

Table[(-3 + 3*(-1)^n + 8*n - 2*(-1)^n*n)/4, {n,1,50}] (* G. C. Greubel, Mar 12 2017 *)

x='x +O('x^50); Vec(x*(1+3*x+4*x^2)/((1-x)^2*(1+x)^2)) \\ G. C. Greubel, Mar 12 2017

a(n) = Sum_{i=1..n} A000034(i)^A000034(n-i+1).
a(2*n) = 3*n; a(2*n+1) = 5*n+1.
From Colin Barker, Jun 16 2012: (Start)
a(n) = (-3+3*(-1)^n+8*n-2*(-1)^n*n)/4.
a(n) = 2*a(n-2)-a(n-4).
G.f.: x*(1+3*x+4*x^2)/((1-x)^2*(1+x)^2). (End)
E.g.f.: (1/2)*(3*(x-1)*sinh(x) + 5*x*cosh(x)). - G. C. Greubel, Mar 12 2017

Definition improved by M. F. Hasler, Jan 13 2012

A113533 Ascending descending base exponent transform of the infinite Fibonacci word (A003842).

Original entry on oeis.org

1, 3, 6, 5, 7, 12, 10, 15, 14, 14, 23, 16, 20, 27, 21, 30, 27, 25, 40, 28, 37, 38, 32, 49, 36, 40, 53, 39, 54, 49, 43, 68, 45, 55, 66, 50, 71, 60, 56, 83, 57, 74, 75, 61, 92, 67, 73, 94, 68, 93, 84, 72, 113, 75, 94, 101, 79, 116, 89, 91, 122, 86, 115, 108, 90

Offset: 1

Jonathan Vos Post, Jan 13 2006

The infinite Fibonacci word b(n) is the fixed point of the morphism 1->12, 2->1, starting from b(1) = 2. This transform a(n) of that sequence b(n) satisfies n <= a(n) <= 4*n, but that is not a tight bound.

			a(1) = A003842(1)^A003842(1) = 1^1 = 1.
a(2) = A003842(1)^A003842(2) + A003842(2)^A003842(1) = 1^2 + 2^1 = 3.
a(3) = 1^1 + 2^2 + 1^1 = 6.
a(4) = 1^1 + 2^1 + 1^2 + 1^1 = 5.
a(5) = 1^2 + 2^1 + 1^1 + 1^2 + 2^1 = 7.
a(6) = 1^1 + 2^2 + 1^1 + 1^1 + 2^2 + 1^1 = 12.
a(7) = 1^2 + 2^1 + 1^2 + 1^1 + 2^1 + 1^2 + 2^1 = 10.
a(8) = 1^1 + 2^2 + 1^1 + 1^2 + 2^1 + 1^1 + 2^2 + 1^1 = 15.
a(9) = 1^1 + 2^1 + 1^2 + 1^1 + 2^2 + 1^1 + 2^1 + 1^2 + 1^1 = 14.
a(10) = 1^2 + 2^1 + 1^1 + 1^2 + 2^1 + 1^2 + 2^1 + 1^1 + 1^2 + 2^1 = 14.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A003842, A005408, A087316, A113122, A113153, A113154, A113208, A113231, A113257, A113258, A113271, A113320, A113336, A113498.

A003842[n_] := n + 1 - Floor[((1 + Sqrt[5])/2)*Floor[2*(n + 1)/(1 + Sqrt[5])]]; Table[Sum[A003842[k]^(A003842[n - k + 1]), {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, May 18 2017 *)

a(n) = Sum_{k=1..n} A003842(k)^(A003842(n-k+1)). - G. C. Greubel, May 18 2017

Corrected and extended by Giovanni Resta, Jun 13 2016

A113535 Ascending descending base exponent transform of the tribonacci substitution (A100619).

Original entry on oeis.org

1, 3, 8, 19, 32, 9, 11, 16, 26, 19, 29, 24, 47, 70, 28, 31, 58, 89, 35, 50, 65, 108, 65, 51, 52, 90, 101, 82, 101, 88, 122, 63, 81, 92, 153, 110, 89, 125, 110, 92, 101, 155, 90, 127, 196, 142, 87, 138, 207, 112, 112, 135, 217, 150, 124, 115, 204, 245, 139, 158, 189, 268, 121, 155, 154

Offset: 1

Jonathan Vos Post, Jan 13 2006

Sirvent comments that in spite of the similarity of this map to the one in A092782, the two sequences have very different properties. They have different complexities, different Rauzy fractals, etc.

