A364500 -id:A364500 - cp's OEIS frontend

A364255 a(n) = gcd(n, A163511(n)).

1, 1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 3, 2, 1, 24, 5, 2, 1, 4, 1, 2, 1, 32, 3, 2, 5, 36, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 10, 1, 4, 1, 2, 11, 8, 3, 2, 1, 4, 1, 2, 1, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 5, 4, 7, 2, 1, 16, 27, 2, 1, 12, 5, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 96, 1, 2, 1, 20, 1, 2, 1, 8, 105

Offset: 0

Antti Karttunen, Jul 16 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A163511, A364257 (Dirichlet inverse), A364258, A364491, A364492, A364493.

Cf. also A364254, A364256, A339969, A364500, A364949.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364255(n) = gcd(n, A163511(n)); \\ Antti Karttunen, Sep 01 2023

from math import gcd
from sympy import nextprime
def A364255(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return gcd(c*p,n) # Chai Wah Wu, Jul 25 2023

From Antti Karttunen, Sep 01 2023: (Start)
a(n) = gcd(n, A364258(n)) = gcd(A163511(n), A364258(n)).
a(n) = n / A364491(n) = A163511(n)/ A364492(n).
(End)

A364499 a(n) = A005940(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 4, 0, 12, 4, 12, 0, -6, -4, 2, 0, 14, 8, 22, 0, 24, 24, 48, 8, 96, 24, 50, 0, -20, -12, -2, -8, 18, 4, 24, 0, 36, 28, 62, 16, 130, 44, 88, 0, 72, 48, 96, 48, 192, 96, 170, 16, 286, 192, 316, 48, 564, 100, 180, 0, -48, -40, -28, -24, -4, -4, 28, -16, 18, 36, 90, 8, 198, 48, 110, 0, 62

Offset: 1

Antti Karttunen, Jul 28 2023

Compare to the scatter plot of A364563.
From Antti Karttunen, Aug 11 2023: (Start)
Can be computed as a certain kind of bitmask transformation of A364568 (analogous to the inverse Möbius transform that is appropriate for A156552-encoding of n).
See A364572, A364573 (and also A364576) for n (apart from those in A029747) where a(n) comes relatively close to the X-axis.
(End)

			A005940(528577) = 528581, therefore a(528577) = 528581 - 528577 = 4. (See A364576).
A005940(2109697) = 2109629, therefore a(2109697) = 2109629 - 2109697 = -68.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A005940, A364500 [= gcd(n,a(n))], A364559, A364572, A364573, A364576.
Cf. A029747 (known positions of 0's), A364540 (positions of terms < 0), A364541 (of terms <= 0), A364542 (of terms >= 0), A364563 [= -a(A364543(n))].
Cf. also A364258, A364568.

nn = 81; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[a[#] - # &, nn] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364499(n) = (A005940(n)-n);

A364499(n) = { my(m=1,p=2,x=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), x += m; z *= p); n>>=1; m <<=1); (z-x)-1; }; \\ Antti Karttunen, Aug 06 2023

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A364499(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(n-1)[2:].split('1')[:0:-1])).items())-n # Chai Wah Wu, Aug 07 2023

a(n) = -A364559(A005940(n)).
For all n >= 1, a(2*n) = 2*a(n).
For all n >= 1, a(A029747(n)) = 0.

A364502 a(n) = A005940(n) / gcd(n, A005940(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 9, 1, 7, 1, 15, 1, 25, 9, 9, 1, 11, 7, 21, 1, 5, 15, 45, 1, 49, 25, 25, 9, 125, 9, 81, 1, 13, 11, 33, 7, 55, 21, 21, 1, 77, 5, 105, 15, 35, 45, 135, 1, 121, 49, 49, 25, 245, 25, 45, 9, 343, 125, 375, 9, 625, 81, 27, 1, 17, 13, 39, 11, 65, 33, 99, 7, 91, 55, 11, 21, 25, 21, 189, 1, 143, 77, 231, 5

Offset: 1

Antti Karttunen, Jul 28 2023

Denominator of n / A005940(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005940, A364500, A364501 (numerators), A364546 (positions of 1's).

Cf. also A364492.

nn = 84; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[a[#]/GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364502(n) = { my(u=A005940(n)); (u / gcd(n, u)); };

A364501 a(n) = n / gcd(n, A005940(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 9, 1, 11, 1, 13, 7, 5, 1, 17, 9, 19, 1, 3, 11, 23, 1, 25, 13, 9, 7, 29, 5, 31, 1, 33, 17, 35, 9, 37, 19, 13, 1, 41, 3, 43, 11, 9, 23, 47, 1, 49, 25, 17, 13, 53, 9, 11, 7, 57, 29, 59, 5, 61, 31, 7, 1, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 5, 19, 7, 13, 79, 1, 81, 41, 83, 3, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 28 2023

Numerator of n / A005940(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A005940, A364500, A364502 (denominators), A364544 (positions of 1's).

