A071615 -id:A071615 - cp's OEIS frontend

A071616 Smallest even number divisible by 2n which is nontotient, i.e., in A005277.

14, 68, 90, 152, 50, 516, 14, 304, 90, 340, 154, 4008, 26, 308, 90, 608, 34, 2412, 38, 680, 714, 308, 230, 10128, 50, 364, 594, 728, 174, 8340, 62, 1984, 594, 68, 350, 7848, 74, 76, 234, 6800, 246, 5124, 86, 968, 90, 644, 94, 20256, 98, 1100, 510, 728, 318

Offset: 1

Labos Elemer, May 27 2002

nonn

a(n) = 2n*A071615(n).

			n=4: 2n=8 and number of terms in invphi(8k) is 5, 6, 10, 7, 9, 11, 3, 8, 17, 10, 6, 17, 3, 6, 17, 9, 9, 21, 0, 12, ... for k=1,2,...,20,...; zero appears first at k=19, so a(4) = 8k = 152.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A071615.

invphi[n_, plist_] := Module[{i, p, e, pe, val}, If[plist=={}, Return[If[n==1, {1}, {}]]]; val={}; p=Last[plist]; For[e=0; pe=1, e==0||Mod[n, (p-1)pe/p]==0, e++; pe*=p, val=Join[val, pe*invphi[If[e==0, n, n*p/pe/(p-1)], Drop[plist, -1]]]]; Sort[val]]; invphi[n_] := invphi[n, Select[1+Divisors[n], PrimeQ]]; a[n_] := For[m=1, True, m++, If[invphi[2n*m]=={}, Return[2n*m]]] (* invphi[n, plist] is list of x with phi(x)=n and all prime divisors of x in plist. *)

Edited and extended by Robert G. Wilson v, May 28 2002 and by Dean Hickerson, Jun 04 2002

A282160 Least k > 1 such that k*n is not a totient number.

Original entry on oeis.org

3, 7, 3, 17, 3, 15, 2, 19, 3, 5, 3, 43, 2, 7, 3, 19, 2, 5, 2, 17, 3, 7, 3, 167, 2, 7, 3, 11, 3, 3, 2, 19, 3, 2, 3, 67, 2, 2, 3, 17, 3, 17, 2, 7, 2, 5, 2, 211, 2, 7, 3, 7, 3, 11, 3, 13, 2, 3, 2, 139, 2, 2, 3, 31, 3, 9, 2, 5, 3, 5, 2, 109, 2, 5, 3, 2, 2, 3, 2, 85, 3, 3, 3, 61

Offset: 1

Altug Alkan, Feb 07 2017

nonn

First occurrence of odd k or zero if impossible: 0, 1, 10, 2, 66, 28, 56, 6, 4, 8, 5244, 460, 272, 0, 232, 64, 7788, 4180, 300, 348, 328, 12, etc. - Robert G. Wilson v, Feb 09 2017

			a(14) = 7 because 7 * 14 = 98 is not a totient number and 7 is the least number that is greater than 1 with this property.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A007617, A067005, A071615, A079695.

TotientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) f[n_] := Block[{k = 2}, While[ TotientQ[k*n], k++]; k]; Array[f, 84] (* Robert G. Wilson v, Feb 09 2017 *)

a(n) = my(k = 2); while (istotient(k*n), k++); k;

a(A079695(n)) = 2. - Michel Marcus, Feb 08 2017

A338266 Least prime p such that p*n is not a totient number.

Original entry on oeis.org

3, 7, 3, 17, 3, 19, 2, 19, 3, 5, 3, 43, 2, 7, 3, 19, 2, 5, 2, 17, 3, 7, 3, 167, 2, 7, 3, 11, 3, 3, 2, 19, 3, 2, 3, 67, 2, 2, 3, 17, 3, 17, 2, 7, 2, 5, 2, 211, 2, 7, 3, 7, 3, 11, 3, 13, 2, 3, 2, 139, 2, 2, 3, 31, 3, 19, 2, 5, 3, 5, 2, 109, 2, 5, 3, 2, 2, 3, 2

Offset: 1

Bernard Schott, Oct 19 2020

nonn

Zhang Ming-Zhi has shown that for every positive integer n, there is a prime p such that p*n is not a totient (see Reference and link, theorem 1).

Differs from A282160, where multiplier p is not requested to be prime, for n = 6, 66, 80, 126, ... those indices where A282160(n) is not prime (see Example).

			a(6) = 19 because 19 * 6 = 114 is not a totient number and 19 is the least prime with this property. Also 15 * 6 = 90 is not either a totient number, so A282160(6) = 15 that is not a prime number.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B36, p. 139.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zhang Ming-Zhi, On Nontotients, J. Number Theory, Vol. 43, No. 2 (1993), pp. 168-172.

Cf. A000010, A002202, A061026, A071615, A079695, A282160.

a(n) = my(p=2); while (istotient(p*n), p = nextprime(p+1)); p; \\ Michel Marcus, Oct 19 2020

a(A079695(n)) = 2.

A338561 a(n) is the least m such that A338266(m) = prime(n), where A338266(m) is the least prime p such that p*m is not a totient number.

Original entry on oeis.org

7, 1, 10, 2, 28, 56, 4, 6, 460, 232, 64, 300, 328, 12, 256, 180, 176, 84, 36, 132, 1400, 984, 2200, 780, 1332, 280, 1664, 1672, 72, 8136, 420, 53244, 1960, 60, 2320, 5928, 264, 936, 24, 32604, 6696, 2976, 2268, 6372, 312, 1380, 48, 320, 2560, 816, 16500, 4860

Offset: 1

Bernard Schott, Nov 02 2020

nonn

Zhang Ming-Zhi has shown that for every positive integer m, there is a prime p such that m*p is not a totient (see Reference, link: theorem 1). A338266 gives the smallest prime p that is such linked to m.

			Prime(2)=3 is the smallest prime such that 3*1=3, 3*3=9, 3*9=27, 3*11=33, 3*15=45,... are not totient (A338266), and 1 is the smallest number of the set {1, 3, 9, 11, 15...} linked to prime(2), so a(2)=1.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B36, p. 139.

Zhang Ming-Zhi, On Nontotients, J. Number Theory, Vol. 43, No. 2 (1993), pp. 168-172.

Cf. A000010, A002202, A007617, A071615, A338266.

f(n) = my(p=2); while (istotient(p*n), p = nextprime(p+1)); p; \\ A338266
a(n) = my(k=1, p=prime(n)); while(f(k) != p, k++); k; \\ Michel Marcus, Nov 03 2020

More terms from Amiram Eldar, Nov 02 2020

Name improved by Amiram Eldar and Michel Marcus, Nov 03 2020

cp's OEIS Frontend

A071616 Smallest even number divisible by 2n which is nontotient, i.e., in A005277.

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A282160 Least k > 1 such that k*n is not a totient number.

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A338266 Least prime p such that p*n is not a totient number.

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A338561 a(n) is the least m such that A338266(m) = prime(n), where A338266(m) is the least prime p such that p*m is not a totient number.

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