A000010 -id:A000010 - cp's OEIS frontend

A175667 Smallest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

1, 5, 7, 34, 11, 13, 58, 17, 19, 55, 23, 65, 106, 29, 31, 85, 0, 37, 0, 41, 43, 115, 47, 119, 125, 53, 133, 145, 59, 61, 0, 388, 67, 274, 71, 73, 298, 0, 79, 187, 83, 203, 346, 89, 209, 235, 0, 97, 394, 101, 103, 169, 107, 109, 253, 113, 458, 295, 0, 287, 0, 0, 127, 514, 131

Offset: 1

Enrique Pérez Herrero, Aug 05 2010

nonn

If p = 2*n+1 is a prime, and if n > 1 then a(n)=p.
From R. J. Mathar, Aug 07 2010: (Start)
First column in the array
1,3,8,10,18,24,30: A020488
5,9,15,28,40,72,84,90,120: A062516
7,21,26,56,70,78,108,126,168,210: A063469
34,45,52,102,140,156,252,360,420: A063470
11,33,88,110,198,264,330,
13,35,39,63,76,104,105,130,228,234,280,312,390,504,540,630,840,
58,98,174,294,
17,51,128,136,170,176,224,260,306,384,408,468,510,528,672,780,1260,
19,57,74,135,152,182,190,222,342,456,546,570,756,1080,
55,82,99,124,165,246,308,350,372,440,792,924,990,1050,1320,
23,69,184,230,414,552,690,
65,117,148,195,238,315,364,380,444,520,684,714,864,936,1092,1140,1170,1560,2520,
... (End)

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A112954, A112955

Cf. A005384, A023212, A043297.

Table[SelectFirst[Range[10^5], EulerPhi@ # == n DivisorSigma[0, #] &] /.
k_ /; MissingQ@ k -> 0, {n, 120}] (* Michael De Vlieger, Aug 09 2017, Version 10.2 *)

From Enrique Pérez Herrero, Jan 01 2012: (Start)
If n > 1 then a(n) >= 2*n+1 or a(n)=0.
If p and q = 2*p+1 are both prime, A005384, then a(p) = 2*p+1.
If p > 3 and q = 4*p+1 are both prime, A023212, then a(p) = 8*p + 2 = 2*q.
If p > 2 is prime and both 2*p+1 and 4*p+1 are composite, A043297, then a(n)=0.
(End)

More terms from R. J. Mathar, Aug 07 2010

Comment corrected by Enrique Pérez Herrero, Aug 12 2010

A229324 Composite squarefree numbers n such that p + tau(n) divides n - phi(n), where p are the prime factors of n, tau(n) = A000005(n) and phi(n) = A000010(n).

Original entry on oeis.org

115, 205, 295, 565, 655, 745, 835, 1195, 1285, 1465, 1555, 1735, 1915, 2005, 2095, 2455, 2545, 2815, 2995, 3085, 3265, 3715, 3805, 3985, 4435, 4705, 4885, 5065, 5155, 5245, 5515, 5965, 6145, 6415, 6505, 6595, 6865, 7045, 7135, 7405, 7495, 7765, 7855, 8035

Offset: 1

Paolo P. Lava, Sep 20 2013

nonn

All terms are apparently multiple of 5.

It appears that a(n) = 5*A061240(n+1). - Michel Marcus, Sep 21 2013

			Prime factors of 2815 are 5, 563 and tau(2815) = 4, phi(2815) = 2248. 2815 - 2248 = 567 and  567 / (5 + 4) = 63, 567 / (563 + 4) = 1.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A000005, A000010, A228299-A228302, A229274-A229276, A229321-A229323.

with (numtheory); P:=proc(q) global a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 then ok:=0; break;
else if not type((n-phi(n))/(a[i][1]+tau(n)),integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(6*10^9);

Deleted first term, changed b-file and comment by Paolo P. Lava, Sep 23 2013

A233867 a(n) = |{0 < m < 2n: m is a square with 2n - 1 - phi(m) prime}|, where phi(.) is Euler's totient function (A000010).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 17 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) For any odd number 2*n - 1 > 4, there is a positive integer k < 2*n such that 2*n - 1 - phi(k) and 2*n - 1 + phi(k) are both prime.
By Goldbach's conjecture, 2*n > 2 could be written as p + q with p and q both prime, and hence 2*n - 1 = p + (q - 1) = p + phi(q).
By induction, phi(k^2) (k = 1,2,3,...) are pairwise distinct.

			a(29) = 1 since 2*29 - 1 = 37 + phi(5^2) with 37 prime.
a(39) = 1 since 2*39 - 1 = 71 + phi(3^2) with 71 prime.
a(66) = 1 since 2*66 - 1 = 89 + phi(7^2) with 89 prime.
a(128) = 1 since 2*128 - 1 = 223 + phi(8^2) with 223 prime.
a(182) = 1 since 2*182 - 1 = 331 + phi(8^2) with 331 prime.
a(413) = 1 since 2*413 - 1 = 823 + phi(2^2) with 823 prime.
a(171) = 3 since 2*171 - 1 = 233 + phi(18^2) = 257 + phi(14^2) = 293 + phi(12^2) with 233, 257, 293 all prime.

