A334101 -id:A334101 - cp's OEIS frontend

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

3, 5, 17, 257, 65537

Offset: 1

It is conjectured that there are only 5 terms. Currently it has been shown that 2^(2^k) + 1 is composite for 5 <= k <= 32 (see Eric Weisstein's Fermat Primes link). - Dmitry Kamenetsky, Sep 28 2008
No Fermat prime is a Brazilian number. So Fermat primes belong to A220627. For proof see Proposition 3 page 36 in "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
This sequence and A001220 are disjoint (see "Other theorems about Fermat numbers" in Wikipedia link). - Felix Fröhlich, Sep 07 2014
Numbers n > 1 such that n * 2^(n-2) divides (n-1)! + 2^(n-1). - Thomas Ordowski, Jan 15 2015
From Jaroslav Krizek, Mar 17 2016: (Start)
Primes p such that phi(p) = 2*phi(p-1); primes from A171271.
Primes p such that sigma(p-1) = 2p - 3.
Primes p such that sigma(p-1) = 2*sigma(p) - 5.
For n > 1, a(n) = primes p such that p = 4 * phi((p-1) / 2) + 1.
Subsequence of A256444 and A256439.
Conjectures:
1) primes p such that phi(p) = 2*phi(p-2).
2) primes p such that phi(p) = 2*phi(p-1) = 2*phi(p-2).
3) primes p such that p = sigma(phi(p-2)) + 2.
4) primes p such that phi(p-1) + 1 divides p + 1.
5) numbers n such that sigma(n-1) = 2*sigma(n) - 5. (End)
Odd primes p such that ratio of the form (the number of nonnegative m < p such that m^q == m (mod p))/(the number of nonnegative m < p such that -m^q == m (mod p)) is a divisor of p for all nonnegative q. - Juri-Stepan Gerasimov, Oct 13 2020
Numbers n such that tau(n)*(number of distinct ratio (the number of nonnegative m < n such that m^q == m (mod n))/(the number of nonnegative m < n such that -m^q == m (mod n))) for nonnegative q is equal to 4. - Juri-Stepan Gerasimov, Oct 22 2020
The numbers of primitive roots for the five known terms are 1, 2, 8, 128, 32768. - Gary W. Adamson, Jan 13 2022
Prime numbers such that every residue is either a primitive root or a quadratic residue. - Keith Backman, Jul 11 2022
If there are only 5 Fermat primes, then there are only 31 odd order groups which have a 2-group automorphism group. See the Miles Englezou link for a proof. - Miles Englezou, Mar 10 2025

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 137-141, 197.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see Table 1, p. 458.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
Richard K. Guy, Unsolved Problems in Number Theory, A3.
Hardy and Wright, An Introduction to the Theory of Numbers, bottom of page 18 in the sixth edition, gives an heuristic argument that this sequence is finite.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 7, 70.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

Anas AbuDaqa, Amjad Abu-Hassan, and Muhammad Imam, Taxonomy and Practical Evaluation of Primality Testing Algorithms, arXiv:2006.08444 [cs.CR], 2020.
Cyril Banderier, Pepin's Criterion For Fermat Numbers (in French)
Kent D. Boklan and John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016.
P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang, On modular categories, arXiv:1310.7050 [math.QA], 2013-2015.
C. K. Caldwell, The Prime Glossary, Fermat number
Miles Englezou, Proof that 5 Fermat primes implies 31 odd order groups with 2-group automorphism groups
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Salah Eddine Rihane, Chèfiath Awero Adegbindin, and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 65.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
Yaryna Stelmakh, Homeomorphisms of the space of non-zero integers with the Kirch topology, arXiv:2101.04676 [math.GN], 2020.
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Pepin's Test
Wikipedia, Fermat number
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A000215, A001146, A159611, A220627, A220570.
Subsequence of A147545 and of A334101. Cf. also A333788, A334092.
Cf. A045544.

