A091439 -id:A091439 - cp's OEIS frontend

A091456 Least number k such that n * phi(k) < k, where phi is Euler's totient function.

2, 6, 30, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130, 1492182350939279320058875736615841068547583863326864530410

Offset: 1

Robert G. Wilson v, Jan 10 2004

nonn

By Mertens' theorem and the Prime Number Theorem log log a(n) ~ n / e^gamma. - Charles R Greathouse IV, Sep 07 2012

Amiram Eldar, Table of n, a(n) for n = 1..13

Subsequence of A002110.

Cf. A000010, A005579, A054741, A073087, A091439 (n * phi(k) <= k).

a(n) = {k = 1; while (n*eulerphi(k) >= k, k++); k;} \\ Michel Marcus, Sep 25 2013

a(n)=my(k=1);forprime(p=2,,if(n*eulerphi(k)Charles R Greathouse IV, Sep 25 2013

a(n) = A002110(A005579(n)). - Amiram Eldar, Nov 30 2024

A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

Original entry on oeis.org

1, 2, 3, 7, 13, 23, 43, 79, 149, 257, 461, 821, 1451, 2549, 4483, 7879, 13859, 24247, 42683, 75037, 131707, 230773, 405401, 710569, 1246379, 2185021, 3831913, 6720059, 11781551, 20657677, 36221753, 63503639, 111333529, 195199289, 342243479, 600036989

Offset: 1

T. D. Noe, Jan 09 2004

nonn

Does the ratio of adjacent terms converge?
It appears that lim_{n->infinity} a(n+1)/a(n) = 1.7532... - Jon E. Schoenfield, Feb 21 2019
For n > 1, a(n) is smallest prime p = prime(k) such that no fewer than (n-1)/n of any p# consecutive integers are divisible by a prime not greater than p. Cf. A053144(k)/A002110(k). - Peter Munn, Apr 29 2017
Also, the smallest prime p such that the sum of the reciprocals of the p-smooth numbers converges to at least n. - Keith F. Lynch, Apr 29 2023
Also, if m is a random integer much larger than the square of a(n), and m is not divisible by any prime less than or equal to a(n), the probability that m is prime is n/log(m). - Keith F. Lynch, Dec 17 2023

			7#/phi(7#) = (2*3*5*7)/(1*2*4*6) = 4.375 >= 4, 5#/phi(5#) = 3.75. Hence a(4) = 7.

Jon E. Schoenfield, Table of n, a(n) for n = 1..44
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A091439, A000010, A002110, A005579, A034386, A053144, A038110, A060753, A164347.

Mathematica
```
prod=1; i=0; Table[While[prod
				
```

al(lim) = local(mm,n,m); mm=3; n=2; m=1; forprime(x=3,lim, n*=x; m*= (x-1); if (n\m >= mm, print1(x","); mm++)); /* This will generate all terms of this sequence from the 3rd onward, up to lim. The computation slows down for large values because of the size of the internal values. */ \\ Fred Schneider, Aug 13 2009, modified by Franklin T. Adams-Watters, Aug 29 2009

a(n) = prime(A005579(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

More terms from David W. Wilson, Sep 28 2005

Sequence reference in name corrected by Peter Munn, Apr 29 2017

A098222 a(n) is the smallest prime q such that floor(sigma(sigma(q))/q) = n.

Original entry on oeis.org

13, 2, 179, 55439, 232792559, 130429015516799

Offset: 1

Labos Elemer, Oct 25 2004

a(7) <= 9854961523502269526351999. If a(7) + 1 is in A025487 then a(7) = 9854961523502269526351999. a(3) + 1 through a(6) + 1 are in A025487. - David A. Corneth, Sep 03 2019

			a(4) = 55439 because floor(sigma(sigma(55439))/55439) = floor(232128/55439) = n = 4.
a(5) = 232792559 because floor(sigma(sigma(a(5)))/a(5)) = floor(5.02561) = 5.

