A070082 -id:A070082 - cp's OEIS frontend

A070095 Number of acute integer triangles with perimeter n and prime side lengths.

0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 1, 3, 0, 2, 0, 2, 0, 2, 1, 3, 0, 3, 0, 2, 0, 2, 0, 3, 0, 2, 1, 2, 0, 2, 1, 3, 0, 1, 0, 3, 0, 3, 0, 2, 0, 3, 1, 4, 0, 3, 0, 3, 0, 1, 1, 3, 0, 3, 1, 4, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: the two consisting of primes ([3,7,7] and [5,5,7]) are also acute, therefore a(17)=2.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070088, A070093, A070097, A070100, A070103, A070120.

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

a(n) = A070088(n) - A070103(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))) * A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, May 13 2019

A070100 Number of integer triangles with perimeter n and prime side lengths which are acute and isosceles.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A059169, A070088, A070092, A070095, A070098, A070126.

A070105 Number of integer triangles with perimeter n and prime side lengths which are obtuse and scalene.

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, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 5, 0, 5, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

a(n) = 0 if n is even. - Robert Israel, Jul 26 2024

Robert Israel, Table of n, a(n) for n = 1..10000
R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A005044, A070088, A070090, A070103, A024156, A070132.

f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  if n::even then return 0 fi;
  for a from 1 to n/3 by 2 do
    if not isprime(a) then next fi;
    bmin:= max(a+1,(n+1)/2-a); if bmin::even then bmin:= bmin+1 fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> isprime(b) and isprime(n-a-b), [seq(b,b=bmin .. bmax,2)]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024

A070106 Number of integer triangles with perimeter n which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 4, 4, 3, 4, 4

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n)=A070101(n)-A024156(n); a(n)=A059169(n)-A070098(n).

			For n=11 there are A005044(11)=4 integer triangles: [1,5,5], [2,4,5], [3,3,5] and [3,4,4]; only one of the two obtuses ([2,4,5] and [3,3,5]) is also isosceles; therefore a(11)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A059169, A070107, A070108, A070133.

A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(k)<=1 until k = 140, for k = 141 there are A005044(141)=432 integer triangles, a(141)=2 as
[37=37<67]: 37+37+67 = 141 and 2*(37^2)<67^2 and 37, 67 are primes,
[41=41<59]: 41+41+59 = 141 and 2*(41^2)<59^2 and 41, 59 are primes.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A059169, A070088, A070092, A070103, A070106, A070135.

A084820 Numbers n such that n, sigma(n) and phi(n) form an integer triangle, where sigma=A000203 is the divisor sum and phi=A000010 the totient.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137

Offset: 1

Reinhard Zumkeller, Jun 04 2003

nonn

a(n)<=A000203(a(n))+A000010(a(n)), A000203(a(n))<=a(n)+A000010(a(n)), A000010(a(n))<=a(n)+A000203(a(n)); values are odd, see A084821 for odd numbers which are not in the sequence.

			n=5, a(5)=9: phi(9)=6, sigma(9)=13: (6,9,13)=(A070080(176), A070081(176), A070082(176)).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Totient Function

Cf. A046022.

Select[Range[1, 140, 2], DivisorSigma[1, #] < EulerPhi[#] + # &] (* Amiram Eldar, Sep 12 2019 *)

is(n)=eulerphi(n)+n>sigma(n) \\ Charles R Greathouse IV, Feb 19 2013

A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A005044, A070088, A070090, A070095, A024154, A070123.

A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 4, 2, 2, 2, 5, 1, 4, 2, 4, 3, 6, 2, 6, 3, 4, 3, 5, 3, 8, 3, 4, 3, 8, 3, 9, 5, 5, 4, 10, 3, 9, 4, 6, 5, 11, 4, 8, 5, 7, 6, 12, 3, 13, 6, 8, 7, 9, 4, 14, 7, 8, 5, 15, 5, 15, 7, 9, 8, 13, 6, 16, 6, 11, 8, 17, 5, 13

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A059169, A051493, A070091, A070094, A070098, A070084, A070125.

A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 1, 3, 2, 1, 1, 1, 0, 4, 2, 2, 2, 4, 1, 3, 2, 3, 2, 4, 1

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A059169, A051493, A070091, A070102, A070106, A070084, A070134.

A070104 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 1, 4, 3, 6, 2, 7, 4, 8, 4, 8, 6, 10, 6, 12, 8, 14, 8, 16, 11, 18, 11, 17, 14, 21, 12, 25, 18, 25, 15, 30, 19, 32, 20, 32, 25, 38, 23, 40, 28, 41, 28, 47, 31, 51, 34, 46, 40, 55, 35, 61, 44, 58, 41, 68

Offset: 1

Reinhard Zumkeller, May 05 2002

Robert Israel, Table of n, a(n) for n = 1..10000
R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A051493, A005044, A070102, A024156, A070084, A070131.

f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  for a from 1 to n/3 do
    if n::even then bmin:= max(a+1,n/2-a+1) else bmin:= max(a+1,(n+1)/2-a) fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> igcd(a,b,n-a-b) = 1, [$bmin .. bmax]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024

cp's OEIS Frontend

A070095 Number of acute integer triangles with perimeter n and prime side lengths.

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A070100 Number of integer triangles with perimeter n and prime side lengths which are acute and isosceles.

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A070105 Number of integer triangles with perimeter n and prime side lengths which are obtuse and scalene.

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A070106 Number of integer triangles with perimeter n which are obtuse and isosceles.

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A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

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A084820 Numbers n such that n, sigma(n) and phi(n) form an integer triangle, where sigma=A000203 is the divisor sum and phi=A000010 the totient.

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PARI

A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

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A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.

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A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

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A070104 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and scalene.

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Maple