Anders Hellström - cp's OEIS frontend

A280532 a(1) = a(2) = 1, a(n) = A014777(a(n-1) + a(n-2)), n >= 3.

1, 1, 6, 13, 37, 31, 605, 1411, 7174, 15567, 608953, 78903, 334535, 611552, 105928, 2557047, 2979162, 3263358, 6242520, 7825254, 37404834, 267494881, 639174488

Offset: 1

Anders Hellström, Jan 13 2017

Dave Andersen, Probability of Finding Strings In Pi
Anders Hellström, Sage program
James Taylor, Irrational Numbers Search Engine (Search 2*10^9 decimal digits of Pi)

Cf. A014777, A032445, A038100, A176341, A232013.

Module[{a = {1, 1}, s = First@ RealDigits[N[Pi, 10^7]]}, Do[AppendTo[a, -1 + SequencePosition[s, IntegerDigits[ a[[n - 1]] + a[[n - 2]] ]][[1, 1]]], {n, 3, 20}]; a] (* Michael De Vlieger, Jan 14 2017 *)

A265131 Decimal expansion of positive x satisfying x^(x^x) = LambertW(1).

Original entry on oeis.org

4, 4, 3, 3, 4, 4, 8, 8, 7, 3, 5, 7, 9, 1, 5, 0, 7, 4, 1, 5, 9, 8, 0, 0, 2, 7, 9, 3, 7, 8, 8, 6, 8, 8, 6, 0, 1, 2, 2, 5, 4, 1, 3, 9, 6, 5, 2, 2, 2, 2, 9, 2, 1, 4, 9, 5, 7, 7, 1, 3, 5, 9, 5, 4, 0, 8, 8, 4, 9, 4, 5, 4, 8, 8, 1, 8, 6, 0, 0, 2, 4, 6, 5, 9, 7, 8, 8, 6, 7, 6, 8, 7, 9, 2, 2, 8, 4, 9, 2, 5, 1, 9, 9, 4, 1, 5, 3, 0, 0, 1, 1, 9, 8, 1

Offset: 0

Anders Hellström, Dec 02 2015

			0.44334488735791507415980027937886886012254139652223...

Wikipedia, Lambert W function
Wikipedia, Omega constant

Cf. A030178, A264807, A264808.

RealDigits[x/.FindRoot[x^(x^x)==ProductLog[1],{x,1},WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Jul 19 2020 *)

default(realprecision,2000);solve(x=0.001,3,x^(x^x)-lambertw(1))

A265009 a(1)=3; for n>1, if n is odd a(n) = spf(Product_{k=1..n-1}(a(k))+1) else a(n) = spf(Product_{k=1..n-1}(a(k))-1), where spf is "smallest prime factor".

Original entry on oeis.org

3, 2, 7, 41, 1723, 5, 14835031, 220078129935929, 241, 23, 79, 101, 23291, 11, 223, 122386298896281959929015788890561251765109069, 38803, 17, 8209, 59, 199, 3340389589, 11527, 13, 47939, 1163, 599, 27198087874669514440553, 181936481, 31, 383, 9623, 739, 33287, 1061, 6493520653, 587, 709, 6548057, 1823, 361789, 20183

Offset: 1

Anders Hellström, Nov 30 2015

nonn

Cf. A000945, A005265.

a[1] = 3; a[n_] := a[n] = FactorInteger[ Product[a[k], {k, n - 1}] + If[OddQ@ n, 1, -1]][[1, 1]]; Array[a, {16}] (* Michael De Vlieger, Nov 30 2015 *)

spf(n)=my(f=factor(n)[1, 1]); f
first(m)=my(v=vector(m)); v[1]=3; for(i=2, m,;v[i]=spf((-1)^(i+1)+prod(j=1, i-1, v[j]))); v

a(20)-a(42) from Hans Havermann, Dec 06 2015

A264936 Decimal expansion of the positive root of x^(x^x) = gamma.

Original entry on oeis.org

4, 5, 5, 5, 5, 7, 0, 3, 6, 7, 0, 1, 9, 5, 8, 4, 2, 9, 0, 0, 4, 9, 5, 0, 0, 0, 4, 7, 0, 4, 0, 7, 0, 5, 7, 7, 0, 4, 9, 0, 3, 9, 3, 3, 9, 6, 9, 1, 2, 1, 7, 1, 7, 0, 6, 0, 7, 3, 0, 2, 3, 6, 0, 9, 6, 7, 6, 4, 2, 5, 3, 0, 9, 7, 2, 2, 4, 7, 6, 2, 2, 4, 4, 8, 9, 0, 0, 2, 5, 0, 1, 4, 0, 1, 0, 5, 0, 7, 7, 6, 8, 5, 3, 7, 8, 6

Offset: 0

Anders Hellström, Nov 28 2015

			0.4555570367019584290049500047040705770490...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Euler-Mascheroni constant

Cf. A001620.

