A335911 -id:A335911 - cp's OEIS frontend

A038550 Products of an odd prime and a power of two (sorted).

3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 122, 124, 127

Offset: 1

Tom Verhoeff

2007. For example, 37 = 18 + 19; 48 = 15 + 16 + 17; 56 = 5 + 6 + 7 + 8 + 9 + 10 + 11. (Edited by M. F. Hasler, Aug 29 2020: "positive" was missing here. If nonnegative integers are allowed, none of the triangular numbers 3, 6, 10, ... would be in the corresponding sequence. If negative integers are also allowed, it would only have powers of 2 (A000079) which are the only positive integers not the sum of more than one consecutive positive integers, since any x > 0 is the sum of 1-x through x.)
Numbers that are the difference of two triangular numbers in exactly two ways.
Numbers with largest odd divisor a prime number. - Juri-Stepan Gerasimov, Aug 16 2016
Numbers k for which A001222(A000265(k)) = 1. - Antti Karttunen, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A001227, A000265, A237593, A279387.
Subsequences: A334101, A335431, A335911.
Subsequence of A093641 and of A336101.

a038550 n = a038550_list !! (n-1)
a038550_list = filter ((== 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

Select[Range[127],DivisorSigma[0,Max[Select[Divisors[#],OddQ]]]-1==1&] (* Jayanta Basu, Apr 30 2013 *)
fQ[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; (Length[p] == 2 && p[[1]] == 2 && e[[2]] == 1) || (Length[p] == 1 && p[[1]] > 2 && e[[1]] == 1)]; Select[Range[2, 127], fQ] (* T. D. Noe, Apr 30 2013 *)
upto=150;Module[{pmax=PrimePi[upto],tmax=Ceiling[Log[2,upto]]}, Select[ Sort[ Flatten[ Outer[ Times, Prime[ Range[ 2,pmax]], 2^Range[0,tmax]]]],#<=upto&]] (* Harvey P. Dale, Oct 18 2013 *)
Flatten@Position[PrimeQ[BitShiftRight[#, IntegerExponent[#, 2]]&/@Range[#]], True]&@127 (* Federico Provvedi, Dec 14 2021 *)

is(n)=isprime(n>>valuation(n,2)) \\ Charles R Greathouse IV, Apr 30 2013

A001227(a(n)) = 2. - Reinhard Zumkeller, May 01 2012
a(n) ~ 0.5 n log n. - Charles R Greathouse IV, Apr 30 2013
A000265(a(n)) is a prime. - Juri-Stepan Gerasimov, Aug 16 2016
Sum_{n>=1} 1/a(n)^s = (2^s*P(s) - 1)/(2^s - 1), for s > 1, where P is the prime zeta function. - Amiram Eldar, Dec 19 2020

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Sep 15 2007

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.

A335910 Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 6, 11, 27, 16, 7, 13, 33, 81, 32, 10, 15, 37, 99, 243, 64, 12, 18, 39, 107, 297, 729, 128, 14, 19, 43, 109, 321, 891, 2187, 256, 17, 21, 45, 111, 327, 963, 2673, 6561, 512, 20, 22, 53, 117, 333, 981, 2889, 8019, 19683, 1024, 24, 23, 54, 121, 351, 999, 2943, 8667, 24057, 59049, 2048, 28, 25, 55, 129, 363, 1053, 2997, 8829, 26001, 72171, 177147

Offset: 0

Antti Karttunen, Jul 01 2020

Array is read by descending antidiagonals with (n,k) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... where A(n,k) is the (k+1)-th solution x to A335885(x) = n. The row indexing (n) starts from 0, and column indexing (k) also from 0.
For any odd prime p that appears on row n, either p-1 or p+1 appears on row n-1.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A335885 is completely additive.

