A000010 -id:A000010 - cp's OEIS frontend

A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

48, 68, 72, 78, 86, 88, 114, 143, 144, 156, 157, 164, 168, 186, 192, 203, 216, 222, 247, 273, 292, 356, 402, 432, 444, 450, 452, 456, 612, 654, 728, 732, 762, 798, 834, 864, 876, 884, 932, 942, 964, 1032, 1054, 1080, 1086, 1124, 1147, 1152, 1194, 1209, 1220

Offset: 1

Michel Marcus, Sep 27 2012

Most of numbers n in this sequence are divisible by 2, and it appears that n/2 belongs to A179188. The other ones are listed in sequence A217141.

Proof of the comment: If n is even and not a multiple of 4 then phi(n)=phi(n/2). If n is a multiple of 4 then phi(n)=2 * phi(n/2). So when k is a multiple of 4 and phi(n)=phi(n+k), then phi(n/2)=phi(n/2+k/2). QED. This also applies to A179186, A179202. - Jud McCranie, Dec 30 2012

Jud McCranie, Table of n, a(n) for n = 1..10000
Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Cf. A000010, A179188, A217141, A007015.

[n: n in [1..3000] | EulerPhi(n) eq EulerPhi(n+12)]; // Vincenzo Librandi, Sep 08 2016

Select[Range[1, 5000], EulerPhi[#] == EulerPhi[# + 12] &] (* Vincenzo Librandi, Jun 24 2014 *)

{op=vector(N=12); for( n=1, 1e4, if( op[n%N+1]+0==op[n%N+1]=eulerphi(n), print1(n-N, ", ")))}

A295503 a(n) = phi(10^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 60, 648, 6000, 64800, 466560, 6637344, 58752000, 648646704, 5890320000, 66663457344, 461894400000, 6458084523072, 60339430569600, 610154104320000, 5529599115264000, 66666634474902192, 441994921381739520, 6666666666666666660, 58301444908800000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352
Eric W. Weisstein, MathWorld: Totient Function
Wikipedia, Euler's totient function

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), this sequence (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A002283.

Array[ EulerPhi[10^# - 1] &, 20] (* Robert G. Wilson v, Nov 22 2017 *)

PARI
```
{a(n) = eulerphi(10^n-1)}
```

a(n) = n*A295497(n).

a(n) = A000010(A002283(n)). - Michel Marcus, Nov 25 2017

A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 11, 19, 30, 44, 69, 103, 157, 229, 341, 491, 722, 1038, 1488, 2128, 3015, 4267, 5989, 8407, 11713, 16289, 22523, 31097, 42729, 58569, 80003, 108957, 147983, 200383, 270693, 364631, 490105, 656961, 878775, 1172653, 1561626, 2074982, 2751648, 3641536, 4810009, 6341365, 8344967

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms

Cf. A000010, A061255, A107742, A159929, A192065, A318975.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(numtheory[phi](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 09 2018

nmax = 46; CoefficientList[Series[Product[(1 + x^k)^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d EulerPhi[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

G.f.: Product_{k>=1} (1 + x^k)^A000010(k).

a(n) ~ exp(3^(5/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi^(2/3))) * Zeta(3)^(1/6) / (2^(1/3) * 3^(1/6) * Pi^(5/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018

A324650 a(n) = A000010(A276086(n)).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 4, 4, 8, 8, 24, 24, 20, 20, 40, 40, 120, 120, 100, 100, 200, 200, 600, 600, 500, 500, 1000, 1000, 3000, 3000, 6, 6, 12, 12, 36, 36, 24, 24, 48, 48, 144, 144, 120, 120, 240, 240, 720, 720, 600, 600, 1200, 1200, 3600, 3600, 3000, 3000, 6000, 6000, 18000, 18000, 42, 42, 84, 84, 252, 252, 168, 168, 336, 336, 1008, 1008

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Terms are duplicated because phi(2*(2n+1)) = phi(2n+1) for all n >= 0.

Cf. A000010, A002110, A276086, A324651 (bisection).
Cf. also A267263, A276150, A324653, A324655 for omega, bigomega, sigma and tau analogs.
Cf. also A290077.

A324650(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m *= (prime(i)-1)*(prime(i)^(((n%nextpr)/pr)-1)); n-=(n%nextpr));pr=nextpr); (m); };

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324650(n) = eulerphi(A276086(n));

a(n) = A000010(A276086(n)).
a(2n+1) = a(2n) for all n >= 0.
For n >= 1, a(A002110(n-1)) = A000040(n)-1.

A366579 a(n) = phi(3^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 4, 12, 40, 120, 288, 1092, 3072, 7776, 23600, 87120, 259200, 797160, 1847104, 5832000, 21523360, 63672480, 152845056, 580921200, 1700870400, 4368821184, 12550120000, 47071589412, 130459631616, 413939700000, 997562438080, 3012122557440, 11159367815680

Offset: 0

Sean A. Irvine, Oct 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..691

Cf. A000010, A002592, A034472, A053285, A057935, A057941, A074476, A274909, A295500, A366577, A366578, A366580.

