A065387 -id:A065387 - cp's OEIS frontend

A145749 Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).

6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 902, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2013, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578

Offset: 1

Farideh Firoozbakht, Nov 01 2008

If n/2 is an odd prime and n+1 is prime then n is in the sequence, the proof is easy. 8,188,285,902,2013,... are terms of the sequence which they aren't of such form. This sequence is a subsequence of A066198.

If p is an odd Sophie Germain prime then 2*p is in the sequence. There is no term of the sequence which is of the form 2*p where p is prime and p isn't Sophie Germain prime. A244438 gives terms of the sequence which isn't of the form 2*p where p is prime. - Farideh Firoozbakht, Aug 14 2014

			10 is in the sequence because phi(10) + sigma(10) = 4 + 18 = 22 and phi(11) + sigma(11) = 10 + 12 = 22 also.
12 is not in the sequence because phi(12) + sigma(12) = 4 + 28 = 32 but phi(13) + sigma(13) = 12 + 14 = 26.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A065387, A066198, A145748, A005384.

Select[Range[2600],DivisorSigma[1,# ]+EulerPhi[ # ]==DivisorSigma[1,#+1]+EulerPhi[ #+1]&]

for(n=1,10^4, s=eulerphi(n)+sigma(n); if(s==eulerphi(n+1)+sigma(n+1), print1(n,", "))) /* Derek Orr, Aug 14 2014*/

{n: A065387(n)=A065387(n+1)}.

A011774 Nonprimes k that divide sigma(k) + phi(k).

Original entry on oeis.org

1, 312, 560, 588, 1400, 23760, 59400, 85632, 147492, 153720, 556160, 569328, 1590816, 2013216, 3343776, 4563000, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 45326160, 52021242, 75007400, 113315400, 137617728, 153587720, 402831360, 699117024

Offset: 1

R. K. Guy

2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023

			a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.

Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13; first 53 terms from Donovan Johnson)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Q.-X. Jin and M. Tang, The 4-Nicol Numbers Having Five Different Prime Divisors, J. Int. Seq. 14 (2011) # 11.7.2.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A065387, A011251, A011254, A055681, A001771, A112729.

Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]

sp(n)=my(f=factor(n));n*prod(i=1, #f[,1], 1-1/f[i,1]) + prod(i=1, #f[,1], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))
p=2;forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", ")));p=q) \\ Charles R Greathouse IV, Mar 19 2012

More terms from David W. Wilson

Corrected by Labos Elemer, Feb 12 2004

A015702 Numbers k where phi(k) + sigma(k) increases to a record value.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 40, 42, 48, 56, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 280, 288, 300, 324, 336, 360, 420, 480, 504, 540, 576, 600, 648, 660, 672, 720, 840, 960, 1008, 1080, 1200, 1260, 1440

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1288 (terms < 10^12)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A000203, A000010, A018892, A018894, A065387.

seq = {}; sm = 0; s = 0; Do[s = EulerPhi[n] + DivisorSigma[1, n];
If[s > sm, sm = s; AppendTo[seq, n]], {n, 1, 1500}]; seq (* Amiram Eldar, Dec 05 2018 *)
DeleteDuplicates[Table[{n,EulerPhi[n]+DivisorSigma[1,n]},{n,1500}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 13 2023 *)

f(n)=eulerphi(n=factor(n))+sigma(n)
r=0;for(n=1,1e6,t=f(n); if(t>r,r=t; print1(n", "))) \\ Charles R Greathouse IV, Nov 27 2013

A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

Original entry on oeis.org

1, 3, 18, 12, 200, 14, 3364, 15, 722, 328, 9801, 42, 25281, 116, 1800, 165, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 172, 98942809, 488, 1547536, 336, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 319, 3150464641

Offset: 1

Labos Elemer, Nov 12 2002

nonn

Values are frequently identical to terms of A077102. Since gcd(a,b) and gcd(a+b,a-b) may differ, so may the smallest solutions. A077102(m) and a(m) differ at m = 1, 2, 4, 8, 16, 28, 32, 40, etc.