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

Cf. A000010, A000040, A002372, A002375, A233542, A233544, A233547, A233654, A233793, A233864.

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

A296078 Least number with the same prime signature as 1+phi(n), where phi = A000010, Euler totient function.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 2, 2, 4, 6, 2, 2, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 6, 6, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 6, 4, 6, 2, 6, 12, 4, 2, 4, 2, 2, 2, 2, 2, 4, 2, 6, 6, 2, 2, 4, 6, 2, 6, 2, 2, 4, 2, 12, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 2, 4, 4

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

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

Cf. A000010, A046523, A039649, A296079, A296080.
Cf. A039698 (positions of 2's).
Cf. also A277906, A296076, A296085, A296092.

f[n_] := Block[{ps = Last@# & /@ FactorInteger[1 + EulerPhi@n]}, Times @@ ((Prime@ Range@ Length@ ps)^ps)]; Array[f, 105] (* Robert G. Wilson v, Dec 11 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A296078(n) = A046523(1+eulerphi(n));

a(n) = A046523(A039649(n)) = A046523(1+A000010(n)).

A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 8, 38, 262, 1732, 16144, 153596, 1660796, 19415384, 264084064, 3664187848, 57366995272, 936097392752, 16131362629568, 302946516251408, 6034409270818576, 125044362929875744, 2756094464546395264, 63280996793936902496

Offset: 0

Vaclav Kotesovec, Sep 05 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A088009, A318769, A318696.
Cf. A000262, A305127, A318695.
Cf. A053529, A028342.
Cf. A000010 (phi), A008683 (mu).

nmax = 20; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[EulerPhi[k]* a[n-k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 29 2022

a(n)/n! ~ 3^(1/4) * exp(2*sqrt(6*n)/Pi) / (Pi * 2^(3/4) * n^(3/4)).
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} mu(gcd(k,j)). - Ilya Gutkovskiy, Aug 17 2021
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 29 2022
E.g.f.: exp( Sum_{n>=1} (mu(n)/n) * x^n/(1 - x^n) ), where mu(n) = A008683(n). - Paul D. Hanna, Jun 24 2023

A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 5, 23, 55, 155, 227, 521, 777, 1263, 1663, 2873, 3449, 5477, 6653, 8453, 10501, 15125, 17069, 23567, 26767, 32059, 36899, 48537, 53145, 65645, 73757, 86879, 96287, 119835, 127035, 155865, 172249, 194029, 212525, 241925, 257477, 306761, 332753, 369257

Offset: 1

Vaclav Kotesovec, Sep 10 2018

nonn

Comment from N. J. A. Sloane, Mar 22 2020: (Start)
Theorem: Sum_{ 1<=i<=n, 1<=j<=n, gcd(i,j)=1 } i*j = a(n).
Proof: From the Apostol reference we know that:
Sum_{ 1<=i<=n, gcd(i,n)=1 } i = n*phi(n)/2 (see A023896).
We use induction on n. The result is true for n=1.
Then a(n) - a(n-1) = 2*Sum_{ i=1..n-1, gcd(i,n)=1 } n*i = n^2*phi(n). QED (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A002088, A011755, A023896, A053191.

Accumulate[Table[k^2*EulerPhi[k], {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*eulerphi(k)); \\ Michel Marcus, Sep 12 2018

a(n) ~ 3*n^4 / (2*Pi^2).

A323333 The Euler phi function values of the powerful numbers, A000010(A001694(n)).

Original entry on oeis.org

1, 2, 4, 6, 8, 20, 18, 16, 12, 42, 32, 24, 54, 40, 36, 110, 100, 64, 48, 156, 84, 80, 72, 120, 162, 128, 96, 272, 108, 294, 342, 168, 160, 144, 252, 220, 200, 256, 506, 192, 500, 216, 360, 312, 486, 336, 320, 812, 288, 240, 930, 440, 324, 400, 512, 660, 600

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Murata's constant Product_{p prime}(1 + 1/(p-1)^2) (A065485).

Sequence is injective: no value occurs more than once. - Amiram Eldar and Antti Karttunen, Sep 30 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A000010, A001694, A002618 (a subsequence), A065485, A082695, A112526, A323332.

