A051612 -id:A051612 - cp's OEIS frontend

A077102 Smallest number m such that GCD(a+b,a-b) = n, where a = sigma(m) and b = phi(m).

4, 1, 18, 21, 200, 14, 3364, 12, 722, 328, 9801, 42, 25281, 116, 1800, 15, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 284, 98942809, 488, 1547536, 364, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 440, 3150464641

Offset: 1

Labos Elemer, Nov 12 2002

nonn

			For n = 10, a(10) = 328, sigma(328) = 630, phi(328) = 160, sigma(328) + phi(328) = 790, sigma(328) - phi(328) = 470, GCD(790,470) = 10.
For n = odd number, a(n) should be either a square or twice a square and so faster search for large values is possible, like e.g., for n = 97: a(97) = 435979^2 is the smallest solution.

Amiram Eldar, Table of n, a(n) for n = 1..82

Cf. A000010, A000203, A051612, A065387, A077099, A077100, A077101.

f[x_] := Apply[GCD, {DivisorSigma[1, x]+EulerPhi[x], DivisorSigma[1, x]-EulerPhi[x]}]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}]; t

lista(len) = {my(v = vector(len), c = 0, k = 1, a, b, i); while(c < len, f = factor(k); a = sigma(f); b = eulerphi(f); i = gcd(a+b,a-b); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 14 2024

a(n) = Min{x; A077099(x) = n}.

A085122 a(n) = PrimePi(sigma(n)-phi(n)) - (PrimePi(sigma(n)) - PrimePi(phi(n))), where PrimePi = A000720, sigma = A000203 and phi = A000010.

Original entry on oeis.org

0, -1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 3, 3, 0, 2, 0, 3, 2, 2, 3, 3, 0, 3, 4, 3, 0, 4, 0, 3, 4, 3, 0, 4, 3, 5, 3, 5, 0, 3, 3, 3, 3, 3, 0, 3, 0, 4, 3, 4, 3, 4, 0, 5, 5, 5, 0, 4, 0, 2, 5, 4, 4, 4, 0, 5, 5, 5, 0, 7, 4, 5, 4, 5, 0, 4, 3, 5, 5, 5, 5, 4, 0, 5, 5, 5, 0, 6, 0, 6, 6, 7, 0, 5, 0, 5, 6, 8, 0, 5, 5, 6, 7, 5, 5, 5, 6, 5, 6, 7, 5, 5, 0, 7

Offset: 1

Labos Elemer, Jul 11 2003

Scatterplot of this sequence shows interesting strata. - Antti Karttunen, Jan 22 2020

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

Cf. A000010, A000203, A000720, A051612, A070803, A070804, A085343.

t=Table[PrimePi[DivisorSigma[1, w]-EulerPhi[w]]- (PrimePi[DivisorSigma[1, w]]-PrimePi[EulerPhi[w]]), {w, 1, 10000}]

A085122(n) = (primepi(sigma(n)-eulerphi(n)) - (primepi(sigma(n))-primepi(eulerphi(n)))); \\ Antti Karttunen, Jan 22 2020

a(n) = A000720(A051612(n)) - (A070803(n) - A070804(n)) = A000720(A051612(n)) - A085343(n). - Antti Karttunen, Jan 22 2020

Name edited by Antti Karttunen, Jan 22 2020

A112730 Numbers k such that the equation sigma(x)-phi(x)=k has at least one solution.