[2^(2^n)+1 : n in [0..4] | IsPrime(2^(2^n)+1)]; // Arkadiusz Wesolowski, Jun 09 2011

Select[Table[2^(2^n) + 1, {n, 0, 4}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

for(i=0,10, isprime(2^2^i+1) && print1(2^2^i+1,", ")) \\ M. F. Hasler, Nov 21 2009

[2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # G. C. Greubel, Mar 07 2019

a(n+1) = A180024(A049084(a(n))). - Reinhard Zumkeller, Aug 08 2010

a(n) = 1 + A001146(n-1), if 1 <= n <= 5. - Omar E. Pol, Jun 08 2018

A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

Original entry on oeis.org

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

Offset: 1

Ali Sada and Robert G. Wilson v, Feb 28 2020

From Antti Karttunen, Apr 07 2020: (Start)
Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).
Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.
The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.
For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.
(End)
A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is {12, 8}, taking 2 iterations to reach 8 = 2^3. So a(15) is 2.
From _Antti Karttunen_, Apr 07 2020: (Start)
Considering all possible paths from 15 to 1 nondeterministic map k -> k-(k/p), where p can be any prime factor of k, we obtain the following graph:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.
It can be seen that there's also alternative route to 8 via 10 (with 10 = 15-(15/3), where 3 is not the largest prime factor of 15), but it's not any shorter than the route via 12.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree labeling the index n, with a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value in a(n) for n = 1..16383.

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for positions of 0 .. 6 in this sequence, and also array A334100.
Cf. A334099 (a right inverse, positions of the first occurrence of each n).
Cf. A334091 (first differences), A335429 (partial sums).
Cf. also A331410 (analogous sequence when using the map k -> k + k/p), A334861, A335877 (their sums and differences), see also A335878 and A335884, A335885.

a[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] -1; Array[a, 100]

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 07 2020

up_to = 2^24;
A329697list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = if(!bitand(n,n-1),0,1+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v329697 = A329697list(up_to);
A329697(n) = v329697[n]; \\ Antti Karttunen, Apr 07 2020

A329697(n) = if(n<=2,0, if(isprime(n), A329697(n-1)+1, my(f=factor(n)); (apply(A329697, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, Apr 19 2020

From Antti Karttunen, Apr 07-19 2020: (Start)
a(1) = a(2) = 0; and for n > 2, a(p) = 1 + a(p-1) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [This is otherwise equal to the definition of A064097, except here we have a different initial condition, with a(2) = 0].
a(2n) = a(A000265(n)) = a(n).
a(p) = 1+a(p-1), for all odd primes p.
If A209229(n) == 1 [when n is a power of 2], a(n) = 0,
  otherwise a(n) = 1 + a(n-A052126(n)) = 1 + a(A171462(n)).
Equivalently, for non-powers of 2, a(n) = 1 + a(n-(n/A078701(n))),
or equivalently, for non-powers of 2, a(n) = 1 + Min a(n - n/p), for p prime and dividing n.
a(n) = A064097(n) - A064415(n), or equally, a(n) = A064097(n) - A054725(n), for n > 1.
a(A019434(n)) = 1, a(A334092(n)) = 2, a(A334093(n)) = 3, etc. for all applicable n.
For all n >= 0, a(A334099(n)) = a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = a(257^n) = a(65537^n) = n.
a(A122111(n)) = A334107(n), a(A225546(n)) = A334109(n).
(End)
From Antti Karttunen, Mar 16 2021: (Start)
a(n) = a(A336466(n)) + A087436(n) = A336396(n) + A087436(n).
a(A053575(n)) = A336469(n) = a(n) - A005087(n).
a(A147545(n)) = A000120(A147545(n)) - 1.
(End)

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

A334100 Square array where the row n lists all numbers k for which A329697(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 6, 9, 19, 16, 10, 11, 21, 43, 32, 12, 13, 23, 47, 127, 64, 17, 14, 27, 49, 129, 283, 128, 20, 15, 29, 57, 133, 301, 659, 256, 24, 18, 31, 59, 139, 329, 817, 1319, 512, 34, 22, 33, 63, 141, 343, 827, 1699, 3957, 1024, 40, 25, 35, 67, 147, 347, 839, 1787, 4079, 9227, 2048, 48, 26, 37, 69, 161, 361, 849, 1849, 4613, 9233, 21599

Offset: 1

Antti Karttunen, Apr 14 2020

Array is read by descending antidiagonals with (n,k) = (0,1), (0,2), (1,1), (0,3), (1,2), (2,1), ... where A(n,k) is the k-th solution x to A329697(x) = n. The row indexing (n) starts from 0, and column indexing (k) from 1.
Any odd prime that appears on row n is 1+{some term 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 A329697 is completely additive.
The binary weight (A000120) of any term on row n is at most 2^n.

