A065655 -id:A065655 - cp's OEIS frontend

A065656 Composite numbers k such that sigma(k)phi(k) + 2(k+1) is a square.

1169, 7777, 41111, 46097, 668167, 846817, 2107519, 3612769, 17424241, 30666527, 37526993, 56323393, 214746055, 383523857, 512376769, 1021934641, 1228492849, 1303949599, 4056001351, 7425397169, 17073544447, 17859428369, 18452226887, 46874737969, 51411954391

Offset: 1

Labos Elemer, Nov 12 2001

nonn

a(n) and square root of phi(a(n))*sigma(a(n)) + 2*a(n) + 2 are close to each other: e.g., a(7) = 2107519 and this square root is 2107458.
Since (p+1)*(p-1) + 2*(p+1) = p*p + 2*p + 1 = (p+1)^2 is a square, all primes are solutions.
73362272287 and 181264312447 are also terms. - Donovan Johnson, Jul 13 2012

			k = 7777: sigma(7777) = 9792, phi(7777) = 6000 and 9792*6000 + 2*7778 = 587675556 = 7666^2.

Cf. A062354, A000203, A000010, A065655.

isok(k) = { !isprime(k) && issquare(sigma(k)*eulerphi(k) + 2*(k + 1)) } \\ Harry J. Smith, Oct 26 2009

a(9)-a(15) from Harry J. Smith, Oct 26 2009
a(16)-a(20) from Donovan Johnson, May 24 2011
a(21)-a(25) from Donovan Johnson, Jul 13 2012

A065550 a(n) = floor(sqrt(phi(w)*sigma(w)+w^2)), where w=10^n.

Original entry on oeis.org

13, 136, 1391, 14030, 140865, 1411444, 14128309, 141352267, 1413868217, 14140409111, 141412724154, 1414170403052, 14141919829640, 141420277272713, 1414208167563878, 14142108649717545, 141421221367320690, 1414212888023339560, 14142132251982630599, 141421339378569021517

Offset: 1

Labos Elemer, Nov 13 2001

nonn

a(n) tends to sqrt(2)*(10^n) when n->oo.

Robert Israel, Table of n, a(n) for n = 1..995

Cf. A000290, A062354, A065655, A065656.

a:= n -> floor(sqrt(2*100^n - 20^n/5 - 50^n/2 + 10^n/10)):
map(a, [$1..100]); # Robert Israel, Dec 03 2024

a[n_] := Floor[Sqrt[EulerPhi[10^n] * DivisorSigma[1, 10^n] + 100^n]]; Array[a, 20] (* Amiram Eldar, Jun 12 2022 *)

a(n) = my(w=10^n); sqrtint(eulerphi(w)*sigma(w)+w^2); \\ Michel Marcus, Mar 23 2020

from sympy import integer_nthroot, totient as phi, divisor_sigma as sigma
def isqrt(n): return integer_nthroot(n, 2)[0]
def a(n): w = 10**n; return isqrt(phi(w)*sigma(w, 1) + w**2)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 12 2022

a(n) = floor(sqrt(A062354(w) + A000290(w))), where w=10^n.

a(n) = floor(10^n * sqrt(2 - 5^(-n-1) - 2^(-n-1) + 10^(-n-1))). - Robert Israel, Dec 03 2024

Corrected and extended by Michel Marcus, Jun 12 2022

A064656 Length of n-th run of odd numbers in A064413.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 09 2001

J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
Index entries for sequences related to EKG sequence

Cf. A064413, A064655, A065655.

More terms from Matthew Conroy, Oct 16 2001

A065552 a(n) = floor(sqrt(phi(10^n)sigma(10^n) + 10^(3n))).

Original entry on oeis.org

1, 32, 1004, 31637, 1000048, 31622932, 1000000496, 31622778176, 1000000004990, 31622776617479, 1000000000049975, 31622776601841868, 1000000000000499938, 31622776601685374362, 1000000000000004999847, 31622776601683809131135, 1000000000000000049999618, 31622776601683793478102215

Offset: 0

Labos Elemer, Nov 13 2001

nonn

Similar results are obtained if the cube is replaced with other odd powers.

Cf. A000010, A000203, A062354, A065655, A065656.

a[n_]:=Floor[Sqrt[EulerPhi[10^n]DivisorSigma[1,10^n]+10^(3n)]]; Array[a,17,0] (* Stefano Spezia, Mar 23 2023 *)

a(0) = 1 prepended by, a(11)-a(15) corrected by, and a(16)-a(17) from Stefano Spezia, Mar 23 2023

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A065656 Composite numbers k such that sigma(k)*phi(k) + 2*(k+1) is a square.

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A065550 a(n) = floor(sqrt(phi(w)*sigma(w)+w^2)), where w=10^n.

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A064656 Length of n-th run of odd numbers in A064413.

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A065552 a(n) = floor(sqrt(phi(10^n)*sigma(10^n) + 10^(3*n))).

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A065656 Composite numbers k such that sigma(k)phi(k) + 2(k+1) is a square.

A065552 a(n) = floor(sqrt(phi(10^n)sigma(10^n) + 10^(3n))).