Take[RealDigits[x /. FindRoot[x^x^x == EulerGamma, {x, 1}, WorkingPrecision -> 120]][[1]], 100] (* G. C. Greubel, Sep 07 2018 *)

default(realprecision,2000); solve(x=0.4,0.5,x^(x^x)-Euler)

More digits from Jon E. Schoenfield, Mar 15 2018

A264849 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 23-gonal: (21n^2 - 19n)/2.

Original entry on oeis.org

1, 30, 648, 6701456, 72020220595275, 970458695858595792221157266, 3377345920936319088412440649783459968197698452784332095, 7477788200541027929765479736500643733301085903714718188060185368351929896324223859775571543015918781111399506

Offset: 1

Anders Hellström, Nov 26 2015

			1, 130, 130648 are 23-gonal.

Chai Wah Wu, Table of n, a(n) for n = 1..11
Wikipedia, Polygonal number

Cf. A051671, A051875 (23-gonal numbers), A061109, A061110, A261696, A264733, A264738, A264776, A264777, A264842, A264848, A264804.

icositrigonal(n)=ispolygonal(n, 23)
first(m)=my(s=""); s="1"; print1(1, ", "); for(i=2, m, n=1; while(!icositrigonal(eval(concat(s, Str(n)))), n++); print1(n, ", "); s=concat(s, Str(n)))

a(5)-a(8) from Chai Wah Wu, Mar 15 2018

A264848 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 19-gonal: (17n^2 - 15n)/2.

Original entry on oeis.org

1, 9, 224, 631909, 58000804596, 61688194098028272863216, 2514637794509678630513616176470588235294117671, 941048382372874985200592647058823529411764708485294117647058823529411764705882352941176469

Offset: 1

Anders Hellström, Nov 26 2015

			1, 19, 19224, 19224631909 are 19-gonal.

Chai Wah Wu, Table of n, a(n) for n = 1..11
Wikipedia, Polygonal number

Cf. A051671, A051871 (19-gonal numbers), A061109, A061110, A261696, A264733, A264738, A264776, A264777, A264804, A264842, A264849.

enneadecagonal(n)=ispolygonal(n, 19)
first(m)=my(s=""); s="1"; print1(1, ", "); for(i=2, m, n=1; while(!enneadecagonal(eval(concat(s, Str(n)))), n++); print1(n, ", "); s=concat(s, Str(n)))

a(5)-a(8) from Chai Wah Wu, Mar 16 2018

A264842 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 13-gonal: (11n^2 - 9n)/2.

Original entry on oeis.org

1, 3, 36, 54765, 123152388, 374848814886363636, 85794018663817263665487289502938826, 107072047880615405294526336549204869795454545454545454545454545454545466

Offset: 1

Anders Hellström, Nov 26 2015

			1, 13, 1336, 133654765 are 13-gonal.

Jon E. Schoenfield, Table of n, a(n) for n = 1..11
Wikipedia, Polygonal number

Cf. A051671, A051865 (13-gonal numbers), A061109, A061110, A261696, A264733, A264738, A264776, A264777, A264848, A264849, A264804.

tridecagonal(n)=ispolygonal(n, 13)
first(m)=my(s=""); s="1"; print1(1, ", "); for(i=2, m, n=1; while(!tridecagonal(eval(concat(s, Str(n)))), n++); print1(n, ", "); s=concat(s, Str(n)))

More terms from Jon E. Schoenfield, Nov 27 2015

A261696 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 17-gonal: (15n^2 - 13n)/2.