			a(1) = A100619(1)^A100619(1) = 1^1 = 1.
a(2) = A100619(1)^A100619(2) + A100619(2)^A100619(1) = 1^2 + 2^1 = 3.
a(3) = 1^3 + 2^2 + 3^1 = 8.
a(4) = 1^1 + 2^3 + 3^2 + 1^1 = 19.
a(5) = 1^1 + 2^1 + 3^3 + 1^2 + 1^1 = 32.
a(6) = 1^1 + 2^1 + 3^1 + 1^3 + 1^2 + 1^1 = 9.
a(7) = 1^2 + 2^1 + 3^1 + 1^1 + 1^3 + 1^2 + 2^1 = 11.
a(8) = 1^1 + 2^2 + 3^1 + 1^1 + 1^1 + 1^3 + 2^2 + 1^1 = 16.
a(9) = 1^1 + 2^1 + 3^2 + 1^1 + 1^1 + 1^1 + 2^3 + 1^2 + 2^1 = 26.
a(10) = 1^1 + 2^2 + 3^1 + 1^2 + 1^1 + 1^1 + 2^1 + 1^3 + 2^2 + 1^1 = 19.
a(11) = 1^2 + 2^1 + 3^2 + 1^1 + 1^2 + 1^1 + 2^1 + 1^1 + 2^3 + 1^2 + 2^1 = 29.
a(12) = 1^3 + 2^2 + 3^1 + 1^2 + 1^1 + 1^2 + 2^1 + 1^1 + 2^1+ 1^3 + 2^2 + 3^1 = 24.

G. C. Greubel, Table of n, a(n) for n = 1..500
V. F. Sirvent, Semigroups and the self-similar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 25-29. [Contains many further references.]

Cf. A100619, A092782, A103269, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208, A113498.

A100619:= Nest[Function[l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {3, 1}, 3 -> {1}})]}], {1}, 8][[1]]; Table[Sum[(A100619[[k]])^(A100619[[n-k+1]]), {k, 1, n}], {n, 1, 100}] (* G. C. Greubel, May 18 2017 *)

a(n) = Sum_{k=1..n} A100619(k)^(A100619(n-k-1)). - G. C. Greubel, May 18 2017

Terms a(13) to a(50) from G. C. Greubel, May 18 2017

Terms a(51) onward added by G. C. Greubel, Jan 03 2019

A113475 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is semiprime (A001358).

Original entry on oeis.org

1, 3, 5, 2, 4, 2, 2, 4, 2, 4, 3, 2, 3, 4, 2, 2, 1, 1, 2, 1, 5, 1, 7, 1, 5, 4, 2, 2, 3, 3, 2, 11, 5, 10, 4, 2, 2, 6, 14, 4, 6, 2, 3, 9, 14, 10, 3, 3, 4, 2, 1, 5, 4, 16, 8, 9, 5, 8, 14, 6, 2, 2, 26, 8, 30, 4, 5, 1, 4, 2, 22, 36, 20, 2, 10, 2, 15, 3, 18, 6, 15

Offset: 1

Jonathan Vos Post, Jan 08 2006

nonn

Previous name was: Least integers so ascending descending base exponent transforms all semiprime.
Semiprime analogy to A113320. The sequence is probably infinite, but it is hard to characterize the asymptotic cost of adding an n-th term. The ascending descending base exponent transform of semiprimes is A113173.
The sequence is infinite because a(n) is the minimum k such that a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is semiprime, and since a(1)=1 this is equal to 1+k+T where T does not depend on k, thus k is the smallest positive value that makes 1+k+T semiprime, which exists because semiprimes are infinite. - Giovanni Resta, Jan 03 2020

			a(1) = 1 by definition.
a(2) = 3 because 3 is the min x such that 1^x + x^1 is semiprime, i.e., 1^3 + 3^1 = 4 = 2*2.
a(3) = 5 because 1^5 + 3^3 + 5^1 = 33 = 3 * 11 is semiprime.
a(4) = 2 because 1^2 + 3^5 + 5^3 + 2^1 = 371 = 7 * 53.
a(5) = 4 because 1^4 + 3^2 + 5^5 + 2^3 + 4^1 = 3147 = 3 * 1049.
a(6) = 2 because 1^2 + 3^4 + 5^2 + 2^5 + 4^3 + 2^1 = 205 = 5 * 41.
a(7) = 2 because 1^2 + 3^2 + 5^4 + 2^2 + 4^5 + 2^3 + 2^1 = 1673 = 7 * 239.
a(8) = 4 because 1^4 + 3^2 + 5^2 + 2^4 + 4^2 + 2^5 + 2^3 + 4^1 = 111 = 3 * 37.