Cf. A000203, A051027, A008333, A025487, A091439, A098219, A098220, A098221.

a(n) >= A091439(n). - David A. Corneth, Sep 03 2019

a(6) from Charles R Greathouse IV, Mar 14 2011

A067206 Numbers n such that the digits of n end in phi(n).

Original entry on oeis.org

1, 1320, 1640, 1768, 1996, 2640, 3960, 13200, 16400, 19984, 19996, 26400, 39600, 132000, 164000, 199996, 264000, 396000, 1320000, 1640000, 1999936, 2640000, 3960000, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996

Offset: 1

Joseph L. Pe, Feb 19 2002

Comments from Farideh Firoozbakht, Dec 30 2006: (Start)
"(1). If n is in the sequence and 10 divides n then for each natural number k, n*10^k is in the sequence. So since 1320, 1640, 2640, 3960 & 16666240 are in the sequence, for each natural number k, 132*10^k, 164*10^k, 264*10^k, 396*10^k & 1666624*10^k are in the sequence. Hence the sequence is infinite.
"(2). If 5*10^k-1 is prime then 4*(5*10^k-1) is in the sequence. So 4*A093945 is a subsequence of this sequence.
"(3). If p=125*10^k-1 is prime then 16*p is in the sequence. For k = 1, 4, 5, 8, 13, 19, 25, 26, 76, 88, 167, 290, 389, ... p is prime.
"(4). If p=3125*10^k-1 is prime then 64*p is in the sequence. For k = 1, 3, 9, 33, 121, 223, 357, 363, 447, ... p is prime." (End)

			The digits of 1768 end in phi(1768) = 768, so 1768 is a term of the sequence.

Pickover, C. "Wonders of Numbers". Oxford Univ. Press, 2001.

Giovanni Resta, Table of n, a(n) for n = 1..66 (terms < 10^12, first 33 terms from Farideh Firoozbakht)
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Cf. A093945, A091439.

Cf. A066663. - R. J. Mathar, Sep 30 2008

(*returns true if a ends in b, false o.w.*) f[a_, b_] := Module[{c, d, e, g, h, i, r}, r = False; c = ToString[a]; d = ToString[b]; e = StringLength[c]; g = StringPosition[c, d]; h = Length[g]; If[h > 0, i = g[[h]]; If[i[[2]] == e, r = True]]; r]; Select[Range[10^5], f[ #, EulerPhi[ # ]] &]

More terms from Farideh Firoozbakht, Dec 30 2006

A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

Original entry on oeis.org

2, 3, 7, 1261, 11242771

Offset: 1

Jaroslav Krizek, Jan 26 2016

a(n) >= A266269(n). - Max Alekseyev, Jan 26 2025

			a(3) = 7 because 7 is the smallest number k such that phi(k) = n*phi(k-1); phi(7) = 6 =3*phi(6) = 3*2.

Sequences of numbers n such that phi(n) = k*phi(n-1): {A001274 + 1} for k=1; A171271 = {A050472 + 1} for k=2; A266268 = {A067143 + 1} for k=3; A268126 = {A172314 + 1} for k=4; {A201253 + 1} for k=5.

Cf. A000010, A091439, A091456, A266269.

Magma
```
a:=func; [a(n):n in[1..5]];
```

a(n) = my(k=2, epk=1, enk); while ((enk=eulerphi(k)) != n*epk, epk = enk; k++); k; \\ Michel Marcus, Feb 20 2020

A066663 Numbers n such that phi(n) is a proper substring of n.

Original entry on oeis.org

1320, 1640, 1768, 1996, 2640, 3960, 13200, 16400, 19984, 19996, 26400, 39600, 132000, 164000, 199996, 264000, 396000, 1320000, 1640000, 1999936, 2640000, 3960000, 13200000, 16400000, 16666240, 17999488, 18515584, 19999984, 19999996

Offset: 1

Vladeta Jovovic, Jan 11 2002

Since the ratio n/phi(n) grows very slowly with n (see A091439), this sequence will be equal to A067206 (apart the initial 1) at least for all the terms smaller than 10^103. - Giovanni Resta, Apr 14 2017

			phi(1320)=320, phi(1640)=640, phi(1768)=768, phi(1996)=996, phi(2640)=640,..., phi(199999999984)=99999999984.