EulerPhi[3^Range[0,30]+1] (* Paolo Xausa, Oct 15 2023 *)

PARI
```
{a(n) = eulerphi(3^n+1)}
```

a(n) = phi(3^n+1) = A000010(A034472(n)).

A366608 a(n) = phi(4^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 16, 48, 256, 800, 3840, 12544, 65536, 186624, 986880, 3345408, 16515072, 52306176, 252645120, 760320000, 4288266240, 13628740608, 64258375680, 218462552064, 1095233372160, 3105655160832, 16510446886912, 56000724240384, 280012271910912, 869940000000000

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A000010, A052539, A057940, A274903, A295501, A366605, A366606, A366607, A366609.

Cf. A053285, A366579, A366618, A366630, A366639, A366658, A366667, A366669, A366690, A366716.

EulerPhi[4^Range[0,30]+1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
{a(n) = eulerphi(4^n+1)}
```

from sympy import totient
def A366608(n): return totient((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = A053285(2*n). - Max Alekseyev, Jan 08 2024

A226561 a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).

Original entry on oeis.org

1, 5, 55, 529, 12501, 94835, 4941259, 67240193, 2324562301, 40039063525, 2853116706111, 35668789979107, 3634501279107037, 66676110291801575, 3503151245145885315, 147575078498173255681, 13235844190181388226833, 236079349222711695887225, 35611553801885644604231623

Offset: 1

Paul D. Hanna, Jun 10 2013

nonn

Compare formula to the identity: Sum_{d|n} phi(d) = n.

			L.g.f.: L(x) = x + 5*x^2/2 + 55*x^3/3 + 529*x^4/4 + 12501*x^5/5 + 94835*x^6/6 + ...
where
exp(L(x)) = 1 + x + 3*x^2 + 21*x^3 + 155*x^4 + 2691*x^5 + 18924*x^6 + 732230*x^7 + 9223166*x^8 + ... + A226560(n)*x^n + ...

Robert Israel, Table of n, a(n) for n = 1..385

Cf. A226560, A226459, A000010, A321349, A332517.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (&+[EulerPhi(k)*(k*x)^k/(1-(k*x)^k): k in [1..2*m]]) )); // G. C. Greubel, Nov 07 2018

f:= n -> add(d^n * numtheory:-phi(d), d = numtheory:-divisors(n)):
map(f, [$1..40]); # Robert Israel, Jun 16 2017

Table[DivisorSum[n, #*EulerPhi[#^n]  &], {n, 1, 30}]  (* or *) With[{nmax = 30}, Rest[CoefficientList[Series[Sum[EulerPhi[k]*(k*x)^k/(1 - (k*x)^k), {k, 1, 2*nmax}], {x, 0, nmax}], x]]]  (* G. C. Greubel, Nov 07 2018 *)

{a(n)=sumdiv(n, d, d^n*eulerphi(d))}
for(n=1,30,print1(a(n),", "))

a(n) = sum(k=1, n, (n/gcd(k, n))^n); \\ Seiichi Manyama, Mar 11 2021

from sympy import totient, divisors
def A226561(n):
    return sum(totient(d)*d**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Logarithmic derivative of A226560.
a(n) = Sum_{d|n} d * phi(d^n).
a(n) = Sum_{d|n} phi(d^(n+1)).
a(n) = Sum_{d|n} phi(d^(n+2))/d.
a(n) = Sum_{d|n} d^(n-k+1) * phi(d^k) for k >= 1.
G.f.: Sum_{k>=1} phi(k)*(k*x)^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{k=1..n} (n/gcd(k,n))^n. - Seiichi Manyama, Mar 11 2021
a(n) = Sum_{k=1..n} gcd(n,k)^n*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021

A233542 Number of ways to write n = k^2 + m with k > 0 and m > 0 such that phi(k^2)*phi(m) - 1 is prime, where phi(.) is Euler's totient function (A000010).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 12 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) Any integer n > 7 can be written as k^2 + m with k > 0 and m > 0 such that phi(k)^2*phi(m) - 1 is prime.
(iii) If n > 1 is not equal to 36, then n can be written as k^2 + m with k > 0 and m > 0 such that sigma(k)^2*phi(m) + 1 is prime, where sigma(k) is the sum of all (positive) divisors of k.
We have verified part (i) of the conjecture for n up to 2*10^7.

			a(6) = 1 since 6 = 1^2 + 5 with phi(1^2)*phi(5) - 1 = 1*4 - 1 = 3 prime.
a(7) = 1 since 7 = 2^2 + 3 with phi(2^2)*phi(3) - 1 = 2*2 - 1 = 3 prime.
a(23) = 1 since 23 = 4^2 + 7 with phi(4^2)*phi(7) - 1 = 8*6 - 1 = 47 prime.
a(42) = 1 since 42 = 6^2 + 6 with phi(6^2)*phi(6) - 1 = 12*2 - 1 = 23 prime.
a(49) = 1 since 49 = 2^2 + 45 with phi(2^2)*phi(45) - 1 = 2*24 - 1 = 47 prime.
a(83) = 1 since 83 = 9^2 + 2 with phi(9^2)*phi(2) - 1 = 54*1 - 1 = 53 prime.
