A002618 -id:A002618 - cp's OEIS frontend

A144824 Triangle read by rows, A054533 * A127648 (matrix product).

1, -1, 2, -1, -2, 6, 0, -4, 0, 8, -1, -2, -3, -4, 20, 1, -2, -6, -4, 5, 12, -1, -2, -3, -4, -5, -6, 42, 0, 0, 0, -16, 0, 0, 0, 32, 0, 0, -9, 0, 0, -18, 0, 0, 54, 1, -2, 3, -4, -20, -6, 7, -8, 9, 40, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 110

Offset: 1

Gary W. Adamson, Sep 21 2008

Row sums = A023896: (1, 1, 3, 4, 10, 16, 21, ...).
Right border = A002618: (1, 2, 6, 8, 20, 12, ...).
Left border = mu(n) = A008683 (n).

			Triangle A054533 starts as follows:
   1;
  -1,  1;
  -1, -1,  2;
   0, -2,  0,  2;
  -1, -1, -1, -1, 4;
   1, -1, -2, -1, 1, 2;
   ...
The first few rows of triangle A144824 are as follows:
   1;
  -1,  2;
  -1, -2,  6;
   0, -4,  0,  8;
  -1, -2, -3, -4, 20;
   1, -2, -6, -4,  5, 12;
  -1, -2, -3, -4, -5, -6, 42;
   ...

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A002260, A002618, A008683, A023896, A054533, A127648.

Triangle read by rows, A054533 * A127648 (matrix product). The operation is equivalent to taking termwise products of row A054533 terms and the natural numbers.
T(n, k) = k * Sum_{d|gcd(n,k)} d * mu(n/d) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Jul 28 2019
a(n) = A002260(n)*A054533(n). - Jinyuan Wang, Jul 29 2019

A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 7, 2, 1, 4, 1, 1, 1, 0, 1, 14, 1, 4, 3, 2, 1, 8, 2, 1, 5, 2, 1, 2, 1, 0, 2, 1, 1, 28, 1, 1, 1, 8, 1, 3, 1, 1, 3, 2, 1, 16, 1, 4, 1, 2, 1, 10, 1, 4, 3, 2, 1, 4, 1, 1, 1, 0, 1, 4, 1, 2, 1, 2, 1, 56, 1, 1, 6, 1, 3, 2, 1, 1, 47, 2, 1, 6, 3, 1, 1, 2, 1, 6, 5, 3, 2, 1, 1, 32, 1, 2, 1, 8, 1, 2, 1

Offset: 1

Vladimir Shevelev, Mar 31 2010

nonn

a(n)=1 if the index n is an odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A006530, A103667, A023503, A023509, A174857, A020639, A002618, A036689.

A006530 := proc(n) option remember; if n = 1 then 1; else max(op(numtheory[factorset](n)) ) ; end if; end proc:
A174869 := proc(n) if n <= 2 then 0; else gpf := A006530(n) ; if gpf = 2 then 0; else for k from 1 do if A006530(n+k) < gpf then return k; end if; end do: end if; end if; end proc:
seq(A174869(n),n=1..120) ; # R. J. Mathar, Aug 10 2010

Block[{s = Array[FactorInteger[#][[-1, 1]] &, 120]}, Array[If[IntegerQ@ Log2[#], 0, Block[{k = 1, n = s[[#]]}, While[n <= s[[# + k]], k++; If[# + k > Length[s], AppendTo[s, FactorInteger[# + k][[-1, 1]] ]] ]; k]] &, 102, 2]] (* Michael De Vlieger, Apr 06 2021 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A174869(n) = if(!bitand(n,n-1), 0, my(gpf=A006530(n)); for(k=1,oo,if(A006530(n+k)Antti Karttunen, Apr 06 2021

More terms from R. J. Mathar, Aug 10 2010

A181538 T(n, k) = sum_(1 <= j <= k) [j | k] j mu(k / j) gcd(n, k), triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 1, 2, 2, 8, 1, 1, 2, 2, 20, 1, 2, 6, 4, 4, 12, 1, 1, 2, 2, 4, 2, 42, 1, 2, 2, 8, 4, 4, 6, 32, 1, 1, 6, 2, 4, 6, 6, 4, 54, 1, 2, 2, 4, 20, 4, 6, 8, 6, 40, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 110, 1, 2, 6, 8, 4, 12, 6, 16, 18, 8, 10, 48

Offset: 1

Peter Luschny, Oct 29 2010

T(n,k) = gcd(n,k) phi(k). Can be seen as a generalization of n -> phi(n^2) [A002618].