			The top left corner of the array:
  n\k |    1     2     3     4     5     6     7     8     9    10
------+----------------------------------------------------------------
   0  |    1,    2,    4,    8,   16,   32,   64,  128,  256,  512, ...
   1  |    3,    5,    6,   10,   12,   17,   20,   24,   34,   40, ...
   2  |    7,    9,   11,   13,   14,   15,   18,   22,   25,   26, ...
   3  |   19,   21,   23,   27,   29,   31,   33,   35,   37,   38, ...
   4  |   43,   47,   49,   57,   59,   63,   67,   69,   71,   77, ...
   5  |  127,  129,  133,  139,  141,  147,  161,  163,  171,  173, ...
   6  |  283,  301,  329,  343,  347,  361,  379,  381,  383,  387, ...
   7  |  659,  817,  827,  839,  849,  863,  883,  889,  893,  903, ...
   8  | 1319, 1699, 1787, 1849, 1977, 1979, 1981, 2021, 2039, 2083, ...
   9  | 3957, 4079, 4613, 4903, 5097, 5179, 5361, 5377, 5399, 5419, ...
etc.
Note that the row 9 is the first one which begins with composite, as 3957 = 3*1319. The next such rows are row 15 and row 22. See A334099.

Index entries for sequences that are permutations of the natural numbers

Cf. A329697.
Cf. A334099 (the leftmost column).
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for the rows 0-6.
Cf. A019434, A334092, A334093, A334094, A334095, A334096 for the primes on the rows 1-6.
Cf. also irregular triangle A334111.

Block[{nn = 16, s}, s = Values@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, 2^nn]; Table[s[[#, k]] &[m - k + 1], {m, nn - Ceiling[nn/4]}, {k, m, 1, -1}]] // Flatten (* Michael De Vlieger, Apr 30 2020 *)

up_to = 105; \\ up_to = 1081; \\ = binomial(46+1,2)
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
memoA334100sq = Map();
A334100sq(n, k) = { my(v=0); if(!mapisdefined(memoA334100sq,[n,k-1],&v),if(1==k, v=0, v = A334100sq(n, k-1))); for(i=1+v,oo,if(A329697(i)==(n-1),mapput(memoA334100sq,[n,k],i); return(i))); };
A334100list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A334100sq(col,(a-(col-1))))); (v); };
v334100 = A334100list(up_to);
A334100(n) = v334100[n];

A334102 Numbers n for which A329697(n) == 2.

Original entry on oeis.org

7, 9, 11, 13, 14, 15, 18, 22, 25, 26, 28, 30, 36, 41, 44, 50, 51, 52, 56, 60, 72, 82, 85, 88, 97, 100, 102, 104, 112, 120, 137, 144, 164, 170, 176, 193, 194, 200, 204, 208, 224, 240, 274, 288, 289, 328, 340, 352, 386, 388, 400, 408, 416, 448, 480, 548, 576, 578, 641, 656, 680, 704, 769, 771, 772, 776, 800, 816, 832, 896, 960, 1096

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Numbers n for which A171462(n) = n-A052126(n) is in A334101.
Numbers k such that A000265(k) is either in A333788 or in A334092.
Each term is either of the form A334092(n)*2^k, for some n >= 1, and k >= 0, or a product of two terms of A334101, whether distinct or not.
Binary weight (A000120) of these terms is always either 2, 3 or 4. It is 2 for those terms that are of the form 9*2^k, 4 for the terms of the form p*q*2^k, where p and q are two distinct Fermat primes (A019434), and 3 for the both terms of the form A334092(n)*2^k, and for the terms of the form (p^2)*(2^k), where p is a Fermat prime > 3.

Row 2 of A334100.
Cf. A000120, A000265, A019434, A052126, A171462, A209229, A329697.
Cf. A333788 (a subsequence), A334092 (primes present), A334093 (primes that are 1 + some term in this sequence).
Squares of A334101 form a subsequence of this sequence. Squares of these numbers can be found (as a subset) in A334104, and the cubes in A334106.

A000265(n) = (n>>valuation(n,2));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2,n-1)); \\ Charfun for A019434, Fermat primes.
isA334102(n) = { n = A000265(n); if(isprime(n), isA019434(A000265(n-1)), if(bigomega(n)!=2,0,factorback(apply(isA019434,factor(n)[,1])))); };

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334102(n) = (2==A329697(n));

A334103 Numbers n for which A329697(n) == 3.

Original entry on oeis.org

19, 21, 23, 27, 29, 31, 33, 35, 37, 38, 39, 42, 45, 46, 53, 54, 55, 58, 61, 62, 65, 66, 70, 73, 74, 75, 76, 78, 83, 84, 89, 90, 92, 101, 103, 106, 108, 110, 113, 116, 119, 122, 123, 124, 125, 130, 132, 140, 146, 148, 150, 152, 153, 156, 166, 168, 178, 180, 184, 187, 202, 205, 206, 212, 216, 220, 221, 226, 232, 238, 241, 244

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Numbers n for which A171462(n) = n-A052126(n) is in A334102.

