A051709 -id:A051709 - cp's OEIS frontend

A297159 a(n) = 3n - 2phi(n) - sigma(n); Difference between the deficiency of n and its Moebius-transform.

0, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 0, 1, 6, 5, 1, 1, 3, 1, 2, 7, 10, 1, -4, 4, 12, 5, 4, 1, 2, 1, 1, 11, 16, 9, -7, 1, 18, 13, -2, 1, 6, 1, 8, 9, 22, 1, -12, 6, 17, 17, 10, 1, 6, 13, 0, 19, 28, 1, -20, 1, 30, 13, 1, 15, 14, 1, 14, 23, 18, 1, -27, 1, 36, 21, 16, 15, 18, 1, -10, 14, 40, 1, -20, 19, 42, 29, 4, 1, -12, 17, 20, 31, 46, 21, -28, 1, 39, 21, 3, 1, 26

Offset: 1

Antti Karttunen, Mar 02 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A000203, A008683, A033879, A083254.

Cf. also A051709, A296074.

a[n_] := 3*n - 2*EulerPhi[n] - DivisorSigma[1, n]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A297159(n) = (3*n - 2*eulerphi(n) - sigma(n));

PARI
```
A297159(n) = -sumdiv(n,d,(d
				
```

from sympy import totient, divisor_sigma
def a(n): return 3*n-2*totient(n)-divisor_sigma(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 02 2018

a(n) = A033879(n) - A083254(n) = 3*n - 2*A000010(n) - A000203(n).
a(n) = -Sum_{d|n, dA008683(n/d)*A033879(d).
Sum_{k=1..n} a(k) = (3/2 - 6/Pi^2 - Pi^2/12) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 04 2023

A278373 Numbers of the form sigma(k) + phi(k) - 2k.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 25, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94, 96, 97, 98, 100, 102, 104, 106, 108, 109, 110, 111, 112, 114

Offset: 1

David W. Wilson, Nov 19 2016

nonn

Empirically, every integer n >= 18 is of the form n = p+q+r for distinct primes p,q,r. If true, then every even number e >= 36 is this sequence, since e = 2(p+q+r) = sigma(pqr) + phi(pqr) - 2pqr, which implies all even e >= 0 are in this sequence.

David W. Wilson, Table of n, a(n) for n = 1..10000

Smallest k with sigma(k) + phi(k) - 2k = a(n) is A278374(n).
Complement is A056996.
Sequence A051709 sorted into ascending order, with duplicates removed.- Antti Karttunen, Dec 09 2016
Cf. A000010 (phi), A000203 (sigma).

Take[#, 85] &@ Union@ Table[DivisorSigma[1, n] + EulerPhi@ n - 2 n, {n, 10^4}] (* Michael De Vlieger, Nov 30 2016 *)

A345051 Numbers k such that A345048(k) is equal to A345049(k).

Original entry on oeis.org

2, 6, 9, 15, 28, 496, 625, 1225, 3993, 8128, 117649, 218491, 857375, 3788435, 4259571, 33550336, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 8589869056, 10214476341

Offset: 1

Antti Karttunen, Jun 06 2021

Numbers k for which A342001(n) * A051709(n) = A173557(n) * A345001(n).

Conjecture: Sequence is a disjoint union of A000396 and A166374, i.e., there are no terms of any other kind.

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

Cf. A051709, A173557, A342001, A345001, A345048, A345049.
Positions of zeros in A345050.
Cf. A000396, A166374 (subsequences).
Cf. also A345003.

PARI
```
isA345051(n) = (0==A345050(n));
```

a(21)-a(27) from Amiram Eldar, Dec 08 2023

A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2n^2, which is A072779(n) - 2n^2.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jul 15 2002

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.

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.

Cf. A000010, A000203, A001157, A051709, A072779.

Cf. A002117, A065465.

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

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) + Product_{p prime} (1 - 1/(p^2*(p+1))) - 2 = A002117 + A065465 - 2 = 0.083570742884... . - Amiram Eldar, Dec 03 2023

A228498 a(n) = sigma(n^2) + phi(n^2) - 2n^2.

