A070982 -id:A070982 - cp's OEIS frontend

A061026 Smallest number m such that phi(m) is divisible by n, where phi = Euler totient function A000010.

1, 3, 7, 5, 11, 7, 29, 15, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 25, 43, 23, 47, 35, 101, 53, 81, 29, 59, 31, 311, 51, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 69, 181, 47, 283, 65, 197, 101, 103, 53, 107, 81, 121, 87, 229, 59, 709, 61, 367, 311, 127, 85

Offset: 1

Melvin J. Knight (knightmj(AT)juno.com), May 25 2001

nonn

Conjecture: a(n) is odd for all n. Verified up to n <= 3*10^5. - Jianing Song, Feb 21 2021

The conjecture above is false because a(16842752) = 33817088; see A002181 and A143510. - Flávio V. Fernandes, Oct 08 2023

			a(48) = 65 because phi(65) = phi(5)*phi(13) = 4*12 = 48 and no smaller integer m has phi(m) divisible by 48.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Ho-joo Lee and Gerald Myerson, Consecutive Integers Whose Totients Are Multiples of n: 10837, The American Mathematical Monthly, Vol. 110, No. 2 (Feb., 2003), pp. 158-159.
Pieter Moree and Hans Roskam, On an arithmetical function related to Euler's totient and the discriminator, Fib. Quart., Vol. 33, No. 4 (1995), pp. 332-340.
József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, pp. 1-8.

Cf. A000010, A066674, A066675, A066676, A066678, A067005.
Cf. A233516, A233517 (records).
Cf. A005179 (analog for number of divisors), A070982 (analog for sum of divisors).

a = ConstantArray[1, 64]; k = 1; While[Length[vac = Rest[Flatten[Position[a, 1]]]] > 0, k++; a[[Intersection[Divisors[EulerPhi[k]], vac]]] *= k]; a  (* Ivan Neretin, May 15 2015 *)

a(n) = my(s=1); while(eulerphi(s)%n, s++); s;
vector(100, n, a(n))

from sympy import totient as phi
def a(n):
  k = 1
  while phi(k)%n != 0: k += 1
  return k
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Feb 21 2021

Sequence is unbounded; a(n) <= n^2 since phi(n^2) is always divisible by n.
If n+1 is prime then a(n) = n+1.
a(n) = min{ k : phi(k) == 0 (mod n) }.
a(n) = a(2n) for odd n > 1. - Jianing Song, Feb 21 2021

A265647 Smallest k such that n divides k(k+1)(k+2)/6.

Original entry on oeis.org

1, 2, 7, 2, 3, 7, 5, 6, 25, 3, 9, 7, 11, 6, 8, 14, 15, 26, 17, 4, 7, 10, 21, 8, 23, 11, 79, 6, 27, 8, 29, 30, 9, 15, 5, 26, 35, 18, 25, 8, 39, 7, 41, 10, 25, 22, 45, 16, 47, 23, 16, 12, 51, 79, 9, 6, 17, 27, 57, 8, 59, 30, 26, 62, 13, 43, 65, 15, 44, 14, 69, 54, 71, 35, 25, 18, 20, 26, 77, 14, 241

Offset: 1

Ctibor O. Zizka, Dec 11 2015

nonn

More generally we can ask for the smallest k such that gcd(n,f(k)) = n. This sequence has f(k) = k*(k+1)*(k+2)/6. For other examples in the OEIS, see the crossrefencess.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000292, A011772, A002034, A047930, A005179, A070982, A038700, A344005, A345989.

Table[k = 1; While[! Divisible[k (k + 1) (k + 2)/6, n], k++]; k, {n, 81}] (* Michael De Vlieger, Dec 11 2015 *)

a(n)=my(k=1);while((k*(k+1)*(k+2)/6)%n>0,k++);k \\ Anders Hellström, Dec 11 2015

first(n) = { my(todo = n, i = 1, res = vector(n)); while(todo > 0, d = select(x -> x <= n, divisors(binomial(i + 2, 3))); for(j = 1, #d, if(res[d[j]] == 0, res[d[j]] = i; todo-- ) ); i++ ); res } \\ David A. Corneth, Mar 22 2021

More terms from Michael De Vlieger, Dec 11 2015

A319068 a(n) is the greatest k such that A000203(k) divides n where A000203 is the sum of divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 4, 7, 2, 1, 1, 11, 9, 13, 8, 7, 1, 17, 1, 19, 4, 1, 1, 23, 1, 9, 2, 13, 1, 29, 25, 31, 2, 1, 4, 22, 1, 37, 18, 27, 1, 41, 1, 43, 8, 1, 1, 47, 4, 1, 2, 9, 1, 53, 1, 39, 49, 1, 1, 59, 1, 61, 32, 31, 9, 5, 1, 67, 2, 13, 1, 71, 1, 73, 8, 37, 4, 45, 1, 79

Offset: 1

Michel Marcus, Sep 09 2018

nonn

Sándor names this function the sum-of-divisors maximum function and remarks that this function is well-defined, since a(n) can be at least 1, and cannot be greater than n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
József Sándor, The sum-of-divisors minimum and maximum functions, Research Report Collection, Volume 8, Issue 1, 2005. See pp. 3-4.

Cf. A000203 (sigma), A070982 (the sum of divisors minimum function).

Right border of A378912.

A319068[n_] := Module[{k = n}, While[!Divisible[n, DivisorSigma[1, k]], k--]; k];
Array[A319068, 100] (* Paolo Xausa, Dec 11 2024 *)

a(n) = {forstep (k=n, 1, -1, if ((n % sigma(k)) == 0, return (k)););}

a(n) = {my(d = divisors(n)); vecmax(vector(#d, i, invsigmaMax(d[i])));} \\ Amiram Eldar, Nov 29 2024, using Max Alekseyev's invphi.gp

a(p+1) = p, for p prime. See Sándor Theorem 2 p. 4.

A233929 Smallest x such that sigma(x) == n-1 (mod n).

Original entry on oeis.org

1, 1, 7, 2, 3, 2401, 5, 4, 7, 18, 21, 9604, 6, 9, 13, 8, 44, 21609, 10, 18, 19, 289, 36, 9604, 14, 162, 57, 72, 12, 2614689, 29, 16, 21, 625, 63, 38416, 22, 4608, 37, 18, 27, 21609, 20, 289, 43, 36, 50, 38416, 33, 196, 111, 162, 157, 28561, 34, 1296, 28, 49

Offset: 1

Michel Marcus, Dec 18 2013

nonn

Right subdiagonal of A074625.

Records values are: 1, 7, 2401, 9604, 21609, 2614689, 21215236, 36324729, 53304601, 338964921, 431642176, 528264256, 1307979556, ... obtained at indices: 1, 3, 6, 12, 18, 30, 60, 210, 288, 384, 534, 630, 732. - Michel Marcus, Dec 22 2013

Michel Marcus and Donovan Johnson, Table of n, a(n) for n = 1..3000 (first 800 terms from Michel Marcus)

Cf. A000203, A070982, A074625.

a(n) = {x = 1; while ((sigma(x) % n) != (n - 1), x++); x;} \\ Michel Marcus, Dec 18 2013

A085961 Sigma of the least numbers k for which sigma is divisible by k (where sigma is the sum of the divisors of k, A000203(k)).

Original entry on oeis.org

1, 4, 3, 4, 15, 6, 7, 8, 18, 20, 44, 12, 13, 28, 15, 32, 68, 18, 38, 20, 42, 44, 138, 24, 150, 78, 54, 28, 174, 60, 31, 32, 132, 68, 140, 36, 74, 38, 39, 40, 164, 42, 258, 44, 90, 138, 282, 48, 98, 150, 102, 104, 212, 54, 110, 56, 57, 174, 354, 60, 183, 124, 63, 128, 195

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2003

nonn

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

Cf. A070982, A000203, A066678.

a(n) = A000203(A070982(n)).

More terms from John W. Layman, Aug 19 2003

A074208 Least k > 1 such that n divides sigma(k) - k.

Original entry on oeis.org

2, 6, 4, 9, 14, 6, 8, 10, 15, 14, 20, 24, 27, 22, 16, 12, 39, 24, 48, 34, 18, 20, 52, 90, 40, 46, 42, 28, 68, 78, 32, 56, 45, 62, 84, 24, 70, 48, 66, 44, 63, 30, 50, 82, 78, 52, 116, 90, 75, 40, 132, 96, 80, 42, 36, 106, 99, 68, 148, 120, 130, 118, 64, 56, 117, 54, 136, 112

Offset: 1

Benoit Cloitre, Sep 17 2002

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A070982.

a = ConstantArray[1, 68]; k = 1; While[Length[vac = Flatten[Position[a, 1]]] > 0, k++; a[[Intersection[Divisors[DivisorSigma[1, k] - k], vac]]] *= k]; a (* Ivan Neretin, May 15 2015 *)
lk[n_]:=Module[{k=2},While[!Divisible[DivisorSigma[1,k]-k,n],k++];k]; Array[lk,70] (* Harvey P. Dale, Oct 23 2017 *)

a(n)=if(n<0,0,s=2; while((sigma(s)-s)%n>0,s++); s)

first(n)=my(res = vector(n), todo = n - 1, k = 2); res[1] = 2; while(todo > 0, d = divisors(sigma(k) - k); for(i=2, #d, if(d[i] <= n && res[d[i]] == 0, res[d[i]] = k; todo--)); k++); res \\ David A. Corneth, Oct 23 2017

Conjecture: Sum_{k=1..n} a(k) = O(n^{2+epsilon}) for any epsilon > 0.

Between n = 90000 and 100000, Sum_{k=1..n} a(k)/n^2 slowly but not monotonically increases from 1.0007 to 1.0023. At n = 10^6, it's about 1.0147. - David A. Corneth, Oct 23 2017

A319067 a(n) is the least k such that n divides A047994(k) where A047994 is the unitary totient function.