Giovanni Resta, Table of n, a(n) for n = 1..120

Cf. A000203, A000010, A009223, A055008, A077099-A077102, A051612, A065387, A023897, A020492.

f[x_] := Apply[GCD, {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}];

a(n)=my(x=n);while(gcd(eulerphi(x),sigma(x))!=n, x++); x \\ Charles R Greathouse IV, Dec 09 2013

a(n) = Min{x; A055008(x)=n}. a(n)=Min{x; gcd(A000203(x), A000010(x))=n}

a(n) = Min{x: A023897(x)= n}, smallest balanced number (A020492) for which the quotient equals n.

A066198 Numbers n where phi changes as fast as sigma, i.e., abs(phi(n+1) - phi(n)) = abs(sigma(n+1) - sigma(n)).

Original entry on oeis.org

2, 6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 854, 862, 886, 902, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2013, 2026, 2038, 2062, 2098, 2206, 2446, 2458

Offset: 1

Joseph L. Pe, Dec 16 2001

nonn

This sequence is the union of two sequences A145748 and A145749. See comment lines of A145749. [Farideh Firoozbakht, Nov 01 2008]

			|phi(7) - phi(6)| = |6 - 2| = |8 - 12| = |sigma(7) - sigma(6)|.
|phi(9) - phi(8)| = |6 - 4| = 2 = |13 - 15| = |sigma(9) - sigma(8)|.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A065387, A145748, A145749.

Select[ Range[ 1, 10^4 ], Abs[ DivisorSigma[ 1, # + 1 ] - DivisorSigma[ 1, # ] ] == Abs[ EulerPhi[ # + 1 ] - EulerPhi[ # ] ] & ]

{ n=0; for (m=1, 10^9, if (abs(eulerphi(m + 1) - eulerphi(m)) == abs(sigma(m + 1) - sigma(m)), write("b066198.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010

More terms from Jason Earls, Jun 05 2002

A077085 Initial values such that if A077080(x)=phi(sigma(x)+phi(x)) is started at these numbers then the sequence does not converge.

Original entry on oeis.org

534, 556, 557, 580, 624, 702, 710, 738, 739, 740, 748, 784, 789, 822, 823, 841, 852, 853, 900, 912, 913, 916, 924, 931, 938, 960, 961, 962, 1020, 1021, 1029, 1032, 1033, 1034, 1065, 1089, 1092, 1093, 1098, 1126, 1136

Offset: 1

Labos Elemer, Oct 28 2002

nonn

These terms are only conjectures.

These terms survive 1000 iterations. - Sean A. Irvine, May 05 2025

Cf. A000010, A000203, A065387, A077080, A077081, A077082, A077083, A077084.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Do[s=Part[NestList[f, n, 100], 100]; If[Greater[s, 10000000], Print[{n, s}]], {n, 1, 10000}]

A067351 Numbers k such that sigma(k) + phi(k) has exactly 2 distinct prime divisors.

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 64, 66, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 87, 89, 91, 92, 93, 95, 96, 97

Offset: 1

Labos Elemer, Jan 17 2002

nonn

			Includes all odd primes and some composites; e.g., 21 and 25, since sigma(21) + phi(21) = 32 + 12 = 44 = 2*2*11; sigma(25) + phi(25) = 31 + 20 = 51 = 3*17.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A001221, A065387, A067349, A067350.

Select[ Range[ 1, 100 ], Length[ FactorInteger[ DivisorSigma[ 1, # ]+EulerPhi[ # ] ] ]==2& ]
Select[Range[500], PrimeNu[EulerPhi[#] + DivisorSigma[1, #]] == 2 &] (* G. C. Greubel, May 08 2017 *)

a(n) = A001221(A000010(n) + A000203(n)) = A001221(A065387(n)) = 2.

Edited by Dean Hickerson, Jan 20 2002

A071390 Least number m such that sigma(m) - phi(m) = n, or 0 if no such m exists.

Original entry on oeis.org

0, 2, 0, 0, 4, 0, 9, 0, 0, 6, 8, 0, 0, 10, 49, 15, 0, 14, 0, 21, 0, 27, 16, 12, 0, 22, 169, 33, 0, 26, 0, 39, 18, 20, 289, 65, 0, 34, 361, 51, 0, 38, 0, 28, 0, 0, 32, 95, 0, 46, 0, 24, 0, 45, 0, 115, 0, 0, 841, 161, 0, 58, 961, 30, 0, 62, 81, 63, 0, 0, 0, 155, 50, 40, 1369, 217, 0, 74

Offset: 1

Labos Elemer, May 23 2002

nonn

For n <> 2, a(n) < n^2/4. - Robert Israel, Apr 02 2020

			n=255: a(255) = 16129 = 127^2, sigma(16129) = 16257, phi(16129) = 16002, 16257 - 16002 = 255 = n. Squares of primes are often solutions (4, 9, 49, 169, 289, 361, etc.).