			The top left corner of the array:
n\k |     0      1      2      3      4      5      6      7      8      9
----+--------------------------------------------------------------------------
  0 |     1,     2,     4,     8,    16,    32,    64,   128,   256,   512, ...
  1 |     3,     5,     6,     7,    10,    12,    14,    17,    20,    24, ...
  2 |     9,    11,    13,    15,    18,    19,    21,    22,    23,    25, ...
  3 |    27,    33,    37,    39,    43,    45,    53,    54,    55,    57, ...
  4 |    81,    99,   107,   109,   111,   117,   121,   129,   131,   135, ...
  5 |   243,   297,   321,   327,   333,   351,   363,   387,   393,   405, ...
  6 |   729,   891,   963,   981,   999,  1053,  1089,  1161,  1177,  1179, ...
  7 |  2187,  2673,  2889,  2943,  2997,  3159,  3267,  3483,  3531,  3537, ...
  8 |  6561,  8019,  8667,  8829,  8991,  9477,  9801, 10449, 10593, 10611, ...
  9 | 19683, 24057, 26001, 26487, 26973, 28431, 29403, 31347, 31779, 31833, ...

Index entries for sequences that are permutations of the natural numbers

Cf. A335885.
Cf. A000079, A335911, A335912 (rows 0-2), A000244 (is very like the leftmost column).
Cf. also arrays A334100, A335430.

up_to = 78-1; \\ = binomial(12+1,2)-1.
memoA335885 = Map();
A335885(n) = if(1==n,0,my(v=0); if(mapisdefined(memoA335885,n,&v), v, my(f=factor(n)); v = 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))))); mapput(memoA335885,n,v); (v)));
memoA335910sq = Map();
A335910sq(n, k) = { my(v=0); if((0==k), v = -1, if(!mapisdefined(memoA335910sq,[n,k-1],&v), v = A335910sq(n, k-1))); for(i=1+v,oo,if(A335885(1+i)==n,mapput(memoA335910sq,[n,k],i); return(1+i))); };
A335910list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A335910sq(col,(a-(col))))); (v); };
v335910 = A335910list(up_to);
A335910(n) = v335910[1+n];
for(n=0,up_to,print1(A335910(n),", "));

A335912 Numbers k for which A335885(k) = 2.

Original entry on oeis.org

9, 11, 13, 15, 18, 19, 21, 22, 23, 25, 26, 29, 30, 35, 36, 38, 41, 42, 44, 46, 47, 49, 50, 51, 52, 58, 60, 61, 67, 70, 72, 76, 79, 82, 84, 85, 88, 92, 93, 94, 97, 98, 100, 102, 104, 113, 116, 119, 120, 122, 134, 137, 140, 144, 152, 155, 158, 164, 168, 170, 176, 184, 186, 188, 191, 193, 194, 196, 200, 204, 208, 217, 223

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers n such that when we start from k = n, and apply in some combination the nondeterministic maps k -> k - k/p and k -> k + k/p, (where p can be any of the odd prime factors of k), then for some combination we can reach a power of 2 in exactly two steps (but with no combination allowing 0 or 1 steps).

			For n = 70 = 2*5*7, if we first take p = 7 and apply the map n -> n + (n/p), we obtain 80 = 2^4 * 5. We then take p = 5, and apply the map n -> n - (n/p), to obtain 80-16 = 64 = 2^16. Thus we reached a power of 2 in two steps (and there are no shorter paths), therefore 70 is present in this sequence.
For n = 769, which is a prime, 769 - (769/769) yields 768 = 3 * 256. For 768 we can then apply either map to obtain a power of 2, as 768 - (768/3) = 512 = 2^9 and 768 + (768/3) = 1024 = 2^10. On the other hand, 769 + (769/769) = 770 and A335885(770) = 4, so that route would not lead to any shorter paths, therefore 769 is a term of this sequence.

Row 2 of A335910.
Cf. A000265, A335911, A335885.
Subsequences of semiprimes (union gives all odd semiprimes present): A144482, A333788, A336115.
Cf. also A334092, A335874, A336116, A336117 and A334102, A335882, A336122.

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))))); };
isA335912(n) = (2==A335885(n));

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A038550 Products of an odd prime and a power of two (sorted).

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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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A335910 Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

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A335912 Numbers k for which A335885(k) = 2.

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