A177888 -id:A177888 - cp's OEIS frontend

A001543 a(0) = 1, a(n) = 5 + Product_{i=0..n-1} a(i) for n > 0.

1, 6, 11, 71, 4691, 21982031, 483209576974811, 233491495280173380882643611671, 54518278368171228201482876236565907627201914279213829353891

Offset: 0

N. J. A. Sloane

nonn

This is the special case k=5 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
S. Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Column k=5 of A177888.

Flatten[{1,RecurrenceTable[{a[1]==6, a[n]==a[n-1]*(a[n-1]-5)+5}, a, {n, 1, 10}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{1},NestList[#(#-5)+5&,6,10]] (* Harvey P. Dale, Oct 10 2016 *)

{
  print1("1, 6");
  n=6;
  m=Mod(5,6);
  for(i=2,9,
    n=m.mod+lift(m);
    m=chinese(m,Mod(5,n));
    print1(", "n)
  )
} \\ Charles R Greathouse IV, Dec 09 2011

a(n) = a(n-1) * (a(n-1) - 5) + 5. - Charles R Greathouse IV, Dec 09 2011

a(n) ~ c^(2^n), where c = 1.696053774403103324180661918166106455311376345474042496749974632237971081462... . - Vaclav Kotesovec, Dec 17 2014

New name from Alonso del Arte, Dec 09 2011

A001544 A nonlinear recurrence: a(n) = a(n-1)^2 - 6*a(n-1) + 6, with a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 13, 97, 8833, 77968897, 6079148431583233, 36956045653220845240164417232897, 1365749310322943329964576677590044473746108255675592519835615233

Offset: 0

N. J. A. Sloane

nonn

This is the special case k=6 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..11
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Seppo Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Column k=6 of A177888. - Alois P. Heinz, Nov 07 2012

Flatten[{1,RecurrenceTable[{a[1]==7, a[n]==a[n-1]*(a[n-1]-6)+6}, a, {n, 1, 10}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{1},NestList[#^2-6#+6&,7,10]] (* Harvey P. Dale, Nov 19 2024 *)

a(n)=if(n<1, n==0, if(n==1, 7, n=a(n-1); n^2-6*n+6))

a(n) ~ c^(2^n), where c = 1.76450357631319101484804524709844019487003729926754942591419313922841785792... . - Vaclav Kotesovec, Dec 17 2014

A067686 a(n) = a(n-1) * a(n-1) - B * a(n-1) + B, a(0) = 1 + B for B = 7.

Original entry on oeis.org

8, 15, 127, 15247, 232364287, 53993160246468367, 2915261353400811631533974206368127, 8498748758632331927648392184620600167779995785955324343380396911247

Offset: 0

Drastich Stanislav (drass(AT)spas.sk), Feb 05 2002

This is the special case k=7 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

Alois P. Heinz, Table of n, a(n) for n = 0..10
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Cf. B=1: A000058 (Sylvester's sequence), B=2: A000215 (Fermat numbers), B=3: A000289, B=4: A000324, B=5: A001543, B=6: A001544.

Column k=7 of A177888.

RecurrenceTable[{a[0]==8, a[n]==a[n-1]*(a[n-1]-7)+7}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)
NestList[#^2-7#+7&,8,10] (* Harvey P. Dale, Jan 26 2025 *)

a(n) ~ c^(2^n), where c = 3.3333858371760195832345950846454963835549715770476958790043961891683146201... . - Vaclav Kotesovec, Dec 17 2014

A110360 Integers with mutual residues of 8.

Original entry on oeis.org

9, 17, 161, 24641, 606981761, 368426853330807041, 135738346255240000293762417728719361, 18424898644107427010977107148874723523180059431182608785043639266493441

Offset: 1

Seppo Mustonen, Sep 04 2005

nonn

This is the special case k=8 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Column k=8 of A177888.

RecurrenceTable[{a[1]==9, a[n]==a[n-1]*(a[n-1]-8)+8}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)

a(n) ~ c^(2^n), where c = 1.8813701045812484604409881785833034768479650739052732570542874567824022000... . - Vaclav Kotesovec, Dec 17 2014

A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 4, 14, 16, 8, 1, 16, 112, 324, 508, 474, 268, 88, 16, 1, 256, 3584, 22912, 88832, 233936, 443936, 628064, 675456, 557492, 353740, 171644, 62878, 17000, 3264, 416, 32, 1, 65536, 1835008, 24576000, 209715200, 1281482752, 5974786048, 22114709504, 66752724992

Offset: 1

Vladimir Shevelev, Dec 11 2010

Length of the first row is 2; for i>=2, length of the i-th row is 2^{i-2}+1.