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000010, A000203, A065387, A051612, A071391.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for m from 2 to N^2/4 do
  v:= numtheory:-sigma(m)-numtheory:-phi(m);
  if v <= N and V[v]=0 then V[v]:= m fi
od:
convert(V,list); # Robert Israel, Apr 02 2020

f[x_] := DivisorSigma[1, x]-EulerPhi[x] t=Table[0, {100}]; Do[c=f[n]; If[c<101&&t[[c]]==0, t[[c]]=n], {n, 1, 1000}]; t

a(n) = Min{x; A000203(x)-A000010(x)=n} or a(n)=0 if no solution exists.

A270837 Numbers k such that sigma(k-1) + phi(k-1) = (5*k-7)/2.

Original entry on oeis.org

3, 5, 7, 9, 17, 33, 65, 67, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 5606129563

Offset: 1

Jaroslav Krizek, Mar 23 2016

nonn

Numbers k such that A065387(k-1) = (5*k-7)/2.
Numbers of the form 2^k + 1 for k >= 1 from A000051 are terms.
Prime terms are in A270779.

			17 is a term because sigma(16)+phi(16) = 31+8 = 39 = (5*17-7)/2.

Giovanni Resta, Table of n, a(n) for n = 1..47 (terms < 10^13)

Cf. A000010, A000051, A000203, A065387, A270779, A270836.

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

Select[Range[10^6], DivisorSigma[1, # - 1] + EulerPhi[# - 1] == (5 # - 7)/2 &] (* Michael De Vlieger, Mar 24 2016 *)

lista(nn) = {for(n=2, nn, if(sigma(n-1) + eulerphi(n-1) == (5*n-7)/2, print1(n, ", "))); } \\ Altug Alkan, Mar 23 2016

a(29)-a(31) from Michel Marcus, Apr 05 2016

a(32)-a(35) from Giovanni Resta, Apr 11 2016

A061367 Composite n such that sigma(n)-phi(n) divides sigma(n)+phi(n).

Original entry on oeis.org

15, 35, 95, 119, 143, 209, 287, 319, 323, 357, 377, 527, 559, 779, 899, 923, 989, 1007, 1045, 1189, 1199, 1343, 1349, 1763, 1919, 2159, 2261, 2507, 2639, 2759, 2911, 3239, 3339, 3553, 3599, 3827, 4031, 4147, 4607, 5049, 5183, 5207, 5249, 5459, 5543, 6439

Offset: 1

Joseph L. Pe, Feb 13 2002

nonn

Primes trivially satisfy the defining condition.

			sigma(15)-phi(15) = 24-8 = 16 divides sigma(15)-phi(15)=24+8 = 32, so 15 is a term of the sequence.

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

Cf. A051612, A065387.

f[n_] := Module[{a = DivisorSigma[1, n], b = EulerPhi[n]}, Mod[(a + b), (a - b)] == 0]; Select[Range[2, 10^4], (f[ # ] && ! PrimeQ[ # ]) &]
cnQ[n_]:=With[{s=DivisorSigma[1,n],p=EulerPhi[n]},Mod[s+p,s-p]==0]; Select[Range[6500],CompositeQ[#]&&cnQ[#]&] (* Harvey P. Dale, Jun 14 2025 *)

It seems that a(n) is asymptotic to c*n^2, 22*n^2. - Benoit Cloitre, Sep 17 2002

cp's OEIS Frontend

A145749 Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).

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A011774 Nonprimes k that divide sigma(k) + phi(k).

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A015702 Numbers k where phi(k) + sigma(k) increases to a record value.

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A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

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A066198 Numbers n where phi changes as fast as sigma, i.e., abs(phi(n+1) - phi(n)) = abs(sigma(n+1) - sigma(n)).

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A077085 Initial values such that if A077080(x)=phi(sigma(x)+phi(x)) is started at these numbers then the sequence does not converge.

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A067351 Numbers k such that sigma(k) + phi(k) has exactly 2 distinct prime divisors.

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A071390 Least number m such that sigma(m) - phi(m) = n, or 0 if no such m exists.

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