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A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2*n^2, which is A072779(n) - 2*n^2. 3
0, 0, 0, 3, 0, 2, 0, 17, 7, 2, 0, 34, 0, 2, 2, 77, 0, 41, 0, 82, 2, 2, 0, 178, 21, 2, 82, 154, 0, 76, 0, 325, 2, 2, 2, 411, 0, 2, 2, 450, 0, 124, 0, 370, 188, 2, 0, 786, 43, 115, 2, 514, 0, 428, 2, 858, 2, 2, 0, 948, 0, 2, 356, 1333, 2, 268, 0, 874, 2, 156, 0, 2047, 0, 2, 220 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

This sequence is interesting because (1) a(n) >= 0, with equality only when n is prime (or 1) and (2) a(n) = 2 if and only if n is the product of two distinct primes. Note for twin primes: let n = m^2 - 1, then m-1 and m+1 are twin primes if and only if a(n) = 2. Note for the Goldbach conjecture: let n = m^2 - r^2, then m-r and m+r are primes that add to 2m if and only if a(n) = 2.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Divisor Function

Eric Weisstein's World of Mathematics, Totient Function

MATHEMATICA

Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n]-2n^2, {n, 100}]

PROG

(PARI) a(n)=sigma(n, 2)+eulerphi(n)*sigma(n)-2*n^2 \\ Charles R Greathouse IV, May 15 2013

CROSSREFS

Cf. A051709, A072779.

Sequence in context: A171759 A073538 A022898 * A124452 A351532 A291786

Adjacent sequences: A072777 A072778 A072779 * A072781 A072782 A072783

KEYWORD

easy,nice,nonn

AUTHOR

T. D. Noe, Jul 15 2002

STATUS

approved

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Last modified January 28 12:41 EST 2023. Contains 359892 sequences. (Running on oeis4.)