Giovanni Resta, Table of n, a(n) for n = 1..65 (terms < 10^12)

Cf. A000010, A036433, A091439, A067206.

Select[Range[2, 10^5], StringPosition[ ToString@ #, ToString@ EulerPhi@ #] != {} &] (* Giovanni Resta, Apr 14 2017 *)

Offset corrected and a(19)-a(29) from Donovan Johnson, May 08 2010

A266269 a(n) is the smallest number k such that phi(k) >= n*phi(k-1).

Original entry on oeis.org

2, 3, 7, 211, 30031, 223092871, 13082761331670031, 3217644767340672907899084554131, 1492182350939279320058875736615841068547583863326864530411, 16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999111

Offset: 1

Jaroslav Krizek, Jan 26 2016

nonn

For the known terms, we have a(n) = 1 + A002110(A256968(n)) = 1 + A091439(n), which likely holds for most (if not all) terms overall. - Max Alekseyev, Jan 26 2025

			a(3) = 7 because 7 is the smallest number k such that phi(k) >= n*phi(k-1); phi(7) = 6 >= 3*phi(6) = 3*2.

Max Alekseyev, Table of n, a(n) for n = 1..12

Cf. A000010, A002110, A091439, A091456, A256968, A266276.

Magma
```
a:=func; [a(n):n in[1..5]];
```

a(n) = {my(k=2, e=1); while(n*e > e=eulerphi(k), k++); k; } \\ Jinyuan Wang, Nov 01 2020

a(6)-a(8) from Jinyuan Wang, Nov 01 2020

a(9)-a(10) from Max Alekseyev, Jan 25 2025

A355667 Least number phi(k) such that n * phi(k) < k, where phi is Euler's totient function.

Original entry on oeis.org

1, 2, 8, 48, 5760, 36495360, 1854081073152000, 400440702414394285778534400000, 165062110921422523175104166476600499887194872217600000000

Offset: 1

Nico Mexis, Jul 13 2022

nonn

a(n) is the Euler totient of A091456(n). Useful for calculating a simple upper bound of the inverse phi function.

			If we know n=276, then all numbers k with phi(k)=n will satisfy k < 5*n since m=a(5) is the first number in the sequence with n < m.

Nico Mexis, Table of n, a(n) for n = 1..13
John Abbott and Nico Mexis, Cyclotomic Factors and LRS-Degeneracy, arXiv:2403.08751 [math.AC], 2024. See pp. 9, 24.

Totients of A091456 or A091439.

// CoCoA 5
define a(n) k := 1; p := 1; while k <= n*EulerTotient(k) do p := NextPrime(p); k := k*p; endwhile; return EulerTotient(k); enddefine;

a[n_] := Module[{k = 1, p = 1, e}, While[k <= n*(e = EulerPhi[k]), p = NextPrime[p]; k *= p]; e]; Array[a, 9] (* Amiram Eldar, Jul 13 2022 *)

a(n)=my(k=1); forprime(p=2, , if(n*eulerphi(k)

a(n) = A000010(A091456(n)).

a(n) = A000010(A091439(n+1)).

cp's OEIS Frontend

A091456 Least number k such that n * phi(k) < k, where phi is Euler's totient function.

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A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

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A098222 a(n) is the smallest prime q such that floor(sigma(sigma(q))/q) = n.

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A067206 Numbers n such that the digits of n end in phi(n).

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A266276 a(n) is the smallest number k such that phi(k) = n*phi(k-1).

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A066663 Numbers n such that phi(n) is a proper substring of n.

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A266269 a(n) is the smallest number k such that phi(k) >= n*phi(k-1).

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A355667 Least number phi(k) such that n * phi(k) < k, where phi is Euler's totient function.