Original entry on oeis.org

2, 5, 7, 10, 11, 14, 15, 16, 18, 20, 22, 23, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 47, 48, 50, 52, 54, 56, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 74, 75, 76, 78, 79, 80, 83, 84, 86, 87, 88, 90, 92, 94, 95, 96, 98, 100, 102, 104, 106, 107, 108, 110, 112

Offset: 1

Farideh Firoozbakht, Dec 12 2005

nonn

It is conjectured that if n>2 then all solutions of the equation sigma(x)-phi(x)=n (*) are less than or equal to (n-1)^2/4 and 2 is the only number n such that the equation (*) has infinitely many solutions. In fact in the case n=2 prime numbers are all solutions of (*). All numbers of the form 2p+1 where p is prime are in the sequence because p^2 is a solution for the equation sigma(x)-phi(x)=2p+1. All numbers of the form 3*2^n-1 are in the sequence because 2^(n+1) is a solution for the equation sigma(x)-phi(x)=3*2^n-1 and etc.
The conjecture in the previous comment was established by Luke Pebody, see the Rivera link.
Theorem (Luke Pebody): If integers n>2 and m satisfy sigma(m)-phi(m)=n then m<=(n-1)^2/4.
Proof: Case I: m=1. Then n=sigma(m)-phi(m)=0 is not more than 2.
Case II: m is prime. Then n=sigma(m)-phi(m)=(m+1)-(m-1)=2 is not more than 2.
Case III: m has at least one nontrivial divisor. Let m=pq where 1=m+p+1. Phi(m) is certainly no greater than the number of integers smaller than or equal to m that are not divisible by p. Thus phi(m)<=m-q. Thus n=sigma(m)-phi(m)>=p+q+1. Finally, the arithmetic mean of two numbers is always greater than their geometric mean, so sqrt(m)=sqrt(pq)<=(p+q)/2<=(n-1)/2. Squaring both sides, m<=(n-1)^2/4.

			5 is in the sequence because 4 is a solution to the equation sigma(x)-phi(x)=5.

Robert Israel, Table of n, a(n) for n = 1..4490 (terms <= 10000)
Carlos Rivera, Puzzle 343. One more Faride's question, The Prime Puzzles and Problems Connection.

Complement of A036446.

Cf. A051612.

N:= 120: # for terms <= N
S:= {}:
for k from 1 to (N-1)^2/4 do
  v:= numtheory:-sigma(k) - numtheory:-phi(k);
  if v > 0 and v <= N then S:= S union {v} fi;
od:
sort(convert(S,list)); # Robert Israel, Jul 21 2025

A228947 a(n) = sigma(n) - phi(n) - n.

Original entry on oeis.org

-1, 0, -1, 1, -3, 4, -5, 3, -2, 4, -9, 12, -11, 4, 1, 7, -15, 15, -17, 14, -1, 4, -21, 28, -14, 4, -5, 16, -27, 34, -29, 15, -5, 4, -11, 43, -35, 4, -7, 34, -39, 42, -41, 20, 9, 4, -45, 60, -34, 23, -11, 22, -51, 48, -23, 40, -13, 4, -57, 92, -59

Offset: 1

M. F. Hasler, Oct 05 2013

While terms with even indices are never negative, this is the case for most terms with odd indices; exceptions are listed in A229978.

Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Cf. A000010, A000203, A051612, A229978.

Table[DivisorSigma[1, n] - EulerPhi[n] - n, {n, 75}] (* Alonso del Arte, Oct 05 2013 *)

PARI
```
A228947(n)=sigma(n)-eulerphi(n)-n
```

a(n) = 0  <=>  n = 2 (conjectured).
a(2n) > 0  for all n > 1.
a(2n+1) > 0  <=>  n in A229978.
a(n) = A051612(n) - n = A000203(n) - A000010(n) - n.
a(p) = 2 - p for p prime. - Alonso del Arte, Oct 05 2013
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 - 6/Pi^2 - 1 = 0.0370069... . - Amiram Eldar, Dec 04 2023

A229978 Numbers k such that (2k+1) + phi(2k+1) <= sigma(2*k+1).

