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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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

Views

Author

Antti Karttunen, Apr 14 2020

Keywords

Comments

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.

Crossrefs

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.

Programs

A334101 Numbers of the form q*(2^k), where q is one of the Fermat primes and k >= 0; Numbers n for which A329697(n) == 1.

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 20, 24, 34, 40, 48, 68, 80, 96, 136, 160, 192, 257, 272, 320, 384, 514, 544, 640, 768, 1028, 1088, 1280, 1536, 2056, 2176, 2560, 3072, 4112, 4352, 5120, 6144, 8224, 8704, 10240, 12288, 16448, 17408, 20480, 24576, 32896, 34816, 40960, 49152, 65537, 65792, 69632, 81920, 98304, 131074, 131584, 139264
Offset: 1

Views

Author

Antti Karttunen, Apr 14 2020

Keywords

Comments

Numbers k that themselves are not powers of two, but for which A171462(k) = k-A052126(k) is [a power of 2].
Numbers k such that A000265(k) is in A019434.
Squares of these numbers can be found (as a subset) in A334102, and the cubes (as a subset) in A334103.

Crossrefs

Row 1 of A334100.
Cf. A000120, A000265, A052126, A171462, A209229, A329697, A334102, A334103.
Cf. A019434 (primes present), A007283, A020714, A110287 (other subsequences).
Subsequence of A018900.

Programs

Formula

For all n, A000120(a(n)) = 2.

A074781 Primes of the form p*2^k + 1 for any k and any prime p.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 149, 167, 173, 179, 193, 227, 233, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 449, 467, 479, 503, 509, 557, 563, 569, 587, 593, 641, 653, 719, 769, 773, 797, 809, 839, 857, 863, 887
Offset: 1

Views

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

Keywords

Comments

From Bernard Schott, Dec 14 2020: (Start)
Equivalently, primes p such that the ratio (p-1)/gpf(p-1) = 2^k where gpf(m) is the greatest prime factor of m, A006530.
Paul Erdős asked if there are infinitely many primes p in this sequence (see R. K. Guy reference). (End)

Examples

			3 = 2*2^0+1 is a term and 2/2 = 1 = 2^0.
7 = 3*2^1+1 is a term and 6/3 = 2 = 2^1.
13 = 3*2^2+1 is a term and 12/3 = 4 = 2^2.
41 = 5*2^3+1 is a term and 40/5 = 8 = 2^3.
113 = 7*2^4+1 is a term and 112/7 = 16 = 2^4.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Links

Crossrefs

Cf. A000079, A006093, A006530, A052126, A339464.
Cf. other ratios : A339463, A339465, A339466.
Subsequences: A039687, A051900, A058500 (this sequence without the Fermat primes), A090866, A147545,

Programs

  • Maple
    alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:
3, op(select(is_a, [$3..919])); # Peter Luschny, Dec 14 2020
  • Mathematica
    Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)

A348041 Square array read by antidiagonals. A(n,k) is the nearest common ancestor of n and k in the Doudna tree (A005940).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 2, 2, 3, 2, 1, 1, 2, 3, 2, 5, 2, 3, 2, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 2, 2, 4, 5, 6, 5, 4, 2, 2, 1, 1, 2, 3, 4, 2, 3, 3, 2, 4, 3, 2, 1, 1, 2, 3, 2, 2, 2, 7, 2, 2, 2, 3, 2, 1, 1, 2, 3, 2, 5, 2, 2, 2, 2, 5, 2, 3, 2, 1
Offset: 1

Views

Author

Antti Karttunen and Peter Munn, Sep 27 2021

Keywords

Comments

Array is symmetric and is read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .
Also the nearest common ancestor of n and k in the tree depicted in A163511 (the mirror image of the Doudna tree).
The first fork in the Doudna tree separates numbers divisible by the square of their largest prime factor (on one main branch) from other numbers greater than 2 (on the other main branch). If n and m are on different main branches then A(n, m) = 2.
In more general terms A(.,.) can be considered as a binary operation that evaluates certain differences between the prime factors of its operands. To start, compare the largest prime factor of each operand with the 2nd largest prime factor. As described above, 2 is the result if these 2 factors are the same in one operand, but are different in the other operand; otherwise 3 is the result if these 2 factors are consecutive primes in one operand, but are nonconsecutive primes in the other operand. Further cases are covered in the examples, but note it is the difference between the indices of the prime numbers that is significant.

