A335876 -id:A335876 - cp's OEIS frontend

A335431 Numbers of the form q*(2^k), where q is one of the Mersenne primes (A000668) and k >= 0.

3, 6, 7, 12, 14, 24, 28, 31, 48, 56, 62, 96, 112, 124, 127, 192, 224, 248, 254, 384, 448, 496, 508, 768, 896, 992, 1016, 1536, 1792, 1984, 2032, 3072, 3584, 3968, 4064, 6144, 7168, 7936, 8128, 8191, 12288, 14336, 15872, 16256, 16382, 24576, 28672, 31744, 32512, 32764, 49152, 57344, 63488, 65024, 65528, 98304, 114688, 126976, 130048, 131056, 131071

Offset: 1

Antti Karttunen, Jun 28 2020

nonn

Numbers of the form 2^k * ((2^p)-1), where p is one of the primes in A000043, and k >= 0.
Numbers k such that A000265(k) is in A000668.
Numbers k for which A331410(k) = 1.
Numbers k that themselves are not powers of two, but for which A335876(k) = k+A052126(k) is [a power of 2].
Conjecture: This sequence gives all fixed points of map n -> A332214(n) and its inverse n -> A332215(n). See also notes in A029747 and in A163511.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000043, A000396 (even terms form a subsequence), A000668 (primes present), A335882, A341622.
Row 1 of A335430.
Positions of 1's in A331410, in A364260, and in A364251 (characteristic function).
Subsequence of A054784.
Cf. also A029747, A163511, A173898, A332214, A332215, A334101, A335876.

qs = 2^MersennePrimeExponent[Range[6]] - 1; max = qs[[-1]]; Reap[Do[n = 2^k*q; If[n <= max, Sow[n]], {k, 0, Log2[max]}, {q, qs}]][[2, 1]] // Union (* Amiram Eldar, Feb 18 2021 *)

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA335431(n) = isA000668(A000265(n));

A332214(a(n)) = A332215(a(n)) = a(n) for all n.

Sum_{n>=1} 1/a(n) = 2 * A173898 = 1.0329083578... - Amiram Eldar, Feb 18 2021

A335885 The length of a shortest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 1, 0, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 2, 2, 1, 3, 0, 3, 3, 2, 1, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 1, 4, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 1, 2, 2, 4, 2, 3, 2, 3, 2, 3

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

The length of a shortest path from n to a power of 2, when using the transitions x -> A171462(x) and x -> A335876(x) in any order.

a((2^e)-1) is equal to A046051(e) = A001222((2^e)-1) when e is either a Mersenne exponent (in A000043), or some other number: 1, 4, 6, 8, 16, 32. For example, 32 is present because 2^32 - 1 = 4294967295 = 3*5*17*257*65537, a squarefree product of five known Fermat primes. - Antti Karttunen, Aug 11 2020

			A335876(67) = 68, and A171462(68) = 64 = 2^6, and this is the shortest path from 67 to a power of 2, thus a(67) = 2.
A171462(15749) = 15748, A335876(15748) = 15872, A335876(15872) = 16384 = 2^14, and this is the shortest path from 15749 to a power of 2, thus a(15749) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000043, A046051, A052126, A171462, A335876, A335875, A335881, A335884, A335904, A335907.

Cf. A000079, A335911, A335912 (positions of 0's, 1's and 2's in this sequence) and array A335910.

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };

\\ Or empirically as:
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
A209229(n) = (n && !bitand(n,n-1));
A335885(n) = if(A209229(n),0,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A171462(u); if(A209229(a), return(k)); b = A335876(u); if(A209229(b), return(k)); newxs = setunion([a],newxs); newxs = setunion([b],newxs)); xs = newxs));

Fully additive with a(2) = 0, and a(p) = 1+min(a(p-1), a(p+1)), for odd primes p.
For all n >= 1, a(n) <= A335875(n) <= A335881(n) <= A335884(n) <= A335904(n).
For all n >= 0, a(A000244(n)) = n, and these also seem to give records.

