A191581 -id:A191581 - cp's OEIS frontend

A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

5, 10, 40, 41, 129, 135, 140, 155, 182, 189, 200, 204, 206, 238, 375, 429, 435, 441, 455, 475, 546, 564, 574, 616, 625, 678, 722, 744, 765, 836, 856, 902, 1035, 1056, 1170, 1188, 1272, 1296, 1344, 1518, 1650, 1764, 1806, 1918, 1925

Offset: 1

Paolo P. Lava, Jun 07 2011

			40-> sum prime factors=2+2+2+5=11; sum anti-divisors=3+9+16+27=55; 55/11=5
129-> sum prime factors=3+43=46; sum anti-divisors=2+6+7+37+86=138; 138/46=3

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

Cf. A001414, A066272, A161917, A191581.

with(numtheory); P:=proc(i) local a,b,j,k,s,n;
for n from 3 to i do b:=ifactors(n)[2];
s:=add(b[k][1]*b[k][2],k=1..nops(b));
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 type(a/s,integer) then print(n); fi; od; end: P(2000);

A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

1, 2, 11, 12, 15, 16, 22, 31, 76, 152, 309, 1576, 375479, 781314, 1114986, 3734218, 24311881, 68133239, 147881549

Offset: 1

Paolo P. Lava, Mar 13 2013

nonn

Tested up to k = 108122.

a(20) > 3*10^8. - Donovan Johnson, Mar 22 2013

			Anti-divisors of 76 are 3, 8, 9, 17 and 51 and their sum is 88.
Anti-divisor of 3 is 2 -> Sum is 2.
Anti-divisors of 8 are 3 and 5 -> Sum is 8.
Anti-divisors of 9 are 2 and 6 -> Sum is 8.
Anti-divisors of 17 are 2, 3, 5, 7 and 11 -> Sum is 28.
Anti-divisors of 51 are 2, 6 and 34 -> Sum is 42.
Finally, 2+8+8+28+42=88.

Cf. A066272, A178029, A191580, A191581, A192282-A192284.

A216213:= proc(q) local a,b,c,j,k,n;
for n from 1 to q do
  a:={}; b:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+k; a:=a union {k}; fi; od;
  c:=0; for j from 1 to nops(a) do for k from 2 to a[j]-1 do if abs((a[j] mod k)-k/2)<1 then c:=c+k; fi; od; od; if b=c then print(n); fi; od; end:
A216213(10^10);

a(13)-a(19) from Donovan Johnson, Mar 22 2013

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A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

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A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.

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cp's OEIS Frontend

A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

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Author

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Maple

A216213 Numbers k such that sigma*(k) = Sum_{j=anti-divisors of k} sigma*(j), where sigma*(k) is the sum of the anti-divisors of k.

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Maple

Extensions

A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.