			Triangle begins:
   1,   1;
   2,   1;
   2,   4,   1;
   4,  14,  16,   8,   1;
  16, 112, 324, 508, 474, 268, 88, 16, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888.

First column gives: A165420.

p:= proc(n) option remember;
       z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
deg:= n-> `if`(n=0, 1, 2^(n-1)):
T:= (n,k)-> coeff(p(n)(z), z, deg(n)-k):
seq(seq(T(n,k), k=0..deg(n)), n=0..6); # Alois P. Heinz, Dec 13 2010

P[0][z_] := z + 1;
P[n_][z_] := P[n][z] = z + Product[P[k][z], {k, 0, n-1}];
row[n_] := CoefficientList[P[n][z], z] // Reverse;
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

Another recursion is: P_n(z)=z+P_(n-1)(z)(P_(n-1)(z)-z).

Private values: P_n(0)=1; P_n(-1)=delta_(n,0)-1; {P_n(1)}=A000058; {P_n(2)}=A000215; {P_n(3)}={A000289(n+1)}; {P_n(4)}={A000324(n+1)}; {P_n(5)}={A001543(n+1)}; {P_n(6)}={A001544(n+1)}; {P_n(7)}={A067686(n)}; {P_n(8)}={A110360(n)}; {P_0(n)}={A000027(n+1)}; {P_1(n)}={A005408(n)}; {P_2(n)}={A056220(n+1)}.

More terms from Alois P. Heinz, Dec 13 2010

A252730 a(n) = P_n(n) with P_0(z) = z+1 and P_n(z) = z + P_{n-1}(z)*(P_{n-1}(z)-z) for n>1.

Original entry on oeis.org

1, 3, 17, 871, 4870849, 483209576974811, 36956045653220845240164417232897, 8498748758632331927648392184620600167779995785955324343380396911247

Offset: 0

Alois P. Heinz, Dec 20 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Main diagonal of A177888.

p:= proc(n) option remember;
      z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
a:= n-> p(n)(n):
seq(a(n), n=0..8);

p[n_] := p[n] = Function[z, z + If[n == 0, 1, p[n-1][z]*(p[n-1][z] - z)]];
a[n_] := p[n][n];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jun 12 2018, from Maple *)

A110368 Integers with mutual residues of 9.

Original entry on oeis.org

10, 19, 199, 37819, 1429936399, 2044718092315659619, 4180872077042990313463432060226288599, 17479691324597767931283328689425028720038746822457352536058485868000785419

Offset: 1

Seppo Mustonen, Sep 04 2005

nonn

This is the special case k=9 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Column k=9 of A177888.

RecurrenceTable[{a[1]==10, a[n]==a[n-1]*(a[n-1]-9)+9}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)

a(n) ~ c^(2^n), where c = 1.9324294501525084771045650938374200605001383645783351474944965038078432359... . - Vaclav Kotesovec, Dec 17 2014

A110383 Integers with mutual residues of 10.

Original entry on oeis.org

11, 21, 241, 55681, 3099816961, 9608865160705105921, 92330289676612360941221747472778199041, 8524882391767151111154918892947398067446166736305624023874497267723631329281

Offset: 1

Seppo Mustonen, Sep 04 2005

nonn

This is the special case k=10 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1..n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Column k=10 of A177888.

RecurrenceTable[{a[1]==11, a[n]==a[n-1]*(a[n-1]-10)+10}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)

a(n) ~ c^(2^n), where c = 1.9797221926746931491020959969764290497942241392143973226882604062455515473... . - Vaclav Kotesovec, Dec 17 2014

cp's OEIS Frontend

A001543 a(0) = 1, a(n) = 5 + Product_{i=0..n-1} a(i) for n > 0.

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A001544 A nonlinear recurrence: a(n) = a(n-1)^2 - 6*a(n-1) + 6, with a(0) = 1, a(1) = 7.

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A067686 a(n) = a(n-1) * a(n-1) - B * a(n-1) + B, a(0) = 1 + B for B = 7.

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A110360 Integers with mutual residues of 8.

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A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

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A252730 a(n) = P_n(n) with P_0(z) = z+1 and P_n(z) = z + P_{n-1}(z)*(P_{n-1}(z)-z) for n>1.

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A110368 Integers with mutual residues of 9.

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A110383 Integers with mutual residues of 10.

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