Cf. A001358, A005408, A113122, A113153, A113154, A113336, A113320, A113271, A113258, A113257, A113231, A087316, A113208.

semipQ[n_] := PrimeOmega[n] == 2; inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! semipQ[ inve@ Append[ Array[a, n - 1], ++k]]]; k]; Array[a, 81] (* Giovanni Resta, Jun 13 2016 *)

lista(n)={my(a=vector(n)); a[1]=1; print1(1, ", "); for(n=2, #a, my(t=sum(i=2, n-1, a[i]^a[n-i+1])); my(k=1); while(2!=bigomega(t+1+k), k++); a[n]=k; print1(k, ", "))} \\ Andrew Howroyd, Jan 03 2020

a(1) = 1. For n>1, a(n) = min {k>0: a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is in A001358}.

Corrected and extended by Giovanni Resta, Jun 13 2016

A113534 Ascending descending base exponent transform of the flipped tribonacci substitution (A092782).

Original entry on oeis.org

1, 3, 6, 7, 20, 10, 39, 12, 26, 19, 20, 43, 21, 78, 24, 53, 30, 57, 43, 88, 61, 59, 56, 43, 90, 42, 155, 46, 109, 53, 122, 75, 105, 114, 73, 122, 62, 197, 63, 172, 71, 136, 96, 183, 140, 122, 139, 86, 179, 81, 304, 83, 185, 98, 153, 162, 160, 261, 121, 192, 107, 236, 126

Offset: 1

Jonathan Vos Post, Jan 13 2006

The flipped tribonacci substitution (A092782) b(n) is the fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 1, starting from b(1) = 1. The transformed sequence a(n) satisfies n <= a(n) <= 27 n but the bound can be determined to be much tighter.

			a(1) = A092782(1)^A092782(1) = 1^1 = 1.
a(2) = A092782(1)^A092782(2) + A092782(2)^A092782(1) = 1^2 + 2^1 = 3.
a(3) = 1^1 + 2^2 + 1^1 = 6.
a(4) = 1^3 + 2^1 + 1^2 + 3^1 = 7.
a(5) = 1^1 + 2^3 + 1^1 + 3^2 + 1^1 = 20.
a(6) = 1^2 + 2^1 + 1^3 + 3^1 + 1^2 + 2^1 = 10.
a(7) = 1^1 + 2^2 + 1^1 + 3^3 + 1^1 + 2^2 + 1^1 = 39.
a(8) = 1^1 + 2^1 + 1^2 + 3^1 + 1^3 + 2^1 + 1^2 + 1^1 = 12.
a(9) = 1^2 + 2^1 + 1^1 + 3^2 + 1^1 + 2^3 + 1^1 + 1^2 + 2^1 = 26.
a(10) = 1^1 + 2^2 + 1^1 + 3^1 + 1^2 + 2^1 + 1^3 + 1^1 + 2^2 + 1^1 = 19.
a(11) = 1^3 + 2^1 + 1^2 + 3^1 + 1^1 + 2^2 + 1^1 + 1^3 + 2^1 + 1^2 + 3^1 = 20.
a(12) = 1^1 + 2^3 + 1^1 + 3^2 + 1^1 + 2^1 + 1^2 + 1^1 + 2^3 + 1^1 + 3^2 + 1^1 = 43.

V. F. Sirvent, Semigroups and the self-similar structure of the flipped tribonacci substitution, Applied Math. Letters, 12 (1999), 25-29. [Contains many further references.]

Cf. A005408, A087316, A092782, A113122, A113153, A113154, A113208, A113231, A113257, A113258, A113271, A113320, A113336, A113498.

A092782[n_] := Nest[Function[l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1}})]}], {1}, n][[1]]; Table[Sum[(A092782[k][[k]])^((A092782[n - k + 1][[n - k + 1]])), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)

a(n) = Sum_{k=1..n} A092782(k)^(A092782(n-k+1)). - G. C. Greubel, May 17 2017

a(3) corrected by Giovanni Resta, Jun 13 2016

a(13) onward from G. C. Greubel, May 18 2017

cp's OEIS Frontend

A113173 Ascending descending base exponent transform of semiprimes (A001358).

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A113170 Ascending descending base exponent transform of odd numbers A005408.

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A113492 Least integers, starting with 1, so ascending descending base exponent transforms all triprimes.

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A113497 Ascending descending base exponent transform of sequence A000034(n) = 1 + n mod 2.

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A113533 Ascending descending base exponent transform of the infinite Fibonacci word (A003842).

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A113535 Ascending descending base exponent transform of the tribonacci substitution (A100619).

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A113475 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is semiprime (A001358).

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