Original entry on oeis.org

7, 22, 31, 37, 52, 67, 82, 94, 97, 112, 115, 127, 136, 142, 148, 157, 172, 178, 187, 199, 202, 214, 217, 220, 232, 241, 247, 262, 277, 283, 292, 304, 307, 322, 325, 337, 346, 352, 367, 382, 388, 397, 409, 412, 427, 430, 442, 445, 451, 457, 472, 487, 502, 517, 532, 535, 547, 562, 577, 592, 598, 607, 622, 637, 643, 652, 661, 667, 682, 697, 712, 724, 727, 742, 757, 772, 787, 802, 808, 817

Offset: 1

M. F. Hasler, Oct 05 2013

It appears that the equation x + phi(x) = sigma(x) has the unique solution x=2. It is easy to show that this is the only even solution to the equation, but for odd solutions this is less obvious. The present sequence is motivated by the observation that for most odd numbers, the l.h.s. is larger than the r.h.s. (while the opposite is the case for all even numbers). (See also formulas in A228947.)
From Amiram Eldar, Dec 23 2024: (Start)
If k is an odd abundant number (A005231), then (k-1)/2 is a term of this sequence.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 9, 99, 981, 9879, 98613, 984293, 9850470, 98496984, 985005850, 9850433480, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0985... . (End)

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

Cf. A000010, A000203, A005231, A051612 and references there, A228947.

Select[Range[1000], DivisorSigma[1, 2*#+1] > EulerPhi[2*#+1] + 2*#+1 &] (* Amiram Eldar, Dec 23 2024 *)

PARI
```
select(n->(2*n+1+eulerphi(2*n+1)
				
```

A270778 Primes p such that sigma(p-1) - phi(p-1) = (3p-5)/2.

Original entry on oeis.org

3, 5, 11, 17, 257, 65537, 119831

Offset: 1

Jaroslav Krizek, Mar 22 2016

Primes p such that A051612(p-1) = (3p-5)/2.
Fermat primes from A019434 are terms.
If a(8) exists, it must be larger than 10^10.
Prime terms from A270836.
Necessary condition: sigma_-1(p-1) < 2. Thus a(n)-1 is a deficient number and a(n) = 2 mod 3 for n > 1. - Charles R Greathouse IV, Apr 01 2016
If a(8) exists, it must be larger than 10^11. - Charles R Greathouse IV, Apr 01 2016
If a(8) exists, it must be larger than 10^13. - Giovanni Resta, Apr 11 2016

			17 is a term because sigma(16)-phi(16) = 31-8 = 23 = (3*17-5)/2.

Cf. A000010, A000203, A051612, A270779, A270836.

[n: n in[1..10^7] | IsPrime(n) and 2*(SumOfDivisors(n-1) - EulerPhi(n-1)) eq 3*n-5]

Select[Prime@ Range[10^6], DivisorSigma[1, # - 1] - EulerPhi[# - 1] == (3 # - 5)/2 &] (* Michael De Vlieger, Mar 23 2016 *)

lista(nn) = forprime(p=2, nn, if (sigma(p-1) - eulerphi(p-1) == (3*p-5)/2, print1(p, ", "))); \\ Michel Marcus, Mar 23 2016

is(n)=my(f=factor(n-1)); sigma(f) - eulerphi(f) == (3*n-5)/2 && isprime(n) \\ Charles R Greathouse IV, Apr 01 2016

A292771 If sigma(n)-phi(n) is even then (sigma(n)-phi(n))/2 otherwise -1.

Original entry on oeis.org

0, 1, 1, -1, 1, 5, 1, -1, -1, 7, 1, 12, 1, 9, 8, -1, 1, -1, 1, 17, 10, 13, 1, 26, -1, 15, 11, 22, 1, 32, 1, -1, 14, 19, 12, -1, 1, 21, 16, 37, 1, 42, 1, 32, 27, 25, 1, 54, -1, -1, 20, 37, 1, 51, 16, 48, 22, 31, 1, 76, 1, 33, 34, -1, 18, 62, 1, 47, 26, 60, 1, -1, 1, 39, 42, 52, 18, 72, 1, 77, -1, 43, 1

Offset: 1

N. J. A. Sloane, Sep 28 2017

sign

a(n) = 1 if and only if n is prime. - Robert Israel, Sep 28 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051612, A000010, A000203, A292770.

f:= proc(n) local v;
   v:= numtheory:-sigma(n)-numtheory:-phi(n);
   if v::even then v/2 else -1 fi
end proc:
map(f, [$1..100]); # Robert Israel, Sep 28 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

A353276 a(n) = phi(n) + tau(n)^omega(n) - sigma(n).