Among the first 2821 terms (terms < 2^31), there are terms with binary weights 2, 3, 4, 5, 6 and 8. For example, 33 is the first term with binary weight 2, and 255 is the first term with binary weight 8.

Antti Karttunen, Table of n, a(n) for n = 1..2821; all terms <= 2^31

Row 3 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334104, A334106.
Cf. A334093 (primes present), A334094.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334103(n) = (3==A329697(n));

A334104 Numbers m for which A329697(m) = 4.

Original entry on oeis.org

43, 47, 49, 57, 59, 63, 67, 69, 71, 77, 79, 81, 86, 87, 91, 93, 94, 95, 98, 99, 105, 107, 109, 111, 114, 115, 117, 118, 121, 126, 131, 134, 135, 138, 142, 143, 145, 149, 151, 154, 155, 157, 158, 159, 162, 165, 167, 169, 172, 174, 175, 179, 181, 182, 183, 185, 186, 188, 190, 195, 196, 198, 210, 214, 218, 219, 222, 225

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Squares of A334102 form a subsequence.

Among the first 12193 terms (terms < 2^31), there are terms with binary weights 2 - 16, except no terms with weight 13, 14 or 15. For example, 1025 is the first term with binary weight 2, and 65535 is the first term with binary weight 16.

			63 = 7*9 is a term as both 7 and 9 are terms of A334102.
65535 = 3*5*17*257 is a term as it is a product of four Fermat primes, thus in four steps all odd primes can be eliminated with p -> (p-1) map.

Antti Karttunen, Table of n, a(n) for n = 1..12193; all terms <= 2^31

Row 4 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334103, A334105, A334106.
Cf. A334094 (primes present).

Position[Array[Length@NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 225], 4][[All, 1]] (* Michael De Vlieger, Apr 30 2020 *)

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334104(n) = (4==A329697(n));

A334106 Numbers n for which A329697(n) == 6.

Original entry on oeis.org

283, 301, 329, 343, 347, 361, 379, 381, 383, 387, 399, 413, 417, 419, 423, 431, 437, 441, 463, 469, 473, 483, 487, 489, 491, 497, 509, 513, 517, 519, 523, 529, 531, 539, 547, 551, 553, 557, 559, 566, 567, 571, 573, 589, 591, 597, 599, 602, 603, 609, 611, 621, 627, 631, 633, 635, 637, 639, 643, 645, 649, 651, 653, 658, 665

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12096; all terms < 2^17

Row 6 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334103, A334104, A334105.
Cf. A334096 (primes present).

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334106(n) = (6==A329697(n));

A334105 Numbers m for which A329697(m) = 5.

Original entry on oeis.org

127, 129, 133, 139, 141, 147, 161, 163, 171, 173, 177, 189, 191, 197, 199, 201, 203, 207, 209, 211, 213, 215, 217, 223, 229, 231, 235, 237, 243, 245, 247, 253, 254, 258, 259, 261, 263, 266, 269, 271, 273, 277, 278, 279, 282, 285, 294, 295, 297, 299, 311, 315, 317, 319, 321, 322, 326, 327, 331, 333, 335, 341, 342, 345, 346, 349, 351

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

			127 = 63*2 + 1 is a term, as 127 is a prime and 63 is in A334104 as A329697(63) = 4.
2^32 -1 = 4294967295 = 3*5*17*257*65537 is a term as it is a product of five Fermat primes, thus in five steps all odd primes can be eliminated with p -> (p-1) map.
Likewise for 1442840405 = 5 * 17 * 257^3. (The first term with binary weight = 24).

Antti Karttunen, Table of n, a(n) for n = 1..50071; all terms < 2^31

Row 5 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334103, A334104, A334106.
Cf. A334095 (primes present).

Position[Array[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 360], 5][[All, 1]] (* Michael De Vlieger, Apr 30 2020 *)

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334105(n) = (5==A329697(n));

cp's OEIS Frontend

A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.

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A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

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

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

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A334100 Square array where the row n lists all numbers k for which A329697(k) = n, read by falling antidiagonals.

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A334102 Numbers n for which A329697(n) == 2.

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A334103 Numbers n for which A329697(n) == 3.

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