A253303 -id:A253303 - cp's OEIS frontend

A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

15, 119, 255, 609, 1295, 1519, 2385, 3479, 4015, 4879, 6305, 9999, 9919, 12319, 14385, 16999, 13345, 28545, 32039, 19199, 38415, 50609, 32239, 50369, 65535, 62839, 50279, 64911, 83505, 96719

Offset: 1

Wolfdieter Lang, Jan 16 2015

The corresponding even legs are given in 4*A253805.
The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253802(n) (odd) and A253803(n) (even).
Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^= A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
Concerning the primitivity question of these triangles see a comment on A253802.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = a(7)^2 + (4*A253805(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253303, A253802, A253805.

A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253805(n))^2,
n >= 1, that is,
a(n) = sqrt(A080175(n) - (4*A253805(n))^2), n >= 1.

A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

Original entry on oeis.org

5, 30, 34, 145, 111, 180, 371, 330, 876, 1560, 1746, 505, 1635, 840, 3014, 3570, 5181, 2249, 1710, 7980, 1379, 3435, 10920, 7230, 2056, 8970, 14490, 11240, 4981, 3900

Offset: 1

Wolfdieter Lang, Jan 16 2015

See A253804 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253804(7)^2 + (4*a(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A080109, A253804, A253802, A253803.

a(n) = sqrt(A080109(n)^2 - A253804(n)^2)/4, n >= 1.

A253302 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to n, when there are at least two such x's.

Original entry on oeis.org

1, 16, 300, 136, 50316, 186, 97748, 1352, 72318, 1340, 1652, 1356

Offset: 1

Michel Marcus, Dec 30 2014

Only integers m that satisfy A048138(m) > 1 are considered here.
Other known terms with odd indices: a(25)=39926, a(49)=46850.
For terms with even indices see A253303.

			For every prime p the sum of proper divisors of p is 1, and the GCD of the primes is 1, so a(1) = 1.

Cf. A001065 (sum of proper divisors), A048138, A152454, A253303.

build(nb) = {vs = vector(nb); nc = nb^2; forcomposite(n=2, nc, val = sigma(n)-n; if (val <= nb, if (vs[val] == 0, vs[val] = -n, vs[val] = gcd(vs[val], n)););); vs[1] = 1; vs;}
lista() = {vd = build(100000); vmax = 20; vr = vector(vmax); for (ig=1, vmax, for (i=1, #vd, w = vd[i]; if (w ==ig, vr[ig] = i; break;););); for (i=1, #vr, if (vr[i] == 0, break, print1(vr[i], ", ")));}

a(1)=1 suggested by Robert Israel, Jan 01 2015

cp's OEIS Frontend

A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

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A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

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Formula

A253302 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to n, when there are at least two such x's.

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