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 1-3 of 3 results.

A257764 Lexicographically largest increasing sequence of primes for which the continued square root map (see A257574) produces the decimal expansion of e (Euler's number).

Original entry on oeis.org

3, 13, 31, 59, 67, 103, 179, 193, 227, 229, 317, 983, 1201, 1213, 1321, 1787, 1811, 2179, 3571, 4817, 5333, 6803, 10433, 12197, 13063, 19391, 21283, 24571, 31817, 42307, 45377, 49957, 61909, 67933, 70573, 74843, 82421, 85909, 91099, 99241, 101293, 109639, 112087
Offset: 1

Views

Author

Chai Wah Wu, May 09 2015

Keywords

Comments

Similar to A257582, but converging to e.

Examples

			sqrt(3) =  1.7320508075688772...
sqrt(3+sqrt(13)) = 2.570126704165378...
sqrt(3+sqrt(13+sqrt(31))) = 2.703522309917472...
sqrt(3+sqrt(13+sqrt(31+sqrt(59)))) = 2.7173508299457327...
sqrt(3+sqrt(13+sqrt(31+sqrt(59+sqrt(67))))) = 2.718217091497069...
sqrt(3+sqrt(13+sqrt(31+sqrt(59+sqrt(67+sqrt(103)))))) = 2.7182780002752187...

Links

  • Chai Wah Wu, Table of n, a(n) for n = 1..1000
  • Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
  • Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Crossrefs

Cf. A001113 (e), A257582 (analog for Pi instead of e), A257809 (analog for delta = 4.6692...), A257574.

Programs

  • PARI
    (CSR(v, s)=forstep(i=#v, 1, -1, s=sqrt(v[i]+s)); s); a=[3]; for(n=1, 50, print1(a[#a]", "); for(i=primepi(a[#a])+1, oo, CSR(concat(a, vector(9, j, prime(i+j))))>=exp(1)&& (a=concat(a, prime(i)))&& break)) \\ The standard precision of 38 digits yields incorrect terms beyond 10433. Increase realprecision to compute larger values. - M. F. Hasler, May 03 2018

Extensions

Edited by M. F. Hasler, May 03 2018

A257809 Lexicographically largest strictly increasing sequence of primes for which the continued square root map produces Feigenbaum's constant delta = 4.6692016... (A006890).

Original entry on oeis.org

13, 67, 97, 139, 293, 661, 1163, 1657, 2039, 3203, 3469, 5171, 6361, 6661, 7393, 7901, 8969, 9103, 9137, 11971, 12301, 13487, 14083, 14699, 15473, 19141, 21247, 28099, 31039, 35423, 39047, 49223, 58427, 61493, 62171, 67699, 71971, 75869, 78857, 81533, 88007, 93199
Offset: 1

Views

Author

Chai Wah Wu, May 10 2015

Keywords

Comments

The continued square root map takes a finite or infinite sequence (x, y, z, ...) to the number CSR(x, y, z,...) = sqrt(x + sqrt(y + sqrt(z + ...))). It is well defined if the logarithm of the terms is O(2^n).
The terms are defined to be the largest possible choice such that the sequence can remain strictly increasing without the CSR growing beyond delta = 4.66920...

Examples

			From _M. F. Hasler_, May 03 2018: (Start)
We look for a strictly increasing sequence of primes (p,q,r,...) such that CSR(p,q,r,...) = sqrt(p + sqrt(q + sqrt(r + ...))) = delta = 4.66920...
The first term must be less than delta^2 ~ 21.8, but p = 19 and also p = 17 are excluded, since CSR(17,19,23,...) > 4.67. It appears that p = 13 does not lead to a contradiction, so this is the largest possible choice for p, whence a(1) = 13.
The second term could be chosen to be q = 17, provided that subsequent terms are large enough to ensure CSR(p, q, r,...) = delta, which is always possible. But one can verify that any q between 19 and 67 is also possible without contradiction. If we try q = 71, then we find that CSR(13, 71, 73, ...) > 4.68. So a(2) = 67, etc. (End)

Links

Crossrefs

Cf. A006890, A257582, A257764, A257574.

Programs

  • PARI
    (CSR(v,s)=forstep(i=#v,1,-1,s=sqrt(v[i]+s));s); a=[13]; for(n=1,50, print1(a[#a]","); for(i=primepi(a[#a])+1,oo, CSR(concat(a,vector(9,j,prime(i+j))))>=delta&& (a=concat(a,prime(i)))&& break)) \\  For delta, see A006890. - M. F. Hasler, May 03 2018

Extensions

Edited by M. F. Hasler, May 02 2018

A257858 An increasing sequence of integers for which the continued square root map (see A257574) produces the decimal expansion of Pi.

Original entry on oeis.org

6, 10, 19, 27, 29, 32, 42, 45, 56, 67, 75, 94, 109, 122, 138, 144, 151, 152, 172, 181, 194, 204, 205, 232, 256, 290, 316, 325, 346, 380, 412, 446, 478, 511, 520, 533, 580, 584, 617, 623, 654, 658, 661, 682, 734, 773, 823, 836, 865, 903, 954, 979, 997, 1008, 1059
Offset: 1

Views

Author

Chai Wah Wu, May 13 2015

Keywords

Examples

			sqrt(6) = 2.449489742783178
sqrt(6+sqrt(10)) = 3.0269254467476365
sqrt(6+sqrt(10+sqrt(19))) = 3.1287879095060176
sqrt(6+sqrt(10+sqrt(19+sqrt(27)))) = 3.140462825727146
sqrt(6+sqrt(10+sqrt(19+sqrt(27+sqrt(29))))) = 3.1414928066743406

Links

  • Chai Wah Wu, Table of n, a(n) for n = 1..1000
  • Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
  • Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Crossrefs

Cf. A257582, A257764, A257809.
Showing 1-3 of 3 results.

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