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.

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A171490 Numbers for which the smallest number of steps to reach 1 in "3x+1" (or Collatz) problem is a prime.

Original entry on oeis.org

1, 5, 7, 12, 14, 16, 29, 51, 56, 58, 60, 64, 65, 67, 74, 75, 78, 83, 87, 90, 100, 102, 104, 106, 109, 115, 118, 119, 122, 128, 130, 132, 134, 141, 142, 147, 161, 166, 173, 176, 187, 188, 200, 212, 219, 221, 231, 234, 239, 241, 251, 259, 264, 293, 313, 314, 316
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 10 2009

Keywords

Comments

Positions of primes in A033491. [R. J. Mathar, Nov 01 2010]

Examples

			1st Collatz sequence with a(1)=1 step starts with 2=prime(1): 2-1;
1st Collatz sequence with a(3)=7 steps starts with 3=prime(2): 3-10-5-16-8-4-2-1;
prime(6)=13 has Collatz sequence with 9 steps: 13-40-20-10-5-16-8-4-2-1, so has the smaller composite 12 < 13: 12-6-3-10-5-16-8-4-2-1 => 9 not a term of sequence;
1st Collatz sequence with a(5)=14 steps starts with 11=prime(5): 11-34-17-52-26-13-40-20-10-5-16-8-4-2-1.

References

  • R. K. Guy, "Collatz's Sequence" in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994
  • Clifford A. Pickover, Wonders of Numbers, Oxford University Press, pp. 116-118, 2001

Links

Crossrefs

Cf. A070905, A033491, A088975, A126727, A060565

Extensions

Terms > 187 from R. J. Mathar, Nov 01 2010
Name edited by Michel Marcus, Jul 07 2018

A337673 a(n) is the sum of all positive integers whose Collatz orbit has length n.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 37, 74, 172, 344, 786, 1572, 3538, 7206, 16252, 33112, 73762, 149967, 330107, 678610, 1498356, 3082302, 6742487, 13855154, 30122440, 62388962, 135783788, 281177482, 608402189, 1259151448, 2711432766, 5646008216, 12172417990, 25339969480, 54409676729, 113159496364
Offset: 0

Views

Author

Markus Sigg, Sep 15 2020

Keywords

Comments

a(n) >= 2^(n-1) as 2^(n-1) has orbit length n.

Examples

			a(6) = 5+32 = 37 as the positive integers whose Collatz orbit has length 6 are {5,32} - the orbit of 5 is 5,16,8,4,2,1, and the orbit of 32 is 32,16,8,4,2,1.

Links

Crossrefs

Cf. A000975, A005186, A153772.
Equals row sums of triangles A088975 and A127824.

Programs

  • PARI
    nextSet(s) = { my(s1 = Set([])); for(i = 1, #s, s1 = setunion(s1, Set([2*s[i]])); if (s[i] > 4 && (s[i]-1) % 3 == 0 && (s[i]-1)/3 % 2 == 1, s1 = setunion(s1, Set([(s[i]-1)/3]))); ); return(s1); }
a(n) = { my(s = Set([1])); for(k = 1, n, s = nextSet(s); ); return(sum(i=1,#s,s[i])); }

Extensions

More terms from David A. Corneth, Sep 15 2020

A171619 Primes in A171490.

Original entry on oeis.org

5, 7, 29, 67, 83, 109, 173, 239, 241, 251, 293, 313, 337, 367, 571, 613, 769, 821, 877, 941, 947, 1031, 1069, 1103, 1511, 1693, 1759, 1901, 2011
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 13 2009

Keywords

Comments

Terms of sequence are primes in growing order where smallest number of steps m to reach 1 in "3x+1" (or Collatz) problem is a prime too.

Examples

			(1) 1st Collatz sequence with 5=prime(3) steps starts with 5=prime(3): 5-16-8-4-2-1, gives a(1)=5.
(2) 1st Collatz sequence with 7=prime(4) steps starts with 3=prime(2): 3-10-5-16-8-4-2-1, gives a(2)=7.
(3) 1st Collatz sequence with 29=prime(10) steps starts with 43=prime(14): 43-130-65-196-98-49-148-74-37-112-56-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1, gives a(3)=29.
(4) List of prime steps m for above a(n): 5, 3, 43, 167, 233, 41, 937, 14831, 9887, 7963, 73063, 45127, 78791, 225023, 6956969, 10998599, 126357223, 859130059, 2845683047, 322623647, 95592191, 8363817307, 28677246203, 38590505339, 35521451596571, 478672174364191, 1168778549494463, 6376392739978081, 103147916159472367.

References

  • R. K. Guy, "Collatz's Sequence" in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994.
  • Clifford A. Pickover, Wonders of Numbers, Oxford University Press, pp. 116-118, 2001.
  • Guenther J. Wirsching, The Dynamical System Generated by the 3n+1 Function, Springer-Verlag, Berlin, 1998.

Links

Crossrefs

Cf. A070905, A033491, A088975, A126727, A060565, A171490.

Extensions

Missing term a(7)=173 inserted by Georg Fischer, Oct 26 2022
a(23)-a(29) (using Eric Roosendaal's data) by Tyler Busby, Feb 11 2023
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