A286570 -id:A286570 - cp's OEIS frontend

A300230 Restricted growth sequence transform of A286570, combining A009194(n) and A046523(n), i.e., gcd(n,sigma(n)) and the prime signature of n.

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

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

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

Cf. A009194, A046523, A286570.

Cf. also A300223, A300226, A300229, A300231, A300232, A300233, A300235.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A009194(n) = gcd(n, sigma(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from A046523
A286570(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n));
write_to_bfile(1,rgs_transform(vector(65537,n,A286570(n))),"b300230.txt");

A173438 Number of divisors d of number n such that d does not divide sigma(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 4, 1, 4, 3, 2, 1, 2, 2, 2, 3, 0, 1, 4, 1, 5, 2, 2, 3, 8, 1, 2, 3, 4, 1, 4, 1, 3, 4, 2, 1, 7, 2, 5, 2, 4, 1, 4, 3, 4, 3, 2, 1, 6, 1, 2, 5, 6, 3, 4, 1, 4, 2, 6, 1, 10, 1, 2, 5, 3, 3, 4, 1, 8, 4, 2, 1, 6, 3, 2, 2, 5, 1, 6, 2, 3, 3, 2, 2, 6, 1, 5

Offset: 1

Jaroslav Krizek, Feb 18 2010

nonn

a(n) = 0 for multiply-perfect numbers (A007691).

			For n = 12, a(12) = 3; sigma(12) = 28, divisors of 12: 1, 2, 3, 4, 6, 12; d does not divide sigma(n) for 3 divisors d: 3, 6 and 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000203, A007691, A009194, A073802, A286570.

A173438 := proc(n)
    local sd,a;
    sd := numtheory[sigma](n) ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(sd,d) <> 0 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015

Table[DivisorSum[n, 1 &, ! Divisible[DivisorSigma[1, n], #] &], {n, 98}] (* Michael De Vlieger, Oct 08 2017 *)

A173438(n) = (numdiv(n) - numdiv(gcd(sigma(n), n))); \\ (See PARI-code in A073802) - Antti Karttunen, Oct 08 2017

a(n) = A000005(n) - A073802(n).

a(n) = tau(n) - tau(gcd(n,sigma(n))). - Antti Karttunen, Oct 08 2017

A286571 Compound filter (prime signature of n & n/gcd(n, sigma(n))): a(n) = P(A046523(n), A017666(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 25, 17, 21, 30, 113, 70, 51, 68, 103, 93, 72, 51, 481, 155, 148, 192, 222, 331, 126, 278, 324, 382, 159, 569, 78, 437, 591, 498, 1985, 126, 237, 786, 2521, 705, 282, 952, 375, 863, 660, 948, 243, 337, 384, 1130, 1759, 1330, 1842, 237, 678, 1433, 520, 1776, 459, 1897, 567, 1772, 2076, 1893, 636, 2713, 8065, 2421, 810, 2280, 1002, 384, 2046

Offset: 1

Antti Karttunen, May 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A017666, A046523, A286360, A286570.

A017666(n) = (n/gcd(n, sigma(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286571(n) = (1/2)*(2 + ((A046523(n)+A017666(n))^2) - A046523(n) - 3*A017666(n));

from sympy import factorint, gcd, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), n/gcd(n, divisor_sigma(n))) # Indranil Ghosh, May 26 2017

(define (A286571 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A017666 n)) 2) (- (A046523 n)) (- (* 3 (A017666 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A017666(n))^2) - A046523(n) - 3*A017666(n)).

cp's OEIS Frontend

A300230 Restricted growth sequence transform of A286570, combining A009194(n) and A046523(n), i.e., gcd(n,sigma(n)) and the prime signature of n.

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A173438 Number of divisors d of number n such that d does not divide sigma(n).

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A286571 Compound filter (prime signature of n & n/gcd(n, sigma(n))): a(n) = P(A046523(n), A017666(n)), where P(n,k) is sequence A000027 used as a pairing function.

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