			1
1,2
1,1,6
1,2,2,8
1,1,2,2,20
1,2,6,4,4,12
1,1,2,2,4,2,42

Peter Luschny, Sequences related to Euler's totient function.

Cf. Row sums of triangle A181540.

Maple
```
A181538 := (n,k) -> igcd(n,k)*phi(k);
```

t[n_, k_] := Block[{j = Divisors@ k}, Plus @@ (#*MoebiusMu[k/#] & /@ j)] GCD[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten (* Robert G. Wilson v, Jan 19 2011 *)

A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

Original entry on oeis.org

1, 1, 3, 9, 20, 52, 105, 253, 536, 1142, 2421, 4999, 10278, 20686, 41512, 81984, 161029, 312681, 603070, 1153284, 2189331, 4129537, 7733317, 14399693, 26644337, 49034811, 89741600, 163411148, 296074694, 533909026, 958416113, 1712893825, 3048468607, 5403248469, 9539609984

Offset: 0

Paul D. Hanna, May 26 2013

nonn

Here phi(n) = A000010(n) is the Euler totient function.

Euler transform of A002618. - Vaclav Kotesovec, Mar 30 2018

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 37*x^4/4 + 101*x^5/5 + 95*x^6/6 + 295*x^7/7 + 293*x^8/8 + 505*x^9/9 +...+ A068963(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A068963, A002618, A061255, A299069, A301986.

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m,d,eulerphi(d^3))*x^m/m)+x*O(x^n)),n)}
for(n=0,35,print1(a(n),", "))

a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - Vaclav Kotesovec, Mar 30 2018

A262750 Least positive integer z such that n - phi(z^2) = x^2 + y^2 for some integers x and y with xyz even and phi(k^2) < n for all 0 < k < z, or 0 if no such z exists, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 4, 1, 1, 2, 2, 4, 1, 2, 4, 1, 1, 2, 2, 1, 2, 3, 4, 4, 1, 2, 2, 5, 1, 2, 6, 1, 2, 3, 2, 1, 1, 2, 4, 1, 1, 2, 4, 4, 1, 2, 4, 4, 1, 2, 2, 1, 1, 2, 5, 4, 3, 3, 2, 4, 1, 2, 6, 1, 1, 2, 8, 1, 2, 3, 4, 1, 1, 2, 2, 6, 3, 3, 4, 1, 1, 2, 2, 5, 1, 2, 4, 4, 1, 2, 2, 4, 6, 3, 8, 4, 1, 2, 2

Offset: 1

Zhi-Wei Sun, Sep 30 2015

nonn

Conjecture: a(n) <= sqrt(n) except for n = 3, 8, 13, 32.

The conjecture in A262747 implies that a(n) > 0 for all n > 0.

			a(68) = 8 since 68 - phi(8^2) = 68 - 32 = 36 = 0^2 + 6^2 with 0*6*8 even and all those phi(k^2) (k = 1,...,7) smaller than 68.
a(5403) = 67 since 5403 - phi(67^2) = 5403 - 4422 = 981 = 9^2 + 30^2 with 9*30*67 even and all those phi(k^2) (k = 1,...,5403) smaller than 5403.

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

Cf. A000010, A001481, A002618, A233867, A262747.

f[n_]:=EulerPhi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[Do[If[f[x]>n,Goto[aa]]; Do[If[SQ[n-f[x]-y^2]&&(Mod[x*y, 2]==0||Mod[n-f[x]-y^2, 2]==0),Print[n," ",x];Goto[bb]], {y, 0, Sqrt[(n-f[x])/2]}]; Continue, {x, 1, n}]; Label[aa];Print[n," ",0];Label[bb]; Continue, {n,1,100}]

A262982 Number of ordered ways to write n as x^4 + phi(y^2) + z*(z+1)/2 with x >= 0, y > 0 and z > 0, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 06 2015

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 6, 20, 26, 40, 82, 89, 105, 305, 416, 470, 725, 6135, 25430, 90285.

