A325563 -id:A325563 - cp's OEIS frontend

A325564 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A325563(n) except f(1) = 0.

1, 2, 2, 3, 2, 4, 2, 5, 4, 6, 2, 7, 2, 8, 6, 9, 2, 10, 2, 11, 8, 12, 2, 13, 2, 14, 10, 15, 2, 16, 2, 17, 4, 18, 8, 19, 2, 20, 4, 21, 2, 22, 2, 23, 16, 24, 2, 25, 8, 26, 18, 27, 2, 28, 2, 29, 4, 30, 2, 31, 2, 32, 22, 33, 6, 34, 2, 35, 2, 36, 2, 37, 2, 38, 16, 39, 2, 40, 2, 41, 2, 42, 2, 43, 18, 44, 2, 45, 2, 46, 2, 47, 32, 48, 20, 49, 2, 50, 34, 51, 2, 52, 2, 53, 16

Offset: 1

Antti Karttunen, May 11 2019

nonn

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j) => A325563(i) = A325563(j).

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

Cf. A325563.

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; };
A325563(n) = if(1==n,n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n,d,if((d>1),my(q = Pol(binary(n/d))*Mod(1, 2)); if(0==(p%q), return(n/d)))));
Aux325564(n) = if(1==n,0,A325563(n));
v325564 = rgs_transform(vector(up_to,n,Aux325564(n)));
A325564(n) = v325564[n];

A032742 a(1) = 1; for n > 1, a(n) = largest proper divisor of n (that is, for n>1, maximum divisor d of n in range 1 <= d < n).

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, May 15 1998

It seems that a(n) = Max_{j=n+1..2n-1} gcd(n,j). - Labos Elemer, May 22 2002
This is correct: No integer in the range [n+1, 2n-1] has n as its divisor, but certainly at least one multiple of the largest proper divisor of n will occur there (e.g., if it is n/2, then at n + (n/2)). - Antti Karttunen, Dec 18 2014
The slopes of the visible lines made by the points in the scatter plot are 1/2, 1/3, 1/5, 1/7, ... (reciprocals of primes). - Moosa Nasir, Jun 19 2022

Rémi Eismann, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Moosa Nasir, Slopes.
Eric Weisstein's World of Mathematics, Proper Divisor.

Cf. A002110, A002808, A005867, A006530, A008578, A020639, A032741, A003961, A052126, A054576, A055396, A060681, A068319, A063928, A130064, A246277, A250245, A250246, A276085, A276086, A276151, A286477, A300236, A302025, A302026, A302032, A302042, A325563, A325567.

Maximal GCD of k positive integers with sum n for k = 2..10: this sequence (k=2,n>=2), A355249 (k=3), A355319 (k=4), A355366 (k=5), A355368 (k=6), A355402 (k=7), A354598 (k=8), A354599 (k=9), A354601 (k=10).

a032742 n = n `div` a020639 n  -- Reinhard Zumkeller, Oct 03 2012

A032742 :=proc(n) option remember; if n = 1 then 1; else numtheory[divisors](n) minus {n} ; max(op(%)) ; end if; end proc: # R. J. Mathar, Jun 13 2011
1, seq(n/min(numtheory:-factorset(n)), n=2..1000); # Robert Israel, Dec 18 2014

f[n_] := If[n == 1, 1, Divisors[n][[-2]]]; Table[f[n], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2010 *)
Join[{1},Divisors[#][[-2]]&/@Range[2,80]] (* Harvey P. Dale, Nov 29 2011 *)
a[n_] := n/FactorInteger[n][[1, 1]]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)
Table[Which[n==1,1,PrimeQ[n],1,True,Divisors[n][[-2]]],{n,80}] (* Harvey P. Dale, Feb 02 2022 *)

a(n)=if(n==1,1,n/factor(n)[1,1]) \\ Charles R Greathouse IV, Jun 15 2011

from sympy import factorint
def a(n): return 1 if n == 1 else n//min(factorint(n))
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jun 21 2022

(define (A032742 n) (/ n (A020639 n))) ;; Antti Karttunen, Dec 18 2014

a(n) = n / A020639(n).
Other identities and observations:
A054576(n) = a(a(n)); A117358(n) = a(a(a(n))) = a(A054576(n)); a(A008578(n)) = 1, a(A002808(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
a(n) = A130064(n) / A006530(n). - Reinhard Zumkeller, May 05 2007
a(m)*a(n) < a(m*n) for m and n > 1. - Reinhard Zumkeller, Apr 11 2008
a(m*n) = max(m*a(n), n*a(m)). - Robert Israel, Dec 18 2014
From Antti Karttunen, Mar 31 2018: (Start)
a(n) = n - A060681(n).
For n > 1, a(n) = A003961^(r)(A246277(n)), where r = A055396(n)-1 and A003961^(r)(n) stands for shifting the prime factorization of n by r positions towards larger primes.
a(n) = A250246(A302042(A250245(n))) = A302026(A302032(A302025(n))).
For all n >= 1, A276085(a(A276086(n))) = A276151(n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Sum_{k>=1} A005867(k-1)/(prime(k)*A002110(k)) = 0.165049... . - Amiram Eldar, Nov 19 2022

Definition clarified by N. J. A. Sloane, Dec 26 2022

A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 25, 31, 37, 41, 47, 55, 59, 61, 67, 73, 87, 91, 97, 103, 109, 115, 117, 131, 137, 143, 145, 157, 167, 171, 185, 191, 193, 203, 211, 213, 229, 239, 241, 247, 253, 283, 285, 299, 301, 313, 319, 333, 351, 355, 357, 361, 369, 375

Offset: 1

David Petry (petry(AT)accessone.com)

nonn

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014
2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - Jianing Song, May 10 2021
For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - Joerg Arndt and Jianing Song, May 10 2021

			x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1377 terms from T. D. Noe)
Index entries for sequences operating on GF(2)[X]-polynomials

Written in binary: A058943.
Number of degree-n irreducible polynomials: A001037, see also A000031.
Multiplication table: A048720.
Characteristic function: A091225. Inverse: A091227. a(n) = A091202(A000040(n)). Almost complement of A091242. Union of A091206 & A091214 and also of A091250 & A091252. First differences: A091223. Apart from a(1) and a(2), a subsequence of A092246 and hence A000069.
Table of irreducible factors of n: A256170.
Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570.
Factorization sentinel: A278239.
Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042.
See A115871 for sequences related to cross-domain congruences.
Functions based on the irreducibles: A305421, A305422.