Original entry on oeis.org

1, 0, 0, -2, 0, 6, 0, -7, -4, 2, 0, 12, 0, -2, 0, -18, 0, 3, 0, 2, -4, -10, 0, 12, -8, -14, -18, -8, 0, 448, 0, -41, -12, -22, -8, 2, 0, -26, -16, -10, 0, 428, 0, -28, -18, -34, 0, -8, -12, -37, -24, -38, 0, -38, -16, -32, -28, -46, 0, 1576, 0, -50, -32, -88, -20, 388, 0, -58, -36, 392, 0, -27, 0, -62, -48, -68, -20, 368

Offset: 1

Antti Karttunen, Apr 27 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A001221, A051612, A110088.

Cf. A110087 (positions of negative terms), A110086 (of terms >= 0), A110085 (of terms > 0).

Array[#1 + #3^#2 - #4 & @@ Flatten@ {EulerPhi[#], PrimeNu[#], DivisorSigma[{0, 1}, #]} &, 78] (* Michael De Vlieger, Apr 27 2022 *)

A353276(n) = (eulerphi(n) + (numdiv(n)^omega(n)) - sigma(n));

a(n) = A110088(n) - A051612(n) = A000010(n) + A000005(n)^A001221(n) - A000203(n).

a(p) = 0 for all primes p.

A226586 Odd values of sigma(n) - phi(n) in the order of appearance and with repetition.

Original entry on oeis.org

5, 11, 7, 23, 33, 11, 47, 79, 15, 73, 95, 171, 67, 129, 177, 23, 191, 355, 309, 27, 315, 385, 283, 289, 383, 723, 35, 739, 393, 39, 687, 801, 489, 1089, 711, 767, 47, 1459, 649, 281, 1599, 969, 801, 607, 1431, 1633, 59, 1971, 2581, 63, 1555, 1535, 1153, 1069, 2931, 1605, 927, 1843, 3319, 2121

Offset: 1

Wesley Ivan Hurt, Jun 28 2013

nonn

Odd values of A051612(n) sorted along n.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051612, A028982.

select(type, [seq(numtheory:-sigma(n)-numtheory:-phi(n), n=1..2000)], odd); # Robert Israel, Aug 11 2019

Select[Table[DivisorSigma[1,n]-EulerPhi[n],{n,2000}],OddQ] (* Harvey P. Dale, Sep 27 2013 *)

sigma(4) - phi(4) = 7 - 2 = 5. Since 5 is the first odd value of sigma(n) - phi(n), it appears first in the list. So a(1) = 5.

cp's OEIS Frontend

A077102 Smallest number m such that GCD(a+b,a-b) = n, where a = sigma(m) and b = phi(m).

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A085122 a(n) = PrimePi(sigma(n)-phi(n)) - (PrimePi(sigma(n)) - PrimePi(phi(n))), where PrimePi = A000720, sigma = A000203 and phi = A000010.

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A112730 Numbers k such that the equation sigma(x)-phi(x)=k has at least one solution.

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A228947 a(n) = sigma(n) - phi(n) - n.

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A229978 Numbers k such that (2*k+1) + phi(2*k+1) <= sigma(2*k+1).

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A270778 Primes p such that sigma(p-1) - phi(p-1) = (3p-5)/2.

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Magma

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A292771 If sigma(n)-phi(n) is even then (sigma(n)-phi(n))/2 otherwise -1.

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

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A229978 Numbers k such that (2k+1) + phi(2k+1) <= sigma(2*k+1).