Compare this with the conjectures in A262311, A262785 and A262813.

			a(2) = 1 since 1 = 0^4 + phi(1^2) + 1*2/2.
a(6) = 1 since 6 = 1^4 + phi(2^2) + 2*3/2.
a(20) = 1 since 20 = 2^4 + phi(1^2) + 2*3/2.
a(26) = 1 since 26 = 0^4 + phi(5^2) + 3*4/2.
a(40) = 1 since 40 = 0^4 + phi(6^2) + 7*8/2.
a(82) = 1 since 82 = 0^4 + phi(9^2) + 7*8/2.
a(89) = 1 since 89 = 3^4 + phi(2^2) + 3*4/2.
a(105) = 1 since 105 = 0^4 + phi(14^2) + 6*7/2.
a(305) = 1 since 305 = 4^4 + phi(12^2) + 1*2/2.
a(416) = 1 since 416 = 4^4 + phi(10^2) + 15*16/2.
a(470) = 1 since 470 = 2^4 + phi(12^2) + 28*29/2.
a(725) = 1 since 725 = 2^4 + phi(3^2) + 37*38/2.
a(6135) = 1 since 6135 = 6^4 + phi(81^2) + 30*31/2.
a(25430) = 1 since 25430 = 5^4 + phi(152^2) + 166*167/2.
a(90285) = 1 since 90285 = 16^4 + phi(212^2) + 73*74/2.

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

Cf. A000010, A000217, A000290, A000583, A002618, A262311, A262781, A262785, A262813, A262941, A262979.

f[n_]:=EulerPhi[n^2]
TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[f[x]>n,Goto[aa]];Do[If[TQ[n-f[x]-y^4],r=r+1],{y,0,(n-f[x])^(1/4)}];Label[aa];Continue,{x,1,n}];Print[n," ",r];Continue,{n,1,100}]

A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

Original entry on oeis.org

0, 6, 16, 42, 54, 132, 120, 270, 286, 450, 360, 952, 546, 1056, 1152, 1674, 1098, 2574, 1440, 3276, 2720, 3312, 2376, 6300, 3534, 5124, 5160, 7672, 4380, 11088, 5184, 10836, 8688, 10314, 9600, 19110, 8322, 13680, 13440, 22590, 11046, 26304, 12452, 24780, 23868, 22968, 15792, 42408, 20349, 34131

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers sigma(n)*pi(n*(n+1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers n*sigma(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct, and all the numbers sigma(n^2)*pi(n^2) (n = 1,2,3,...) are also pairwise distinct.
(iv) All the numbers n*phi(n)*sigma(n^2) = phi(n^2)*sigma(n^2) (n = 1,2,3,...) are pairwise distinct, where phi(.) is Euler's totient function.
We have verified that the terms a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263319 for a similar conjecture.

			a(1) = 0 since sigma(1)*pi(1^2) = 1*0 = 0.
a(2) = 6 since sigma(2)*pi(2^2) = 3*2 = 6.

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

Cf. A000010, A000203, A000290, A000720, A002618, A038107, A065764, A263317, A263319.

[#PrimesUpTo(n^2)*SumOfDivisors(n): n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=DivisorSigma[1,n]*PrimePi[n^2]
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = sigma(n)*primepi(n^2); \\ Michel Marcus, Oct 15 2015

A283795 Triangle T(n,k) read by rows: the number of q-circulant n X n {0,1}-matrices where each row sum and each column sum equals k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 14, 8, 1, 1, 20, 40, 40, 20, 1, 1, 12, 42, 44, 42, 12, 1, 1, 42, 126, 210, 210, 126, 42, 1, 1, 32, 136, 224, 350, 224, 136, 32, 1, 1, 54, 216, 546, 756, 756, 546, 216, 54, 1, 1, 40, 260, 480, 1200, 1032, 1200, 480, 260, 40, 1, 1, 110, 550, 1650, 3300, 4620, 4620, 3300, 1650, 550, 110, 1, 1, 48, 324, 992, 2538, 3168

Offset: 0

R. J. Mathar, Mar 16 2017

q-circulant matrices are constructed by fixing the first row and obtaining the remaining n-1 rows by circularly shifting values by q columns, any q from 0 to n-1.
The triangle is symmetric in each row because flipping 1's and 0's in a matrix gives also a circulant matrix with n-k ones in each row and column.
The number of 1-circulant matrices with k zeros in each row and each column is apparently given by Pascal's Triangle.
Is the column k=1 given by A002618?

			The triangle starts in row n=0 and column k=0 as:
1 rsum= 1
1 1 rsum= 2
1 2 1 rsum= 4
1 6 6 1 rsum= 14
1 8 14 8 1 rsum= 32
1 20 40 40 20 1 rsum= 122
1 12 42 44 42 12 1 rsum= 154
1 42 126 210 210 126 42 1 rsum= 758
1 32 136 224 350 224 136 32 1 rsum= 1136
1 54 216 546 756 756 546 216 54 1 rsum= 3146

P. Zellini, On some properties of circulant matrices, Lin. Alg. Applic. 26 (1979) 31-43

Cf. A045655.

A306411 a(n) = phi(n^6) = n^5*phi(n).

Original entry on oeis.org

1, 32, 486, 2048, 12500, 15552, 100842, 131072, 354294, 400000, 1610510, 995328, 4455516, 3226944, 6075000, 8388608, 22717712, 11337408, 44569782, 25600000, 49009212, 51536320, 141599546, 63700992, 195312500, 142576512, 258280326, 206524416, 574312172, 194400000, 858874530, 536870912, 782707860, 726966784

Offset: 1

Jianing Song, Feb 13 2019

The number of elements of the wreath product of C_n and S_6 with cycle partition equal to (6*n) is equal to 5!*a(n), where C_n is the cyclic group of order n, S_6 the symmetric group on 6 elements. - Josaphat Baolahy, Mar 13 2024

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

Eulerphi(n^e): A000010 (e=1), A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), this sequence (e=6), A239442 (e=7), A306412 (e=8), A239443 (e=9).

Array[EulerPhi[#] #^5 &, 34] (* Michael De Vlieger, Feb 17 2019 *)

PARI
```
a(n) = n^5 * eulerphi(n)
```

Multiplicative with a(p^e) = (p - 1)*p^(6*e-1).
Dirichlet g.f.: zeta(s - 6) / zeta(s - 5).
Sum_{k=1..n} a(k) ~ 6*n^7 / (7*Pi^2). See A239443 for a more general formula.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p/(p^7 - p^6 - p + 1)) = 1.03396580456393429553879930771676667947490034699829164744357501993310897305... - Vaclav Kotesovec, Sep 20 2020

A306412 a(n) = phi(n^8) = n^7*phi(n).

Original entry on oeis.org

1, 128, 4374, 32768, 312500, 559872, 4941258, 8388608, 28697814, 40000000, 194871710, 143327232, 752982204, 632481024, 1366875000, 2147483648, 6565418768, 3673320192, 16089691302, 10240000000, 21613062492, 24943578880, 74906159834, 36691771392, 122070312500

Offset: 1

Jianing Song, Feb 13 2019

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

Eulerphi(n^e): A000010 (e=1), A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), A306411 (e=6), A239442 (e=7), this sequence (e=8), A239443 (e=9).

Table[n^7*EulerPhi[n], {n, 1, 25}] (* Amiram Eldar, Dec 06 2020 *)

PARI
```
a(n) = n^7 * eulerphi(n)
```

Multiplicative with a(p^e) = (p - 1)*p^(8*e-1).
Dirichlet g.f.: zeta(s - 8) / zeta(s - 7).
Sum_{k=1..n} a(k) ~ 2*n^9 / (3*Pi^2). See A239443 for a more general formula.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^9 - p^8 - p + 1)) = 1.00807702579309679541... - Amiram Eldar, Dec 06 2020

cp's OEIS Frontend

A144824 Triangle read by rows, A054533 * A127648 (matrix product).

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A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

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A181538 T(n, k) = sum_(1 <= j <= k) [j | k] j mu(k / j) gcd(n, k), triangle read by rows.

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A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

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A262750 Least positive integer z such that n - phi(z^2) = x^2 + y^2 for some integers x and y with x*y*z even and phi(k^2) < n for all 0 < k < z, or 0 if no such z exists, where phi(.) is Euler's totient function given by A000010.

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A262982 Number of ordered ways to write n as x^4 + phi(y^2) + z*(z+1)/2 with x >= 0, y > 0 and z > 0, where phi(.) is Euler's totient function given by A000010.

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A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

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Magma

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A283795 Triangle T(n,k) read by rows: the number of q-circulant n X n {0,1}-matrices where each row sum and each column sum equals k.

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A262750 Least positive integer z such that n - phi(z^2) = x^2 + y^2 for some integers x and y with xyz even and phi(k^2) < n for all 0 < k < z, or 0 if no such z exists, where phi(.) is Euler's totient function given by A000010.