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* Robert G. Wilson v, Aug 12 2011 *)
Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus -> 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* Jean-François Alcover, Nov 21 2016 *)

is(n)=polisirreducible(Pol(binary(n))*Mod(1,2)) \\ Charles R Greathouse IV, Mar 22 2013

A325559 Numbers n such that for any divisor d of n, and some integer k, A048720(d,k) = n only for trivial cases d=1 and d=n.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 115, 117, 121, 127, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181, 185, 191, 193, 197, 199, 203, 205, 209, 211, 213, 223, 227, 229, 233

Offset: 1

Antti Karttunen, May 11 2019

nonn

These are numbers n such that there are only two divisor pairs (d, n/d) [namely, the trivial pairs (1, n) and (n, 1)] that satisfy the condition that when their binary expansions are converted to (0,1)-polynomials (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then their product is the (0,1)-polynomial similarly converted from n, when the multiplication is done over field GF(2).

Differs from A206074 for the first time at n=173, where a(173) = 555, a value missing from A206074, while the first three terms of A206074 not present in this sequence are k = 689, 781 and 913, for all of which A325560(k) = 3, not 2.

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A048720, A206074.
Positions of 2's in A325560, positions of 1's in A325563 (after the initial 1), fixed points of A325643 (after the initial 1).
Some subsequences: A257688 (after its initial 1), A325386 (the remaining terms).

A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };
isA325559(n) = (2 == A325560(n));

A325643 a(1) = 1; for n > 1, a(n) is the least divisor d > 1 of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 7, 2, 23, 2, 25, 2, 3, 2, 29, 2, 31, 2, 3, 2, 7, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 55, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 69, 2, 71, 2, 73, 2, 3, 2, 77, 2, 79, 2, 81, 2, 83, 2, 5, 2, 87, 2, 89, 2, 91, 2, 31, 2, 5, 2, 97, 2, 3, 2, 101, 2, 103, 2, 3

Offset: 1

Antti Karttunen, May 11 2019

nonn

For n > 1, a(n) is the least divisor d of n that is larger than 1 and for which it holds that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then that polynomial is a divisor of (0,1)-polynomial similarly converted from n, when the division is done over GF(2). See the example.

			For n = 21 = 3*7, 3 is not the answer because X^1 + 1 does not divide X^4 + X^2 + 1 (21 is "10101" in binary) over GF(2). However, the next larger divisor 7 works because X^4 + X^2 + 1 = (X^2 + X^1 + 1)^2 when multiplication is done over GF(2) (note that A048720(7,7) = 21). Thus a(21) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A048720, A325559 (fixed points after 1), A325563, A325641, A325642.

A325643(n) = if(1==n,n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n,d,if((d>1),my(q = Pol(binary(d))*Mod(1, 2)); if(0==(p%q), return(d)))));

a(2n) = 2.
For all n >= 1, a(A325559(n)) = A325559(n).
For all n >= 1, n = a(n) * A325641(n) = A048720(a(n), A325642(n)).

A325560 a(n) is the number of divisors d of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 3, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 3, 4, 3, 9, 2, 4, 3, 8, 2, 6, 2, 6, 6, 4, 2, 10, 3, 4, 4, 6, 2, 8, 2, 8, 3, 4, 2, 12, 2, 4, 6, 7, 3, 6, 2, 6, 2, 6, 2, 12, 2, 4, 5, 6, 2, 6, 2, 10, 2, 4, 2, 9, 4, 4, 2, 8, 2, 12, 2, 6, 3, 4, 4, 12, 2, 6, 4, 6, 2, 8, 2, 8, 5

Offset: 1

Antti Karttunen, May 11 2019

nonn

a(n) is the number of divisors d of n such that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), that polynomial is a divisor of the (0,1)-polynomial similarly converted from n, when the polynomial division is done over field GF(2).

			39 = 3*13 has four divisors 1, 3, 13, 39, of which all other divisors except 13 are counted because we have A048720(1,39) = A048720(39,1) = A048720(3,29) = 39, but A048720(13,u) is not equal to 39 for any u, thus a(39) = 3. See also the example in A325563.

Cf. A000005, A048720, A091220, A325559 (positions of 2's), A325563, A325565.

A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };

For all n, A325565(n) <= a(n) <= min(A000005(n), A091220(n)).

cp's OEIS Frontend

A325564 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A325563(n) except f(1) = 0.

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A032742 a(1) = 1; for n > 1, a(n) = largest proper divisor of n (that is, for n>1, maximum divisor d of n in range 1 <= d < n).

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A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

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A325559 Numbers n such that for any divisor d of n, and some integer k, A048720(d,k) = n only for trivial cases d=1 and d=n.

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A325643 a(1) = 1; for n > 1, a(n) is the least divisor d > 1 of n such that A048720(d,k) = n for some k.

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A325560 a(n) is the number of divisors d of n such that A048720(d,k) = n for some k.

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