A063530 -id:A063530 - cp's OEIS frontend

A063533 Hypotenuses of special Pythagorean triples constructed from twin primes as follows: {u, w}={p,p+2}; side a=2p(p+2), side b=(p+2)^2-p^2 and the terms of sequence are values of c=a(n)=p^2+(p+2)^2=phi(a/2)+1+sigma(a/2)+1.

34, 74, 290, 650, 1802, 3530, 7202, 10370, 20810, 23330, 38090, 45002, 64802, 73730, 78410, 103970, 115202, 145802, 159050, 194690, 242210, 352802, 373250, 426890, 544970, 649802, 720002, 763850, 824330, 871202, 1312202, 1351370

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Sum of the numbers on the corners of the square array that lists the numbers from 1..A014574(n)^2 in increasing order by rows. - Wesley Ivan Hurt, May 27 2023

			a(6) is obtained as follows: u = p = 41, w = p+2 = 43; a = 2*41*43 = 2*1763 = 3526; b = 43*2-41^2 = 1849-1681 = 168; c = 43^2+41^2 = 1849+1681 = 3530 = 1+phi(1763)+1+sigma(1763) = 1680+1848+2 = a(6); and 3526^2+168^2 = 3530^2.

Cf. A001359, A006512, A000010, A000203, A014574, A037074, A063530-A063532.

a(n) = 2 + A000203(A037074(n)) + A000010(A037074(n)) = A001359(n)^2 + A006512(n)^2.

a(n) = 2*(A014574(n)^2 + 1). - Wesley Ivan Hurt, May 27 2023

A063531 Numbers k such that sigma(k) + 1 is a square.

Original entry on oeis.org

2, 7, 8, 14, 15, 23, 32, 33, 35, 47, 54, 56, 57, 60, 72, 78, 79, 84, 87, 92, 95, 120, 123, 124, 128, 138, 143, 154, 165, 167, 174, 184, 190, 196, 213, 223, 235, 242, 252, 253, 258, 267, 295, 312, 315, 319, 323, 327, 348, 359, 375, 378, 380, 393, 412, 423, 439

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Numbers k such that A000203(k) = -1 + m^2 for some m.

			If k = p(p+2) is a product of twin primes (from A037074), then sigma(k) + 1 = 1 + (p+1)(p+3) = (p+2)^2, square of the larger twin. Other solutions can be either special primes = m^2 - 2 or composites like 120: sigma(120) = 120 + 60 + ... + 1 = 360 = 19^2 - 1. Square number solution is, e.g., 196: sigma(196) = 399 = 20^2 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A028871, A037074, A063530, A088580.

Select[Range[500],IntegerQ[Sqrt[DivisorSigma[1,#]+1]]&] (* Harvey P. Dale, Jul 02 2021 *)

{ n=0; for (a=1, 10^9, if (issquare(sigma(a) + 1), write("b063531.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

Minor edits from Franklin T. Adams-Watters, Aug 29 2009

A089952 Numbers n such that n+1 and phi(n)+1 are both perfect squares.

Original entry on oeis.org

15, 24, 35, 143, 168, 323, 575, 675, 728, 899, 1599, 1763, 2600, 3599, 3720, 5183, 5775, 10403, 11663, 19043, 19320, 21315, 22499, 29928, 32399, 36863, 38024, 39203, 47523, 51528, 51983, 54288, 57599, 67080, 72899, 79523, 93635, 97343

Offset: 1

Joseph L. Pe, Jan 17 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000290.

Intersection of A005563 and A063530.

Select[Range[10^5], IntegerQ[Sqrt[ # + 1]] && IntegerQ[Sqrt[EulerPhi[ # ] + 1]] &]

cp's OEIS Frontend

A063533 Hypotenuses of special Pythagorean triples constructed from twin primes as follows: {u, w}={p,p+2}; side a=2p(p+2), side b=(p+2)^2-p^2 and the terms of sequence are values of c=a(n)=p^2+(p+2)^2=phi(a/2)+1+sigma(a/2)+1.

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A063531 Numbers k such that sigma(k) + 1 is a square.

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A089952 Numbers n such that n+1 and phi(n)+1 are both perfect squares.

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Author

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Crossrefs

Programs

Mathematica