A344994 -id:A344994 - cp's OEIS frontend

A051709 a(n) = sigma(n) + phi(n) - 2n.

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 7, 0, 9, 0, 10, 2, 2, 0, 20, 1, 2, 4, 12, 0, 20, 0, 15, 2, 2, 2, 31, 0, 2, 2, 26, 0, 24, 0, 16, 12, 2, 0, 44, 1, 13, 2, 18, 0, 30, 2, 32, 2, 2, 0, 64, 0, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0

Offset: 1

Jud McCranie and Carlos Rivera

nonn

Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013
Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - T. D. Noe, Aug 01 2002
It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - Eric Desbiaux, Jul 04 2013
a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - Jianing Song, Jun 27 2021

			a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.)
Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).

Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=sigma(n)+eulerphi(n)-2*n \\ Charles R Greathouse IV, Jul 05 2013

A051709(n) = -sumdiv(n,d,(dAntti Karttunen, Mar 02 2018

Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002
From Antti Karttunen, Mar 02 2018: (Start)
a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]
a(n) = -Sum_{d|n, dA008683(n/d)*A001065(d). (End)
Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 03 2023

A344995 Numbers k such that A051709(k)/A173557(k) is a positive natural number and a divisor of k.

Original entry on oeis.org

4, 6, 12, 28, 42, 312, 456, 496, 588, 828, 1080, 1216, 1242, 1377, 1560, 4560, 5964, 7320, 7480, 7584, 8128, 11400, 13728, 14784, 23760, 33462, 59400, 59520, 124020, 147840, 188600, 277648, 321000, 543552, 1288224, 1510272, 1596048, 1964544, 2038140, 3323736, 3611520, 3780672, 3909816, 6137440, 9034032, 10783890

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A051709, A173557.
Subsequence of A344994.
Cf. also A344755, A345002.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA344995(n) = { my(u=A051709(n),t=A173557(n),r=u/t); ((u>0)&&(1==denominator(r)&&!(n%r))); };

A345054 Odd numbers k such that A173557(k) divides nonzero A051709(k).

Original entry on oeis.org

27, 243, 1377, 2187, 3125, 19683, 28125, 55233, 68445, 177147, 195625, 203125, 239805, 253125, 453125, 823543, 907137, 1323297, 1378125, 1464561, 1594323, 1953125, 2278125, 3341637, 3572829, 5255361, 5877117, 9034497, 9819837, 11701053, 14348907, 17453125, 19460393, 20503125, 22209633, 26010621, 30074733, 44910045

Offset: 1

Antti Karttunen, Jun 06 2021

nonn

Question: Are there any common terms with A345051?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Odd terms in A344994.

Cf. A051709, A173557, A345051.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA345054(n) = if(!(n%2),0,my(u=A051709(n)); ((u>0)&&(0==(u%A173557(n)))));

cp's OEIS Frontend

A051709 a(n) = sigma(n) + phi(n) - 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A344995 Numbers k such that A051709(k)/A173557(k) is a positive natural number and a divisor of k.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A345054 Odd numbers k such that A173557(k) divides nonzero A051709(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI