A053191 -id:A053191 - cp's OEIS frontend

A053211 Cototients of consecutive pure powers of primes.

2, 4, 3, 8, 5, 9, 16, 7, 32, 27, 11, 25, 64, 13, 81, 128, 17, 49, 19, 256, 23, 125, 243, 29, 31, 512, 121, 37, 41, 43, 1024, 729, 169, 47, 343, 53, 625, 59, 61, 2048, 67, 289, 71, 73, 79, 2187, 361, 83, 89, 4096, 97, 101, 103, 107, 109, 529, 113, 1331, 3125, 127

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Cototients of prime powers do not remain always prime powers, but are primes if their exponent is 2.

			The 10th pure power of prime (but not a prime) is 81, so a(10) = 81 - EulerPhi(81) = 81 - 54 = 27. For n=p^2, a(n)=p.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^20, showing even a(n) in blue, 3 | a(n) in green, and prime a(n) in red, else black.

Cf. A000010, A051953, A001248, A002618, A036689, A053650, A053191, A053192, A246547.

Map[# - EulerPhi@ # &, Select[Range[16200], And[! PrimeQ@ #, PrimePowerQ@ #] &]] (* Michael De Vlieger, Jun 11 2018 *)
With[{nn = 2^14}, Map[Times @@ Map[#1^(#2 - 1) & @@ FactorInteger[#][[1]]] &, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] ] (* Michael De Vlieger, Mar 11 2023 *)

a(n) = A051953(A025475(n+1)) = cototient(p^k) = p^(k-1).

A057789 a(n) = Sum_{k = 1..n, gcd(k,n)=1} k*(n-k).

Original entry on oeis.org

0, 1, 4, 6, 20, 10, 56, 44, 84, 60, 220, 92, 364, 182, 280, 344, 816, 318, 1140, 520, 840, 770, 2024, 760, 2100, 1300, 2196, 1540, 4060, 1240, 4960, 2736, 3520, 2992, 4760, 2580, 8436, 4218, 5928, 4240, 11480, 3612, 13244, 6380, 8040, 7590, 17296, 6128

Offset: 1

Leroy Quet, Nov 04 2000

Equal to convolution sum over positive integers, k, where k<=n and gcd(k,n)=1, except in first term, where the convolution sum is 1 instead of 0.

			Since 1, 3, 5 and 7 are relatively prime to 8 and are <= 8, a(8) = 1*(8-1) +3*(8-3) +5*(8-5) +7*(8-7) = 44.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000292, A023896, A053818, A053819, A002618, A053191, A023900.

f:= proc(n) local i;
  2*add(`if`(igcd(i,n)=1, i*(n-i),0),i=1..n/2)
end proc:
f(2):= 1:
map(f, [$1..100]); # Robert Israel, Sep 29 2019

a[n_] := 2 Sum[Boole[CoprimeQ[k, n]] k (n - k), {k, 1, n/2}];
a[2] = 1;
Array[a, 100] (* Jean-François Alcover, Aug 16 2020, after Maple *)

a(n) = sum(k=1, n, if (gcd(n,k)==1, k*(n-k))); \\ Michel Marcus, Sep 29 2019

From Robert Israel, Sep 29 2019: (Start)
If n is prime, a(n) = A000292(n-1).
If n/2 is an odd prime, a(n) =  A000292(n-2)/2.
If n/3 is a prime other than 3, a(n) = A000292(n-3)*2*n/(3*(n-2)). (End)
From Ridouane Oudra, Mar 21 2024: (Start)
a(n) = n*A023896(n) - A053818(n) ;
a(n) = (2/3)*(n*A023896(n) - A053819(n)/n) ;
a(n) = (n/6)*(A002618(n) - A023900(n)) ;
a(n) = (1/6)*(A053191(n) - n*A023900(n)). (End)
Sum_{k=1..n} a(k) ~ n^4 / (4*Pi^2). - Amiram Eldar, Apr 11 2024

A368197 Triangle read by rows: T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

Original entry on oeis.org

1, 4, 4, 18, 0, 9, 32, 8, 0, 24, 100, 0, 0, 0, 25, 72, 72, 36, 0, 0, 36, 294, 0, 0, 0, 0, 0, 49, 256, 64, 0, 96, 0, 0, 0, 96, 486, 0, 144, 0, 0, 0, 0, 0, 99, 400, 400, 0, 0, 100, 0, 0, 0, 0, 100, 1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121

Offset: 1

Mats Granvik, Dec 16 2023

Row n has sum n^3. The number of nonzero terms in row n appears to be A000005(n). It appears that Sum_{k=1..n} T(n,k)*A023900(k) = A063524(n). Main diagonal appears to be A062775. First column appears to be A053191.

It appears that when p > 2 in f(x,y,z,p) = x^p + y^p - z^p and T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z,p), n) = k], then Sum_{k=1..n} T(n,k)*A023900(k) is not equal to A063524(n). - Mats Granvik, May 07 2024

			Triangle begins:
     1;
     4,   4;
    18,   0,   9;
    32,   8,   0,  24;
   100,   0,   0,   0,  25;
    72,  72,  36,   0,   0,  36;
   294,   0,   0,   0,   0,   0,  49;
   256,  64,   0,  96,   0,   0,   0,  96;
   486,   0, 144,   0,   0,   0,   0,   0,  99;
   400, 400,   0,   0, 100,   0,   0,   0,   0, 100;
  1210,   0,   0,   0,   0,   0,   0,   0,   0,   0, 121;
  ...

Cf. A000578, A000005, A023900, A063524, A062775, A053191, A367689.

nn = 11; p = 2; f = x^p + y^p - z^p; Flatten[Table[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {z, 1, n}], {k, 1, n}], {n, 1, nn}]]

T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

A085646 Sum of the entries in the character table of the group GL(2,Z_n).

Original entry on oeis.org

1, 5, 24, 52, 120, 120, 336, 496, 654, 600, 1320, 1248, 2184, 1680, 2880

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 11 2003

Cf. A000252, A062354, A053191.

For an odd prime p : a(p) = p*(p^2-1).

A288572 a(n) = smallest positive integer k such that (n^3)^k == 1 mod (n+1)^3.

Original entry on oeis.org

1, 6, 16, 50, 6, 98, 64, 54, 50, 242, 48, 338, 98, 150, 256, 578, 54, 722, 400, 294, 242, 1058, 192, 1250, 338, 486, 784, 1682, 150, 1922, 1024, 726, 578, 2450, 432, 2738, 722, 1014, 1600, 3362, 294, 3698, 1936, 1350, 1058, 4418, 768, 4802, 1250, 1734, 2704, 5618, 486, 6050, 3136, 2166, 1682, 6962

Offset: 1

N. J. A. Sloane, Jun 28 2017

nonn

From Robert Israel, Jun 28 2017: (Start)
Multiplicative order of n^3 mod (n+1)^3.
a(n) divides A053191(n+1).
a(n) is even for n >= 2.
If n == 3 (mod 4) then a(n) == 0 (mod 16), otherwise it appears that a(n) == 2 or 6 (mod 16).
If n == 0 or 4 (mod 6), then a(n) = 2*n^2.
(End)

			5^3 = 125, 6^3 = 216, 125^6 = 14551915228366851806640625 == 1 mod 216, so a(5) = 6.
(6^3)^98 == 1 mod 7^3, so a(6) = 98.

Robert Israel, Table of n, a(n) for n = 1..10000
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. See page 45.
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. [Cached copy] See page 45.

Cf. A053191.

seq(numtheory:-order(n^3, (n+1)^3), n=1..300); # Robert Israel, Jun 28 2017

a(n) = my(k=1); while(Mod(n^3, (n+1)^3)^k!=1, k++); k \\ Felix Fröhlich, Jun 28 2017

Empirical: a(n+36) = 3*a(n+24) - 3*a(n+12) + a(n) for n >= 2. - Robert Israel, Jun 28 2017

Empirical g.f.: x*(x^36 + 2*x^35 + 2*x^33 + 2*x^32 + 6*x^31 + 16*x^30 + 50*x^29 + 6*x^28 + 98*x^27 + 64*x^26 + 54*x^25 + 47*x^24 + 236*x^23 + 48*x^22 + 332*x^21 + 92*x^20 + 132*x^19 + 208*x^18 + 428*x^17 + 36*x^16 + 428*x^15 + 208*x^14 + 132*x^13 + 95*x^12 + 338*x^11 + 48*x^10 + 242*x^9 + 50*x^8 + 54*x^7 + 64*x^6 + 98*x^5 + 6*x^4 + 50*x^3 + 16*x^2 + 6*x + 1)/(-x^36 + 3*x^24 - 3*x^12 + 1). - Colin Barker, Jun 30 2017

Corrected and more terms from Robert Israel, Jun 28 2017

A086782 Sum of the degrees of the irreducible representations of the group GL(3,Z_n).

Original entry on oeis.org

1, 28, 468, 1876, 12400, 13104, 100548

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 03 2003

Cf. A086768, A053191.

For a prime p : a(p) = phi(p^3)*(p^3 - 1) = p^2*(p-1)^2*(p^2+p+1)

cp's OEIS Frontend

A053211 Cototients of consecutive pure powers of primes.

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A057789 a(n) = Sum_{k = 1..n, gcd(k,n)=1} k*(n-k).

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A368197 Triangle read by rows: T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.

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A085646 Sum of the entries in the character table of the group GL(2,Z_n).

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A288572 a(n) = smallest positive integer k such that (n^3)^k == 1 mod (n+1)^3.

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A086782 Sum of the degrees of the irreducible representations of the group GL(3,Z_n).

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