EulerPhi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after Harvey P. Dale at A001694 *)

lista(nn) = apply(x->eulerphi(x), select(x->ispowerful(x), vector(nn, k, k))); \\ Michel Marcus, Jan 11 2019

A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 10, 100, 720, 9792, 90900, 990000, 9090900, 94117632, 681410880, 9897840000, 86925373920, 979102080000, 9080325951840, 95255567232000, 712493107200000, 9926748531589120, 90004044661864320, 989999010000000000, 9090909090909090900, 97910150554895155200

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000010, A003021, A057934, A062397, A119704, A295503, A344897, A366668.

Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366658, A366667, A366690, A366716.

EulerPhi[10^Range[0,20] + 1] (* Paul F. Marrero Romero, Nov 10 2023 *)

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

a(n) = A000010(A062397(n)). - Paul F. Marrero Romero, Nov 10 2023

A055008 Numbers k such that gcd(phi(k), sigma(k)) = 1 with phi = A000010, sigma = A000203.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 32, 36, 50, 64, 81, 100, 121, 128, 144, 225, 242, 256, 289, 324, 400, 484, 512, 529, 576, 578, 625, 729, 800, 841, 900, 1024, 1058, 1089, 1156, 1250, 1296, 1600, 1681, 1682, 1936, 2025, 2048, 2116, 2209, 2304, 2312, 2401, 2500, 2601

Offset: 1

Labos Elemer, May 31 2000

nonn

The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020
Conjecture: Every term is a square or twice a square. - Jason Yuen, May 16 2024
The conjecture is true: If k is neither a square nor twice a square (i.e., in A028983), then sigma(k) is even. Since gcd(phi(k), sigma(k)) = 1, then phi(k) must be odd, but phi(k) is odd only for k = 1 and 2. - Amiram Eldar, May 19 2024

			For n = 484, phi(484) = 220 = 2*2*5*11, sigma(484) = 931 = 7*7*19, and gcd(220,931) = 1.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.

Cf. A000010, A000203, A009223, A028983.

Select[Range@ 2700, CoprimeQ[EulerPhi@ #, DivisorSigma[1, #]] &] (* Michael De Vlieger, Feb 05 2017 *)
Select[With[{max = 51}, Union[Array[#^2 &, max], Array[2*#^2 &, Floor[max / Sqrt[2]]]]], CoprimeQ[EulerPhi[#], DivisorSigma[1, #]] &] (* Amiram Eldar, May 19 2024 *)

is(n)=gcd(sigma(n),eulerphi(n))==1 \\ Charles R Greathouse IV, Feb 19 2013

Incorrect comment removed by Charles R Greathouse IV, Feb 19 2013

A065093 Convolution of A000010 with itself.

Original entry on oeis.org

1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120, 124, 188, 196, 276, 272, 404, 380, 544, 532, 716, 668, 968, 860, 1184, 1120, 1472, 1332, 1896, 1624, 2204, 2036, 2656, 2352, 3284, 2752, 3684, 3356, 4324, 3744, 5192, 4312, 5720, 5180, 6540, 5628, 7768, 6388, 8476

Offset: 1

Vladeta Jovovic, Nov 11 2001

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.

Cf. A000010, A000385, A055507, A065474, A330319.

Column k=2 of A340995.

Table[Sum[EulerPhi[j]*EulerPhi[n-j], {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Aug 18 2021 *)

{ for (n=1, 1000, a=sum(k=1, n, eulerphi(k)*eulerphi(n+1-k)); write("b065093.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

a(n) = Sum_{k=1..n} phi(k)*phi(n+1-k), where phi is Euler totient function (A000010).
G.f.: (1/x)*(Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2)^2. - Ilya Gutkovskiy, Jan 31 2017
a(n) ~ (n^3/6) * c * Product_{primes p|n+1} ((p^3-2*p+1)/(p*(p^2-2))), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Ingham, 1927). - Amiram Eldar, Jul 13 2024

cp's OEIS Frontend

A175667 Smallest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

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A229324 Composite squarefree numbers n such that p + tau(n) divides n - phi(n), where p are the prime factors of n, tau(n) = A000005(n) and phi(n) = A000010(n).

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A233867 a(n) = |{0 < m < 2*n: m is a square with 2*n - 1 - phi(m) prime}|, where phi(.) is Euler's totient function (A000010).

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A296078 Least number with the same prime signature as 1+phi(n), where phi = A000010, Euler totient function.

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A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

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A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.

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A323333 The Euler phi function values of the powerful numbers, A000010(A001694(n)).

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A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

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A233867 a(n) = |{0 < m < 2n: m is a square with 2n - 1 - phi(m) prime}|, where phi(.) is Euler's totient function (A000010).