Original entry on oeis.org

1, 7, 689, 6797, 67984832, 6798483348333332, 8455610150480042707742277762479, 707328322040172689545426423113211907561874137758547957769721082

Offset: 1

Anders Hellström, Nov 26 2015

From Chai Wah Wu, Mar 16 2018: (Start)
There are some interesting patterns observed in the terms. Terms a(5), a(6), a(9), a(10), a(11), a(12), ... share the same prefix of 6798483...
From terms a(n) for n > 5, there seems to a pattern of how they are constructed from previous terms. a(6) is formed by inserting 3483...3 between the penultimate digit and the last digit of a(5). Then a(7) and (8) do not follow this pattern.
The digits of a(9) and a(6) match until the last digit of a(6). Next, a(10), a(11) and (12) are formed from a(9), a(10) and a(11) resp. by inserting 3483...3. Then this pattern is interrupted by a(13) and a(14), and continue again for a(15) ..., etc.
(End)

			1, 17, 17689, 176896797 are 17-gonal.

Chai Wah Wu, Table of n, a(n) for n = 1..11
Wikipedia, Polygonal number
Chai Wah Wu, terms < 10^70000 to illustrate the pattern observed (see comments)

Cf. A051671, A051869 (17-gonal numbers), A061109, A061110, A264733, A264738, A264776, A264777, A264842, A264848, A264849, A264804.

heptadecagonal(n)=ispolygonal(n, 17)
first(m)=my(s=""); s="1"; print1(1, ", ");for(i=2, m, n=1; while(!heptadecagonal(eval(concat(s, Str(n)))), n++); print1(n, ", "); s=concat(s, Str(n)))

a(6)-a(8) from Chai Wah Wu, Mar 16 2018

A264808 Decimal expansion of the positive root of x^(x^x) = e.

Original entry on oeis.org

1, 6, 0, 1, 0, 7, 5, 1, 5, 3, 9, 2, 3, 7, 2, 8, 7, 7, 9, 3, 2, 7, 0, 6, 4, 7, 6, 7, 7, 4, 7, 8, 3, 5, 4, 5, 9, 5, 3, 1, 5, 9, 5, 8, 0, 4, 9, 6, 3, 4, 5, 2, 8, 9, 0, 7, 7, 2, 1, 8, 2, 0, 9, 9, 1, 3, 1, 1, 6, 0, 2, 2, 7, 1, 5, 4, 0, 9, 0, 3, 2, 2, 0, 5, 8, 5, 4, 9

Offset: 1

Anders Hellström, Nov 25 2015

			1.6010751539237287793270647677478354595315958049634528907721820991...

Cf. A001113, A264807.

default(realprecision,2000);solve(x=0.01, 2, x^(x^x)-exp(1))

A264807 Decimal expansion of the positive root of x^(x^x) = Pi.

Original entry on oeis.org

1, 6, 5, 0, 1, 7, 9, 7, 5, 0, 7, 6, 8, 8, 4, 8, 3, 5, 9, 7, 4, 4, 6, 8, 3, 1, 1, 9, 9, 2, 0, 4, 7, 6, 3, 7, 2, 6, 1, 8, 6, 8, 9, 1, 2, 5, 3, 4, 2, 1, 4, 8, 2, 4, 3, 2, 0, 8, 0, 8, 0, 6, 8, 1, 2, 8, 8, 4, 6, 1, 4, 3, 0, 7, 0, 7, 4, 1, 2, 7

Offset: 1

Anders Hellström, Nov 25 2015

			1.65017975076884835974468311992047637261868912534214824320808068128846...

Cf. A000796, A264808.

RealDigits[x /. FindRoot[x^(x^x) == Pi, {x, 1}, WorkingPrecision -> 105]][[1]] (* Amiram Eldar, Jun 25 2023 *)

default(realprecision,2000);solve(x=0.01, 2, x^(x^x)-Pi)

cp's OEIS Frontend

User: Anders Hellström

A280532 a(1) = a(2) = 1, a(n) = A014777(a(n-1) + a(n-2)), n >= 3.

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Mathematica

A265131 Decimal expansion of positive x satisfying x^(x^x) = LambertW(1).

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A265009 a(1)=3; for n>1, if n is odd a(n) = spf(Product_{k=1..n-1}(a(k))+1) else a(n) = spf(Product_{k=1..n-1}(a(k))-1), where spf is "smallest prime factor".

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Programs

Mathematica

PARI

Extensions

A264936 Decimal expansion of the positive root of x^(x^x) = gamma.

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Programs

Mathematica

PARI

Extensions

A264849 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 23-gonal: (21n^2 - 19n)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A264848 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 19-gonal: (17n^2 - 15n)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A264842 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 13-gonal: (11n^2 - 9n)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A261696 a(n) is least number > 0 such that the concatenation of a(1) ... a(n) is 17-gonal: (15n^2 - 13n)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI