A066452 -id:A066452 - cp's OEIS frontend

A241004 Numbers n such that anti-phi(sigma(n)) = n, where anti-phi is A066452 and sigma is the sum of anti-divisors of n (A066417).

90, 137, 162, 581, 714, 773, 3735, 4557, 71028

Offset: 1

Paolo P. Lava, Aug 07 2014

Like A001229 but using anti-phi(n) (A066452) and sigma*(n) (A066417).

			90 is in the sequence: Anti-divisors of 90 are 4, 12, 20, 36, 60 and their sum is 132. Anti-phi of 132 is 90.

Cf. A001229, A066417, A066452.

isA241004 := proc(n)
    simplify( n = A066452(A066417(n))) ;
end proc:
for n from 1 do
    if isA241004(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Aug 07 2014

sad(n) = my(k); if(n>1, k=valuation(n, 2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2, 0);
antidiv(n) = {my(v = []); for (k=2, n-1, if (abs((n % k) - k/2) < 1, v = concat(v, k));); v;}
antiphi(n) = {my(vad = antidiv(n)); my(nbad = 0); for (j=1, n-1, isad = 1; for (k=1, #vad, if ((j % vad[k]) == 0, isad = 0; break); ); nbad += isad;); nbad;}
isok(n) = n == antiphi(sad(n)); \\ Michel Marcus, Feb 25 2016

a(9) from Michel Marcus, Feb 25 2016

A058838 a(n) = 1 + sum of the anti-divisors of n.

Original entry on oeis.org

1, 1, 3, 4, 6, 5, 11, 9, 9, 15, 13, 14, 20, 17, 19, 15, 29, 29, 19, 25, 23, 37, 35, 24, 40, 25, 43, 47, 25, 37, 43, 59, 49, 31, 53, 33, 51, 71, 53, 56, 42, 67, 57, 41, 87, 59, 61, 57, 73, 81, 43, 95, 89, 53, 75, 57, 75, 97, 91, 108, 58, 79, 113, 47, 85

Offset: 1

Jon Perry, Dec 28 2001

nonn

See A066272 for definition of anti-divisor.

			Consider n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} respectively and quotients {7, 5, 1}, {12, 4}, {1}; so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 1 + 28 = 29.

Jon Perry, Anti-divisors [Broken link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A066417, A066241, A066452.

a(n) = A066417(n) + 1.

A066418 Numbers k for which phi(k) + anti-phi(k) = k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 12, 15, 27, 30, 40, 44, 57, 117, 128, 171, 236, 399, 408, 510, 1623, 3597, 3915, 4616, 4684, 7335, 10197, 10768, 14144, 32768, 39387, 76035, 77097, 106605, 162450, 196080, 219966, 391696

Offset: 1

Jon Perry, Dec 28 2001

Anti-phi(n) (A066452) is the number of numbers coprime to all the anti-divisors of n.

See A066272 for definition of anti-divisor.

			The anti-divisors of 7 are 1, 2, 3 and 5. Therefore of the integer 1-6, only 1 is coprime to 2, 3 and 5, therefore anti-phi(7)=1. phi(7)=6, therefore anti-phi(7)+phi(7)=7

Jon Perry, Anti-phi function [Wayback Machine link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A066416, A066417, A058838, A066241, A066272, A066452.

a(21)-a(34) from Nathaniel Johnston, Apr 20 2011

a(35)-a(39) from Amiram Eldar, Jan 12 2020

A241003 Numbers k such that anti-phi(k) = anti-phi(k+1).

Original entry on oeis.org

2, 8, 14, 20, 27, 32, 284, 297, 362, 717, 842, 1322, 1377, 1725, 1802, 1917, 1982, 2222, 2637, 3410, 4094, 4149, 4850, 5288, 5642, 5654, 5660, 5690, 5750, 5937, 5949, 6237, 7017, 7245, 7377, 7490, 8097, 8217, 8277, 8462, 8774, 9117, 9542, 9897, 10034, 11409, 11810

Offset: 1

Paolo P. Lava, Aug 07 2014

nonn

Like A001274 but using anti-phi, as defined in A066452, instead of phi, per A000010.

			anti-phi(2) = anti-phi(3) = 1.
anti-phi(8) = anti-phi(9) = 4.
anti-phi(14) = anti-phi(15) = 7. Etc.

Amiram Eldar, Table of n, a(n) for n = 1..512

Cf. A000010, A001274, A066418, A066452.

for n from 1 do
    if A066452(n) = A066452(n+1) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Aug 07 2014

A241006 Number of positive numbers

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 3, 2, 2, 5, 3, 5, 4, 9, 2, 4, 5, 6, 6, 6, 6, 10, 5, 8, 6, 5, 8, 8, 9, 12, 7, 10, 7, 12, 9, 8, 9, 13, 13, 9, 9, 14, 10, 11, 10, 18, 13, 13, 16, 12, 12, 18, 13, 18, 13, 13, 14, 12, 17, 16, 15, 41, 15, 16, 14, 18, 22, 15, 18, 16, 16, 22, 20, 24, 15, 19, 25, 21

Offset: 2

R. J. Mathar, Aug 07 2014

nonn

Note that a different sequence could be defined by "Number of positive numbers < n that do not have any anti-divisor as a factor," which gives A066452. Consider for example n=10 with anti-divisors {3,4,7} and the number 2. 2 is not coprime to the anti-divisor 4 and does not contribute to a(10), whereas 2 does not have 4 as a factor and contributes to A066452.

			10 has anti-divisors {3,4,7}. The positive integers that are <10 and coprime to
all of them are {1,5}, so a(10)=2. The integers 2, 3, 4, 6, 7, 8 and 9
are not coprime to all of {3,4,7} and do not contribute to the count.

Cf. A066452.

A241006 :=proc(n)
    local a,ad,i,isco ;
    a := 0 ;
    ad := antidivisors(n) ; # implemented in A066272
    for i from 1 to n-1 do
        isco := true;
        for adiv in ad do
            if igcd(adiv,i) > 1 then
                isco := false;
                break;
            end if;
        end do:
        if isco then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:

cp's OEIS Frontend

A241004 Numbers n such that anti-phi(sigma*(n)) = n, where anti-phi is A066452 and sigma* is the sum of anti-divisors of n (A066417).

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A058838 a(n) = 1 + sum of the anti-divisors of n.

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A066418 Numbers k for which phi(k) + anti-phi(k) = k.

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A241003 Numbers k such that anti-phi(k) = anti-phi(k+1).

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Maple

A241006 Number of positive numbers

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Author

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Programs

Maple

A241004 Numbers n such that anti-phi(sigma(n)) = n, where anti-phi is A066452 and sigma is the sum of anti-divisors of n (A066417).