A130799 -id:A130799 - cp's OEIS frontend

A246841 Sum of digits of all the anti-divisors of n.

0, 0, 2, 3, 5, 4, 10, 8, 8, 14, 12, 13, 19, 16, 9, 5, 19, 19, 9, 15, 13, 27, 25, 14, 21, 15, 24, 28, 15, 9, 24, 31, 21, 12, 16, 14, 23, 34, 25, 28, 23, 30, 29, 22, 32, 22, 24, 20, 27, 26, 15, 40, 34, 16, 20, 20, 29, 42, 45, 35, 12, 24, 40, 10, 21, 32, 60, 49

Offset: 1

Paolo P. Lava, Sep 05 2014

Sum of the digits of the terms in row n of A130799.

First occurrence of k, or 0 if k never appears: 0, 3, 4, 6, 5, 0, 0, 8, 15, 7, 0, 11, 12, 10, 20, 14, 75, 69, 13, 48, 25, 44, 37, 27, 23, 50, 22, 28, 43, 42, 32, 45, 92, 38, 60, 82, 208, 81, 110, 52, 72, 58, 97, 73, 59, 77, 255, 85, 68, 127, ...

			Anti-divisors of 20 are 3, 8, 13 and the sum of their digits is 3 + 8 + 1 + 3 = 15.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A034690, A066272.

# function antidivisors defined in A066272. transforms is https://oeis.org/transforms.txt
read("transforms");
A246841 := proc(n)
    a :=0 ;
    for adiv in antidivisors(n) do
        a := a+digsum(adiv) ;
    end do:
    a ;
end proc:
seq(A246841(n),n=1..30) ; # R. J. Mathar, Sep 07 2014

A248787 Numbers x such that sigma(x) = rev(sigma(x)), where sigma(x) is the sum of the divisors of x, sigma(x) the sum of the anti-divisors of x and rev(x) the reverse of x.

Original entry on oeis.org

20, 26, 36531, 42814, 4513010, 63033577

Offset: 1

Paolo P. Lava, Oct 14 2014

No further terms up to 10^6.

a(7) > 10^10. - Hiroaki Yamanouchi, Mar 18 2015

			Antidivisors of 20 are 3,8,13 and their sum is 24, while sigma(20) = 42.
Antidivisors of 26 are 3,4,17 and their sum is 24, while sigma(26) = 42.
Antidivisors of 36531 are 2, 6, 18, 22, 54, 66, 82, 162, 198, 246, 594, 738, 902, 1782, 2214, 2706, 6642, 8118, 24354 and their sum is sigma*(36531) = 48906, while sigma(36531) = 60984.

Diana Mecum, Anti-divisors from 3 to 500

Cf. A000203, A066417, A178029.

with(numtheory):T:=proc(w) local x,y,z; y:=w; z:=0;
for x from 1 to ilog10(w)+1 do z:=10*z+(y mod 10); y:=trunc(y/10); od; z; end:
P:=proc(q) local a,j,k,n; for n from 1 to q do
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if T(a)=sigma(n) then print(n); fi; od; end: P(10^10);

rev(n) = subst(Polrev(digits(n)), x, 10);
sad(n) = k=valuation(n, 2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
isok(n) = sigma(n) == rev(sad(n)); \\ Michel Marcus, Dec 07 2014

a(5) from Chai Wah Wu, Dec 06 2014

a(6) from Hiroaki Yamanouchi, Mar 18 2015

A261488 Number of triples (x, y, x mod y) such that x > y are divisors of n and x mod y is an anti-divisor of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Aug 20 2015

nonn

An anti-divisor of n is a number d in the range [2,n-1] which does not divide n and is either a (necessarily odd) divisor of 2n-1 or 2n+1, or a (necessarily even) divisor of 2n.

a(n) = 0 if n is a prime power.

			a(45) = 2 with triples (5, 3, 5 mod 3) and (15, 9, 15 mod 9) since 3, 5, 9, and 15 are divisors of 45 and 5 mod 3 = 2 and 15 mod 9 = 6 are anti-divisors of 45.

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

Cf. A027750, A130799, A260073, A261476.

a(n)=my(d=divisors(n)); sum(i=1,#d-1, sum(j=i+1,#d, my(z=d[j]%d[i]); z && n%z && if(z%2, (2*n+1)%z==0 || (2*n-1)%z==0, (2*n)%z==0))) \\ Charles R Greathouse IV, Aug 26 2015

A286917 Numbers k such that there is an anti-divisor d of k satisfying sigma(d) = k.

Original entry on oeis.org

3, 4, 13, 32, 40, 60, 121, 364, 1093, 3200, 3280, 9841, 15120, 16380, 29282, 29524, 88573, 91728, 264992, 265720, 797161, 2391484, 7174453, 21523360, 40098240, 64570081, 71495424, 78427440, 193690562, 193710244, 229909120, 581130733, 689727360, 1743392200, 5230176601

Offset: 1

Paolo P. Lava, May 16 2017

nonn

As powers of 3 are in the sequence (larger than 1), the sequence is infinite. - David A. Corneth, Jul 20 2020

			Anti-divisors of 60 are 7, 8, 11, 17, 24, 40 and sigma(24) = 60.

Cf. A000203, A014224, A066272, A081756, A130799.

with(numtheory): P:= proc(q) local a,k,n; for n from 3 to q do a:=[];
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=[op(a),k]; fi; od;
for k from 1 to nops(a) do if n=sigma(a[k]) then print(n); break; fi; od;
od; end: P(10^4); # Paolo P. Lava, May 16 2017

isok(n) = {ad = select(t->n%t && tMichel Marcus, May 20 2017

sigma(3^m) is in the sequence, as is sigma(3^m*(3^(m + 1) - 2)) for prime 3^(m + 1) - 2. - David A. Corneth, Jul 20 2020

More terms from Michel Marcus, May 20 2017
a(22)-a(26) from Jinyuan Wang, Jul 20 2020
a(27)-a(35) from David A. Corneth, Jul 20 2020

cp's OEIS Frontend

A246841 Sum of digits of all the anti-divisors of n.

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A248787 Numbers x such that sigma(x) = rev(sigma*(x)), where sigma(x) is the sum of the divisors of x, sigma*(x) the sum of the anti-divisors of x and rev(x) the reverse of x.

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A261488 Number of triples (x, y, x mod y) such that x > y are divisors of n and x mod y is an anti-divisor of n.

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PARI

A286917 Numbers k such that there is an anti-divisor d of k satisfying sigma(d) = k.

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A248787 Numbers x such that sigma(x) = rev(sigma(x)), where sigma(x) is the sum of the divisors of x, sigma(x) the sum of the anti-divisors of x and rev(x) the reverse of x.