A335884 The length of a longest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 3, 1, 3, 2, 3, 0, 3, 2, 3, 2, 3, 3, 4, 1, 4, 3, 3, 2, 4, 3, 4, 0, 4, 3, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 4, 4, 5, 1, 4, 4, 4, 3, 4, 3, 5, 2, 4, 4, 5, 3, 5, 4, 4, 0, 5, 4, 5, 3, 5, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 2, 4, 4, 5, 3, 5, 4, 5, 3, 5, 4, 5, 4, 5, 5, 5, 1, 5, 4, 5, 4, 5, 4, 5, 3, 5

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

The length of a longest path from n to a power of 2, when using the transitions x -> A171462(x) and x -> A335876(x).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A052126, A171462, A335875, A335876, A335881, A335885, A335904, A335908.

Cf. A335883 (position of the first occurrence of each n).

A335884(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+max(A335884(f[k,1]-1),A335884(f[k,1]+1))))); };

\\ Or empirically as:
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
A209229(n) = (n && !bitand(n,n-1));
A335884(n) = if(A209229(n),0,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, newxs=Set([]); if(!#xs, return(k-1)); for(i=1,#xs,u = xs[i]; a = A171462(u); if(!A209229(a), newxs = setunion([a],newxs)); b = A335876(u); if(!A209229(b), newxs = setunion([b],newxs))); xs = newxs));

Fully additive with a(2) = 0, and a(p) = 1+max(a(p-1), a(p+1)), for odd primes p.
For all n >= 1, A335904(n) >= a(n) >= A335881(n) >= A335875(n) >= A335885(n).
For all n >= 0, a(A335883(n)) = n.

A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 4, 1, 4, 2, 3, 0, 3, 2, 5, 2, 3, 4, 6, 1, 4, 4, 3, 2, 6, 3, 4, 0, 5, 3, 4, 2, 8, 5, 5, 2, 6, 3, 8, 4, 4, 6, 8, 1, 4, 4, 4, 4, 8, 3, 6, 2, 6, 6, 10, 3, 8, 4, 4, 0, 6, 5, 9, 3, 7, 4, 7, 2, 11, 8, 5, 5, 6, 5, 8, 2, 4, 6, 10, 3, 5, 8, 7, 4, 9, 4, 6, 6, 5, 8, 7, 1, 6, 4, 6, 4, 9, 4, 9, 4, 5

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000244, A052126, A087436, A171462, A335875, A335876, A335881, A335884, A335885, A335905, A335906, A335915, A336118.

A335904(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A335904(f[k,1]-1)+A335904(f[k,1]+1)))); };

Totally additive with a(2) = 0, and for odd primes p, a(p) = 1 + a(p-1) + a(p+1).
a(n) = A336118(n) +  A087436(n).
For all n >= 1, a(A335915(n)) = A336118(n).
For all n >= 1, a(n) >= A335884(n) >= A335881(n) >= A335875(n) >= A335885(n).
For all n >= 0, a(3^n) = n.

A335905 Number of distinct integers encountered on all possible paths from n to any first encountered powers of 2 (that are excluded from the count), when using the transitions x -> x - (x/p) and x -> x + (x/p) in any order, where p is the largest prime dividing x.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 3, 2, 3, 1, 3, 2, 4, 0, 4, 3, 5, 2, 4, 3, 4, 1, 6, 3, 6, 2, 6, 4, 5, 0, 5, 4, 6, 3, 6, 5, 5, 2, 6, 4, 7, 3, 7, 4, 5, 1, 6, 6, 7, 3, 9, 6, 7, 2, 8, 6, 7, 4, 6, 5, 7, 0, 7, 5, 7, 4, 6, 6, 7, 3, 7, 6, 9, 5, 8, 5, 7, 2, 10, 6, 7, 4, 9, 7, 9, 3, 10, 7, 7, 4, 8, 5, 11, 1, 7, 6, 8, 6, 11, 7, 10, 3, 9