Examples

			The top left 17x17 corner of the array:
  n/k |  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17
------+-------------------------------------------------------------
    1 |  1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,
    2 |  1, 2, 2, 2, 2, 2, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,
    3 |  1, 2, 3, 2, 3, 3, 3, 2, 2,  3,  3,  3,  3,  3,  3,  2,  3,
    4 |  1, 2, 2, 4, 2, 2, 2, 4, 4,  2,  2,  2,  2,  2,  2,  4,  2,
    5 |  1, 2, 3, 2, 5, 3, 5, 2, 2,  5,  5,  3,  5,  5,  3,  2,  5,
    6 |  1, 2, 3, 2, 3, 6, 3, 2, 2,  3,  3,  6,  3,  3,  6,  2,  3,
    7 |  1, 2, 3, 2, 5, 3, 7, 2, 2,  5,  7,  3,  7,  7,  3,  2,  7,
    8 |  1, 2, 2, 4, 2, 2, 2, 8, 4,  2,  2,  2,  2,  2,  2,  8,  2,
    9 |  1, 2, 2, 4, 2, 2, 2, 4, 9,  2,  2,  2,  2,  2,  2,  4,  2,
   10 |  1, 2, 3, 2, 5, 3, 5, 2, 2, 10,  5,  3,  5,  5,  3,  2,  5,
   11 |  1, 2, 3, 2, 5, 3, 7, 2, 2,  5, 11,  3, 11,  7,  3,  2, 11,
   12 |  1, 2, 3, 2, 3, 6, 3, 2, 2,  3,  3, 12,  3,  3,  6,  2,  3,
   13 |  1, 2, 3, 2, 5, 3, 7, 2, 2,  5, 11,  3, 13,  7,  3,  2, 13,
   14 |  1, 2, 3, 2, 5, 3, 7, 2, 2,  5,  7,  3,  7, 14,  3,  2,  7,
   15 |  1, 2, 3, 2, 3, 6, 3, 2, 2,  3,  3,  6,  3,  3, 15,  2,  3,
   16 |  1, 2, 2, 4, 2, 2, 2, 8, 4,  2,  2,  2,  2,  2,  2, 16,  2,
   17 |  1, 2, 3, 2, 5, 3, 7, 2, 2,  5, 11,  3, 13,  7,  3,  2, 17,
.
The nearest common ancestor of 7 and 15 in the Doudna tree (see diagram in the links and A005940) is 3, thus A(7,15) = A(15,7) = 3.
The nearest common ancestor of 12 and 15 in the Doudna tree is 6, thus A(12,15) = A(15,12) = 6.
The nearest common ancestor of 4 and 27 is 4 because 27 is a descendant of 4 in the Doudna tree, thus A(4,27) = A(27,4) = 4.
Example without reference to the Doudna tree: (Start)
The method below works in general for A(.,.) considered as a binary operation, but we use A(20, 42) as our example.
(1) Write each operand as a product of primes in nondecreasing order, convert to a tuple of prime indices, decrement each index, take first differences, then reverse the order:
  20 = 2*2*5 = prime(1) * prime(1) * prime(3) -> (1,1,3) -> (0,0,2) -> (0,0,2) -> (2,0,0);
  42 = 2*3*7 = prime(1) * prime(2) * prime(4) -> (1,2,4) -> (0,1,3) -> (0,1,2) -> (2,1,0).
(2) Truncate each tuple after the first elements that differ between them (or at the length of the shorter tuple):
  (2,0,0) -> (2,0); (2,1,0) -> (2,1).
(3) Choose the lesser tuple: (2,0).
(4) Determine which number would generate this tuple by the process from step (1):
  10 = 2*5 = prime(1) * prime(3) -> (1,3) -> (0,2) -> (0,2) -> (2,0).
This gives A(20, 42) = 10.
(End)

Links

Crossrefs

Cf. A000027 (main diagonal).
Cf. A000040, A005940, A052126, A061395, A064989, A070003, A102750, A156552, A163511, A241917, A347879 [= a(n,sigma(n))], A348040, A348041, A348042, A348043, A348044 [= a(n,n^2)].
Cf. also A341510, A347380, A347381.

Programs

  • PARI
    up_to = 105;
Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
Abinprefix(n,k) = { my(digs=binary(n)); fromdigits(vector(k,i,digs[i]),2); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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}; \\ From A156552
A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
A348041sq(x,y) = A005940(1+Abinprefix(A156552(x),A348040sq(x,y)));
A348041list(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] = A348041sq(col,(a-(col-1))))); (v); };
v348041 = A348041list(up_to);
A348041(n) = v348041[n];
  • PARI
    \\ A348041sq can be defined also as:
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A348041sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, h=y); while(k>1, listput(lista,k); k = A252463(k)); stemvec = Vecrev(Vec(lista)); while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A252463(h)));

Formula

A(n, 1) = A(1, n) = 1; otherwise if A241917(n) <> A241917(m) then A(n, m) = A000040(1 + min(A241917(2*n), A241917(2*m))); otherwise A(n, m) = x * A000040(A061395(x)+A241917(n)), where x = A(A052126(n), A052126(m)).
A(i, j) = A(j, i).
A(n, n) = n.
A(2, n) = 2 for all n > 1.
A(p, q) = min(p, q) for any primes p and q.
A(A070003(n), A102750(m)) = 2.
A(u^2, v^2) = A(u, v)^2.
A(4k+2, 6k+3) = A064989(2k+1) for all k >= 1.

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

Views

Author

Antti Karttunen, Jun 29 2020

Keywords

Comments

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

Links

Crossrefs

Cf. A052126, A171462, A335875, A335876, A335881, A335885, A335904, A335908.
Cf. A335883 (position of the first occurrence of each n).