a(188) = 1 since 188 = 6^2 + 152 with phi(6^2)*phi(152) - 1 = 12*72 - 1 = 863 prime.
a(327) = 1 since 327 = 5^2 + 302 with phi(5^2)*phi(302) - 1 = 20*150 - 1 = 2999 prime.
a(557) = 1 since 557 = 12^2 + 413 with phi(12^2)*phi(413) - 1 = 48*348 - 1 = 16703 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000203, A233544, A233547, A233549.

a[n_]:=Sum[If[PrimeQ[EulerPhi[k^2]*EulerPhi[n-k^2]-1],1,0],{k,1,Sqrt[n-1]}]
Table[a[n],{n,1,100}]

A242959 Numbers n such that 5^A000010(n) == 1 (mod n^2).

Original entry on oeis.org

2, 20771, 40487, 41542, 80974, 83084, 161948, 643901, 1255097, 1287802, 1391657, 1931703, 2510194, 2575604, 2783314, 3765291, 3863406, 4174971, 5020388, 5151208, 5566628, 7530582, 7726812, 8349942, 10040776, 11133256, 15061164, 15308227, 15453624, 16699884

Offset: 1

Felix Fröhlich, May 27 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..479 (terms < 10^15, first 75 terms from Felix Fröhlich)

Cf. A077816, A242958, A242960.

If a(n) is prime, then a(n) is in A123692.

Select[Range[167*10^5],PowerMod[5,EulerPhi[#],#^2]==1&] (* Harvey P. Dale, Jun 02 2020 *)

for(n=2, 10^9, if(Mod(5, n^2)^(eulerphi(n))==1, print1(n, ", ")))

A306071 Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).

Original entry on oeis.org

8, 0, 7, 3, 3, 0, 8, 2, 1, 6, 3, 6, 2, 0, 5, 0, 3, 9, 1, 4, 8, 6, 5, 4, 2, 7, 9, 9, 3, 0, 0, 3, 1, 1, 3, 4, 0, 2, 5, 8, 4, 5, 8, 2, 5, 0, 8, 1, 5, 5, 6, 6, 4, 4, 0, 1, 8, 0, 0, 5, 2, 0, 7, 7, 0, 4, 4, 1, 3, 8, 1, 4, 8, 4, 9, 3, 7, 5, 1, 8, 6, 4, 9, 6, 9, 5, 6, 0, 9, 3, 5, 0, 9, 6, 2, 9, 4, 8, 3, 7, 6, 5, 0, 1, 1, 8

Offset: 0

Amiram Eldar, Jun 19 2018

The constant A that appears in the asymptotic formulae for the sums of the bi-unitary divisor function (A306069) and the bi-unitary totient function (A306070).
The product in Suryanarayana's 1972 paper has a error that was corrected in his 1975 paper.
The probability that 2 randomly selected numbers will be unitary coprime (i.e. their largest common unitary divisor is 1). - Amiram Eldar, Aug 27 2019

			0.80733082163620503914...

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 508), preprint, arXiv:1310.7053 [math.NT] (2014) (see p. 21).

Cf. A000010, A001221, A286324, A306069, A306070.

cc = CoefficientList[Series[Log[1 - (p - 1)/(p^2*(p + 1))] /. p -> 1/x, {x, 0, 36}], x]; f = FindSequenceFunction[cc]; digits = 20; A = Exp[NSum[f[n + 1 // Floor]*(PrimeZetaP[n]), {n, 2, Infinity}, NSumTerms -> 16 digits, WorkingPrecision -> 16 digits]]; RealDigits[A, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
$MaxExtraPrecision = 1000; Do[Print[Zeta[2] * Exp[-N[Sum[q = Expand[(2*p^2 - 2*p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}], 120]]], {t, 300, 1000, 100}] (* Vaclav Kotesovec, May 29 2020 *)

prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ Amiram Eldar, Mar 18 2021

Equals Product_{p prime} (1 - (p-1)/(p^2 * (p+1))).

Equals zeta(2) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4).

a(1)-a(20) from Jean-François Alcover, Jun 19 2018
a(20)-a(24) from Jon E. Schoenfield, May 27 2019
More terms from Vaclav Kotesovec, May 29 2020

cp's OEIS Frontend

A217139 Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010.

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A295503 a(n) = phi(10^n-1), where phi is Euler's totient function (A000010).

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A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).

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A324650 a(n) = A000010(A276086(n)).

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A366579 a(n) = phi(3^n+1), where phi is Euler's totient function (A000010).

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A366608 a(n) = phi(4^n+1), where phi is Euler's totient function (A000010).

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A226561 a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).

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