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 1-5 of 5 results.

A341515 The Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

1, 5, 2, 15, 3, 11, 5, 45, 4, 125, 7, 33, 11, 245, 6, 135, 13, 77, 17, 375, 10, 605, 19, 99, 9, 845, 8, 735, 23, 17, 29, 405, 14, 1445, 15, 231, 31, 1805, 22, 1125, 37, 1331, 41, 1815, 12, 2645, 43, 297, 25, 275, 26, 2535, 47, 539, 21, 2205, 34, 4205, 53, 51, 59, 4805, 20, 1215, 33, 1859, 61, 4335, 38, 3125, 67, 693
Offset: 1

Views

Author

Antti Karttunen, Feb 14 2021

Keywords

Comments

Collatz-conjecture can be formulated via this sequence by postulating that all iterations of a(n), starting from any n > 1, will eventually reach the cycle [2, 5, 3].

Links

Crossrefs

Cf. A005940, A006370, A064989, A156552, A329603, A341510, A347115 (Möbius transform),
Sequences related to iterations of this sequence: A352890, A352891, A352892, A352893, A352894, A352896, A352897, A352898, A352899.
Cf. A341516 (a variant).

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));

Formula

If n is odd, then a(n) = A064989(n), otherwise a(n) = A329603(n) = A341510(n,2*n).
a(n) = A005940(1+A006370(A156552(n))).

A352892 Next even term in the trajectory of map x -> A341515(x), when starting from x=n; a(1) = 1. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

1, 2, 2, 6, 2, 2, 2, 12, 4, 8, 2, 14, 2, 18, 6, 24, 2, 6, 2, 54, 10, 50, 2, 28, 4, 98, 8, 150, 2, 2, 2, 48, 14, 242, 6, 70, 2, 338, 22, 108, 2, 8, 2, 294, 12, 578, 2, 56, 4, 20, 26, 726, 2, 12, 10, 300, 34, 722, 2, 26, 2, 1058, 20, 96, 14, 18, 2, 1014, 38, 32, 2, 140, 2, 1682, 18, 1734, 6, 50, 2, 216, 16, 1922, 2, 686
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2022

Keywords

Links

Crossrefs

Cf. A000040, A005940, A064989, A156552, A329603, A341515, A348717.
Cf. also A139391, A352893, A352894, A352896, A352897, A352898, A352899, A353267.
Coincides with A353268 on even n, and with A348717 on odd n.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
  • PARI
    A352892(n) = if(1==n, n, n = A341515(n); while(n%2, n = A341515(n)); (n)); \\ A slower alternative.

Formula

a(n) = A348717(A341515(n)).
For all n >= 1, a(2n) = A353268(2n), a(2n-1) = A348717(2n-1).
a(p) = 2 for all primes p.
For n > 1, a(n) = A005940(1+A139391(A156552(n))).

A352894 Number of iterations of map x -> A352892(x) needed to reach x < n when starting from x=n, or 0 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 35, 1, 40, 1, 2, 1, 5, 1, 5, 1, 1, 1, 2, 1, 6, 1, 34, 1, 1, 1, 2, 1, 1, 1, 33, 1, 6, 1, 1, 1, 17, 1, 35, 1, 1, 1, 6, 1, 1, 1, 3, 1, 35, 1, 4, 1, 1, 1, 5, 1, 10, 1, 3, 1, 10, 1, 4, 1, 1, 1, 6, 1, 24, 1, 34, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2022

Keywords

Links

Crossrefs

Cf. A000040, A005940, A064989, A139391, A156552, A286380, A329603, A341515, A352890, A352891, A352892, A352893.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
A352894(n) = if(n<=2, 0, my(k=0,x=n); while(x>=n, x = A352892(x); k++); (k));

Formula

a(2n+1) = 1 for n >= 1.
For n >= 1, a(n) <= A352891(n).
For n >= 1, a(n) <= A352893(n).

A352890 Number of iterations of map x -> A341515(x) needed to reach x <= 2 when starting from x=n, or -1 if such number is never reached. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 7, 2, 5, 3, 16, 8, 19, 4, 14, 5, 12, 6, 17, 6, 9, 7, 20, 20, 26, 8, 15, 9, 27, 17, 13, 9, 7, 10, 106, 13, 121, 7, 111, 11, 122, 27, 34, 12, 21, 13, 27, 15, 35, 14, 104, 10, 23, 28, 28, 15, 18, 21, 102, 122, 36, 16, 29, 17, 156, 21, 107, 14, 14, 18, 122, 123, 109, 19, 112, 20, 113, 10, 123, 8, 28, 21, 35
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2022

Keywords

Comments

The unbroken ray in the scatter plot corresponds to primes.

Links

Crossrefs

Cf. A000040, A005940, A006577, A064989, A156552, A329603, A341515.
Cf. also A352891, A352892, A352893, A352894.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A352890(n) = { my(k=0); while(n>2, n = A341515(n); k++); (k); };

Formula

If n <= 2, a(n) = 0, otherwise a(n) = 1 + a(A341515(n)).
For n > 1, a(n) = A006577(A156552(n)).
For n >= 1, a(A000040(n)) = n-1.
For n >= 1, a(n) >= A352891(n).
For n >= 1, a(n) >= A352893(n).

A352899 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A352892(n), except f(n) = -n when <= 2.

Original entry on oeis.org

1, 2, 3, 4, 3, 3, 3, 5, 6, 7, 3, 8, 3, 9, 4, 10, 3, 4, 3, 11, 12, 13, 3, 14, 6, 15, 7, 16, 3, 3, 3, 17, 8, 18, 4, 19, 3, 20, 21, 22, 3, 7, 3, 23, 5, 24, 3, 25, 6, 26, 27, 28, 3, 5, 12, 29, 30, 31, 3, 27, 3, 32, 26, 33, 8, 9, 3, 34, 35, 36, 3, 37, 3, 38, 9, 39, 4, 13, 3, 40, 41, 42, 3, 43, 21, 44, 45, 46, 3, 8, 12
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2022

Keywords

Comments

Restricted growth sequence transform of function f(n) = -n if n < 3, and otherwise f(n) = A352892(n).
For all i, j:
A305801(i) = A305801(j) => A352898(i) = A352898(j) => a(i) = a(j),
a(i) = a(j) => A352893(i) = A352893(j),
a(i) = a(j) => A352896(i) = A352896(j).

Links

Crossrefs

Cf. A305801, A329603, A341515, A348717, A352892, A352893, A352896, A352898.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
Aux352899(n) = if(n<=2,-n,A352892(n));
v352899 = rgs_transform(vector(up_to, n, Aux352899(n)));
A352899(n) = v352899[n];
Showing 1-5 of 5 results.

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