Original entry on oeis.org

1, 3, 4, 5, 11, 7, 8, 9, 19, 11, 23, 13, 27, 24, 16, 17, 103, 19, 191, 33, 43, 23, 47, 25, 101, 27, 76, 29, 59, 31, 32, 45, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 69, 112, 47, 283, 49, 197, 101, 103, 53, 107, 76, 253, 72, 229, 59, 709, 61, 367, 96, 64, 85, 131

Offset: 1

Michel Marcus, Sep 09 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, The Unitary Totient Minimum and Maximum Functions, Studia Universitatis Babes-Bolyai Mathematica, Volume L, Number 2, June 2005.

Cf. A047994 (unitary totient).

Cf. A005179 (analog for number of divisors), A061026 (analog for Euler totient), A070982 (analog for sum of divisors).

a047994(n) = prod(i=1, #n=factor(n)~, n[1, i]^n[2, i]-1);
a(n) = my(k=1); while(a047994(k) % n, k++); k;

a(p-1) = p for p prime. See Sándor link Theorem 2 p. 95.

A283495 Smallest k such that there is a number whose divisors sum to k*n.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 4, 1, 1, 2, 1, 2, 4, 1, 2, 1, 2, 2, 6, 1, 6, 3, 2, 1, 6, 2, 1, 1, 4, 2, 4, 1, 2, 1, 1, 1, 4, 1, 6, 1, 2, 3, 6, 1, 2, 3, 2, 2, 4, 1, 2, 1, 1, 3, 6, 1, 3, 2, 1, 2, 3, 2, 6, 1, 2, 2, 4, 1, 7, 1, 2, 2, 4, 1, 2, 1, 2, 2, 4, 1, 3, 3, 2, 2, 23, 1, 1, 4, 1, 3, 6, 1

Offset: 1

Juri-Stepan Gerasimov, Mar 08 2017

nonn

Smallest k >=1 such that (number of numbers whose divisor sum to k*n) = m:
m \n|  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9  | 10 | 11 | 12 |
---------------------------------------------------------------------------
0   |  2  |  1  |  3  |  4  |  1  | 11  |  3  |  2  |  1  |  1 |  1 | 23 |
1   |  1  |  2  |  1  |  1  |  3  |  1  |  1  |  1  |  2  |  2 |  4 |  3 |
2   | 12  |  6  |  4  |  3  | 16  |  2  |  8  |  4  |  2  |  8 | 12 |  1 |
3   | 24  | 12  |  8  |  6  | 12  |  4  |  6  |  3  | 10  |  6 | ...|    |
...

			a(2) = 2 because (number of numbers whose divisor sum to 2*2) = 1.

Cf. A000203, A054973, A283486, A070982.

Cf. A007369 (numbers n such that a(n) > 1).

a(n)=my(k=oo,m,t); while(mCharles R Greathouse IV, Mar 09 2017

Corrected by Charles R Greathouse IV, Mar 09 2017

A283625 Smallest k such that 2n - 1 divides sigma(k^2), or 0 if no such k exists.

Original entry on oeis.org

1, 7, 121, 2, 91, 9, 3, 847, 12667700813876161, 7, 14, 32, 116281, 1729, 343, 4, 63, 242, 47, 21, 1369, 79, 11011, 2048, 22, 88673905697133127, 4826809, 961, 7, 4782969, 13, 182, 363, 29, 224, 25, 16, 813967, 18, 23, 53599, 3486784401, 1532791798479015481, 4459

Offset: 1

Juri-Stepan Gerasimov, Mar 12 2017

nonn

			a(3)=121 because 3*2 - 1 = 5 divides sigma(121^2) = 16105, and sigma(n^2) is not divisible by 5 for n < 121.

Cf. A000203, A070982, A227470.

a(n) = my(k = 1); while(1,if(sigma(k^2)%(2*n - 1)==0, return(k), k+=1)); \\ Indranil Ghosh, Mar 13 2017

a(9), a(26), a(42)-a(44) from Giovanni Resta, Mar 12 2017

cp's OEIS Frontend

A061026 Smallest number m such that phi(m) is divisible by n, where phi = Euler totient function A000010.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A265647 Smallest k such that n divides k*(k+1)*(k+2)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A319068 a(n) is the greatest k such that A000203(k) divides n where A000203 is the sum of divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A233929 Smallest x such that sigma(x) == n-1 (mod n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A085961 Sigma of the least numbers k for which sigma is divisible by k (where sigma is the sum of the divisors of k, A000203(k)).

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A074208 Least k > 1 such that n divides sigma(k) - k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A319067 a(n) is the least k such that n divides A047994(k) where A047994 is the unitary totient function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A283495 Smallest k such that there is a number whose divisors sum to k*n.

Views

Author

Keywords

Comments

A265647 Smallest k such that n divides k(k+1)(k+2)/6.