Programs

  • PARI
    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))))); };
  • PARI
    \\ 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));

Formula

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.

A065966 Numbers k such that phi(k) / 2 is prime.

Original entry on oeis.org

5, 7, 8, 9, 10, 11, 12, 14, 18, 22, 23, 46, 47, 59, 83, 94, 107, 118, 166, 167, 179, 214, 227, 263, 334, 347, 358, 359, 383, 454, 467, 479, 503, 526, 563, 587, 694, 718, 719, 766, 839, 863, 887, 934, 958, 983, 1006, 1019, 1126, 1174, 1187, 1283, 1307, 1319
Offset: 1

Views

Author

Joseph L. Pe, Dec 08 2001

Keywords

Comments

This is probably an infinite sequence, but a proof would be nice. Are there infinitely many consecutive terms of the sequence which are also consecutive integers? (For example, 7, 8 and 46, 47.)
Apart from 8, 9, 12 and 18, all the terms of the sequence are safe primes or twice safe primes. It is not known if there are infinitely many safe primes (cf. A005385, A005384). For consecutive terms of the sequence which are also consecutive integers see A066179. - Vladeta Jovovic, Dec 16 2001

Examples

			phi(46)/2 = 22/2 = 11, a prime.

Links

Crossrefs

Cf. A000010, A005385, A005384, A066179, A006530, A052126, A068211, A068212.

Programs

  • Mathematica
    Select[Range[1400],PrimeQ[EulerPhi[#]/2]&] (* Harvey P. Dale, Feb 11 2020 *)
  • PARI
    for(n=3,5000, if(isprime(eulerphi(n)/2),print1(n,",")))
  • PARI
    { n=0; for (m=3, 10^9, if (isprime(eulerphi(m)/2), write("b065966.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 05 2009

Formula

Numbers k such that A068212(k) = 2.

Extensions

More terms from Jason Earls, Dec 09 2001
Edited by Charles R Greathouse IV, Mar 18 2010

A284600 a(n) = n/(largest prime power dividing n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 12, 1, 2, 7, 1, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 12, 5, 2, 3, 8, 1, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 30 2017

Keywords

Comments

a(n) = smallest positive number k such that n/k is a prime power.

Examples

			a(12) = 3 because 12 = 2^2*3 therefore 12/(largest prime power dividing 12) = 12/4 = 3.

Links

Crossrefs

Cf. A000961, A003557, A007913, A034699, A052126, A121289, A278568.
Has same beginning as A052128 and A114536 but is strictly different from those two sequences.

Programs

  • Maple
    f:= n ->  n /max(map(t -> t[1]^t[2], ifactors(n)[2])):
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Apr 09 2017
  • Mathematica
    Join[{1}, Table[n/Last[Select[Divisors[n], PrimePowerQ[#1] &]], {n, 2, 90}]]
  • Python
    from sympy import lcm
def a003418(n): return 1 if n<1 else lcm(range(1, n + 1))
def a(n):
    m=1
    while True:
        if a003418(m)%n==0: return m
        else: m+=1
print([n//a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

Formula

a(n) = n/A034699(n).
a(n) = 1 if n is a prime power (A000961).
a(n) = 2 if n is a twice odd prime power (A278568).

A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2*A300841(n)) = 2*a(n), a(A300841(n)) = A003961(a(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 11, 6, 13, 10, 17, 9, 19, 14, 15, 23, 29, 22, 31, 25, 21, 26, 37, 8, 41, 34, 33, 35, 43, 12, 47, 38, 39, 46, 49, 55, 53, 58, 51, 18, 59, 20, 61, 65, 77, 62, 67, 57, 71, 74, 69, 85, 73, 28, 91, 30, 87, 82, 79, 27, 83, 86, 121, 95, 119, 44, 89, 115, 93, 50, 97, 42, 101, 94, 111, 145, 143, 52, 103, 133, 107, 106, 109, 45, 161
Offset: 1

Views

Author

Antti Karttunen, Apr 15 2018

Keywords

Comments

Because "Fermi-Dirac factorization" is fundamentally different from ordinary prime factorization (as no exponents larger than 1 are allowed) this pair of permutations mapping between them is not always very intuitive. For example, we have ("as expected") A302776(n) = A302023(A052126(A302024(n))), while on the other hand, we have A302792(n) = A300841(A302023(A032742(A302024(n)))), where an additional shift-operator A300841 is needed for "correction".

Links

Crossrefs

Cf. A302023 (inverse).
Cf. A050376, A052331, A003961, A005940, A300840, A300841, A302791.

Programs

  • PARI
    up_to = 32768;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A302024(n) = A005940(1+A052331(n));

Formula

a(n) = A005940(1+A052331(n)).
a(A050376(n)) = A000040(n).
A001221(a(n)) = A302790(n).
A001222(a(n)) = A064547(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

Views

Author

Antti Karttunen, Apr 14 2020

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

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Author

Antti Karttunen, Apr 14 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    Position[Array[Length@NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 225], 4][[All, 1]] (* Michael De Vlieger, Apr 30 2020 *)
  • PARI
    A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334104(n) = (4==A329697(n));
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