Cf. also A364491.

nn = 89; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[#/GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364501(n) = (n / gcd(n, A005940(n)));

A364501(n) = { my(orgn=n,p=2,rl=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), rl++; if(1==(n%4), z *= p^min(rl,valuation(orgn,p)); rl=0)); n>>=1); (orgn/z); };

A365463 a(n) = gcd(n, A356867(n)), where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 2, 3, 1, 1, 6, 1, 8, 9, 1, 1, 3, 1, 2, 3, 2, 1, 18, 1, 4, 3, 1, 1, 24, 25, 2, 27, 1, 1, 3, 1, 4, 3, 2, 7, 9, 1, 2, 3, 5, 1, 6, 1, 4, 9, 1, 1, 6, 1, 50, 3, 4, 1, 54, 11, 2, 3, 1, 1, 12, 1, 2, 9, 1, 5, 3, 1, 4, 3, 10, 1, 72, 1, 2, 75, 1, 1, 6, 1, 16, 81, 1, 1, 3, 5, 2, 3, 2, 1, 9, 91, 2, 3, 1, 5, 12, 1, 2, 9, 5, 1, 6, 1, 8, 21

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A007949, A356867, A364957 (Dirichlet inverse), A365462, A365464, A365465.

Cf. also A364500.

up_to = 19683;
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v356867 = A356867list(up_to);
A356867(n) = v356867[n];
A365463(n) = gcd(n, A356867(n));

a(n) = gcd(n, A365462(n)) = gcd(A356867(n), A365462(n)).
a(n) = n / A365464(n) = A356867(n) / A365465(n).
For all n >= 1, A007949(a(n)) = A007949(n), A011655(a(n)) = A011655(n).

A365432 a(n) = A156552(A364502(n)), where A364502(n) = A005940(n) / gcd(n, A005940(n)), and A156552 is the inverse of offset-0 version of Doudna-sequence A005940.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 0, 8, 0, 10, 0, 12, 6, 6, 0, 16, 8, 18, 0, 4, 10, 22, 0, 24, 12, 12, 6, 28, 6, 30, 0, 32, 16, 34, 8, 36, 18, 18, 0, 40, 4, 42, 10, 20, 22, 46, 0, 48, 24, 24, 12, 52, 12, 22, 6, 56, 28, 58, 6, 60, 30, 14, 0, 64, 32, 66, 16, 68, 34, 70, 8, 72, 36, 16, 18, 12, 18, 78, 0, 80, 40, 82, 4, 40, 42, 42, 10

Offset: 1

Antti Karttunen, Sep 07 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005940, A364500, A341520, A365430, A365431 (rgs-transform).

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
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364502(n) = { my(u=A005940(n)); (u / gcd(n, u)); };
A365432(n) = A156552(A364502(n));

For all n >= 1, a(n) <= n-1 and A341520(a(n), A365430(n)) = n-1.

A365430 a(n) = A156552(gcd(n, A005940(n))), and A156552 is the inverse of offset-0 version of Doudna-sequence A005940.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 7, 0, 9, 0, 11, 0, 1, 2, 15, 0, 1, 0, 19, 8, 1, 0, 23, 0, 1, 2, 3, 0, 5, 0, 31, 0, 1, 0, 3, 0, 1, 2, 39, 0, 17, 0, 3, 4, 1, 0, 47, 0, 1, 2, 3, 0, 5, 4, 7, 0, 1, 0, 11, 0, 1, 6, 63, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 10, 3, 16, 5, 0, 79, 0, 1, 0, 35, 4, 1, 2, 7, 0, 9, 8, 3, 0, 1, 4, 95, 0, 1, 34, 3

Offset: 1

Antti Karttunen, Sep 07 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005940, A364500, A341520, A365432.

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
A364500(n) = { my(orgn=n,p=2,rl=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), rl++; if(1==(n%4), z *= p^min(rl,valuation(orgn,p)); rl=0)); n>>=1); (z); };
A365430(n) = A156552(A364500(n));

a(n) = A156552(A364500(n)).

For all n >= 1, a(n) <= n-1 and A341520(a(n), A365432(n)) = n-1.

cp's OEIS Frontend

A364255 a(n) = gcd(n, A163511(n)).

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A364499 a(n) = A005940(n) - n.

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A364502 a(n) = A005940(n) / gcd(n, A005940(n)).

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A364501 a(n) = n / gcd(n, A005940(n)).

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A365463 a(n) = gcd(n, A356867(n)), where A356867 is Sycamore's Doudna variant D(3).

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A365432 a(n) = A156552(A364502(n)), where A364502(n) = A005940(n) / gcd(n, A005940(n)), and A156552 is the inverse of offset-0 version of Doudna-sequence A005940.

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A365430 a(n) = A156552(gcd(n, A005940(n))), and A156552 is the inverse of offset-0 version of Doudna-sequence A005940.

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