Original entry on oeis.org

0, 1, 1, 7, 1, 31, 1, 31, 13, 57, 1, 163, 1, 91, 73, 127, 1, 307, 1, 321, 111, 183, 1, 691, 31, 241, 121, 535, 1, 1261, 1, 511, 211, 381, 157, 1591, 1, 463, 273, 1377, 1, 2163, 1, 1131, 781, 651, 1, 2803, 57, 1467, 421, 1513, 1, 2791, 273, 2311, 507, 993, 1, 6253, 1, 1123, 1227, 2047, 343, 4711, 1, 2445, 703

Offset: 1

Wesley Ivan Hurt, Aug 23 2013

If n is a prime, p, then a(p) = 1. Proof: a(p) = sigma(p^2) + phi(p^2) - 2p^2 = p^2 + p + 1 + p^2*( 1-(1/p) ) - 2p^2 = p^2 + p + 1 + p^2 - p - 2p^2 = 1.

			a(6) = 31; sigma(6^2) + phi(6^2) - 2*6^2 = 91 + 12 - 72 = 31.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A051709 (sequence at n instead of n^2).

Cf. A000010, A000203, A001105, A002117, A002618, A065764.

with(numtheory); seq(sigma(k^2) + phi(k^2) - 2*k^2, k=1..20);

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

vector(100, n, sigma(n^2)+eulerphi(n^2)-2*n^2) \\ Altug Alkan, Oct 28 2015

a(n) = A051709(n^2).
a(n) = A000203(n^2) + A000010(n^2) - 2*n^2.
a(n) = A065764(n) + A002618(n) - A001105(n).
Sum_{k=1..n} a(k) ~ ((5*zeta(3) + 2)/ Pi^2 - 2/3) * n^3. - Amiram Eldar, Dec 03 2023

More terms from Antti Karttunen, Oct 30 2017

A324048 a(n) = A000203(n) - A083254(n) = n + sigma(n) - 2*phi(n).

Original entry on oeis.org

0, 3, 3, 7, 3, 14, 3, 15, 10, 20, 3, 32, 3, 26, 23, 31, 3, 45, 3, 46, 29, 38, 3, 68, 16, 44, 31, 60, 3, 86, 3, 63, 41, 56, 35, 103, 3, 62, 47, 98, 3, 114, 3, 88, 75, 74, 3, 140, 22, 103, 59, 102, 3, 138, 47, 128, 65, 92, 3, 196, 3, 98, 95, 127, 53, 170, 3, 130, 77, 166, 3, 219, 3, 116, 119, 144, 53, 198, 3, 202, 94, 128, 3

Offset: 1

Antti Karttunen, Feb 13 2019

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A000203, A001065, A051709, A051612, A051953, A083254, A297159.

a[n_] := n + DivisorSigma[1, n] - 2 * EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A083254(n) = (2*eulerphi(n)-n);
A324048(n) = (sigma(n) - A083254(n));

a(n) = {my(f = factor(n)); n + sigma(f) - 2*eulerphi(f);} \\ Amiram Eldar, Dec 04 2023

a(n) = A000203(n) - A083254(n) = n + A000203(n) - 2*A000010(n).
a(n) = A051612(n) + A051953(n).
a(n) = A297159(n) + 2*A001065(n).
Sum_{k=1..n} a(k) = (Pi^2/12 - 6/Pi^2 + 1/2) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 04 2023

cp's OEIS Frontend

A297159 a(n) = 3*n - 2*phi(n) - sigma(n); Difference between the deficiency of n and its Moebius-transform.

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A278373 Numbers of the form sigma(k) + phi(k) - 2k.

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A345051 Numbers k such that A345048(k) is equal to A345049(k).

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A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2*n^2, which is A072779(n) - 2*n^2.

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A228498 a(n) = sigma(n^2) + phi(n^2) - 2n^2.

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A324048 a(n) = A000203(n) - A083254(n) = n + sigma(n) - 2*phi(n).

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Formula

A297159 a(n) = 3n - 2phi(n) - sigma(n); Difference between the deficiency of n and its Moebius-transform.

A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2n^2, which is A072779(n) - 2n^2.