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

			From 9 one can reach with the transitions x -> A171462(x) (leftward arrow) and x -> A335876(x) (rightward arrow) the following three numbers, when one doesn't expand any power of 2 (in this case, 4, 8 and 16, that are not included in the count) further:
       9
      / \
     6   12
    / \ / \
  (4) (8) (16)
thus a(9) = 3.
From 10 one can reach with the transitions x -> A171462(x) and x -> A335876(x) the following two numbers (10 & 12), when one doesn't expand any powers of 2 (8 and 16 in this case, not counted) further:
  10
   |\
   | \
   | 12
   | /\
   |/  \
  (8)  (16)
thus a(10) = 2.
For n = 9, the numbers encountered are 6, 9, 12, thus a(9) = 3.
For n = 67, the numbers encountered are 48, 60, 66, 67, 68, 72, 96, thus a(67) = 7.
For n = 105, the numbers encountered are 48, 72, 90, 96, 105, 108, 120, 144, 192, thus a(105) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A171462, A335876, A335906.

Cf. also A332809, A335884, A335885, A335904.

A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
A209229(n) = (n && !bitand(n,n-1));
A335905(n) = if(A209229(n),0,my(xs=Set([n]),allxs=xs,newxs,a,b,u); for(k=1,oo, newxs=Set([]); if(!#xs, return(#allxs)); allxs = setunion(allxs,xs); for(i=1,#xs,u = xs[i]; a = A171462(u); if(!A209229(a), newxs = setunion([a],newxs)); b = A335876(u); if(!A209229(b), newxs = setunion([b],newxs))); xs = newxs));

A335906 Number of distinct integers encountered on all possible paths from n to any first encountered powers of 2 (that are included in the count), when using the transitions x -> x - (x/p) and x -> x + (x/p) in any order, where p is the largest prime dividing x.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 4, 1, 6, 4, 5, 3, 5, 4, 7, 1, 7, 6, 8, 4, 7, 5, 6, 3, 9, 5, 10, 4, 9, 7, 8, 1, 8, 7, 9, 6, 9, 8, 8, 4, 9, 7, 10, 5, 11, 6, 7, 3, 9, 9, 11, 5, 13, 10, 10, 4, 12, 9, 10, 7, 9, 8, 11, 1, 10, 8, 10, 7, 9, 9, 10, 6, 10, 9, 13, 8, 11, 8, 10, 4, 15, 9, 10, 7, 13, 10, 13, 5, 14, 11, 10, 6, 12, 7, 15, 3, 10, 9, 12, 9, 15, 11, 14, 5, 13

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

			From 9 one can reach with the transitions x -> A171462(x) (leftward arrow) and x -> A335876(x) (rightward arrow) the following six numbers, when one doesn't expand any power of 2 further:
       9
      / \
     6   12
    / \ / \
   4   8   16
thus a(9) = 6.
From 10 one can reach with the transitions x -> A171462(x) and x -> A335876(x) the following for numbers, when one doesn't expand any power of 2 further:
  10
   |\
   | \
   | 12
   | /\
   |/  \
   8   16
thus a(10) = 4.
From 15 one can reach with the transitions x -> A171462(x) and x -> A335876(x) the following seven numbers, when one doesn't expand any power of 2 further:
        15
       /  \
      /    \
    12<----18
    / \      \
   /   \      \
  8     16<----24
                \
                 \
                  32
thus a(15) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A171462, A335876, A335905.

A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
A209229(n) = (n && !bitand(n,n-1));
A335906(n) = { my(xs=Set([n]),allxs=xs,newxs,a,b,u); for(k=1,oo, newxs=Set([]); if(!#xs, return(#allxs)); allxs = setunion(allxs,xs); for(i=1,#xs,u = xs[i]; if(!A209229(u), newxs = setunion([A171462(u)],newxs); newxs = setunion([A335876(u)],newxs))); xs = newxs); };

cp's OEIS Frontend

A335431 Numbers of the form q*(2^k), where q is one of the Mersenne primes (A000668) and k >= 0.

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A335885 The length of a shortest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

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A335884 The length of a longest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

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A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

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A335905 Number of distinct integers encountered on all possible paths from n to any first encountered powers of 2 (that are excluded from the count), when using the transitions x -> x - (x/p) and x -> x + (x/p) in any order, where p is the largest prime dividing x.

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A335906 Number of distinct integers encountered on all possible paths from n to any first encountered powers of 2 (that are included in the count), when using the transitions x -> x - (x/p) and x -> x + (x/p) in any order, where p is the largest prime dividing x.

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