A237016 -id:A237016 - cp's OEIS frontend

A236998 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) is a square}|, where phi(.) is Euler's totient function.

0, 0, 1, 0, 0, 1, 2, 0, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 3, 2, 2, 4, 3, 1, 3, 1, 3, 1, 1, 2, 2, 1, 4, 4, 3, 3, 1, 1, 5, 2, 3, 7, 2, 5, 3, 4, 3, 2, 7, 3, 2, 3, 4, 6, 2, 1, 7, 5, 3, 2, 2, 4, 4, 2, 6, 4, 3, 5, 5, 7, 4, 3, 2, 6, 4, 2, 7, 5, 5, 4, 4, 2, 4, 8, 2, 7, 5, 7, 3, 3, 8, 6, 7, 5, 7, 3, 9, 3, 7, 5

Offset: 1

Zhi-Wei Sun, Feb 02 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 8.
(ii) If n > 20, then phi(k)*phi(n-k) + 1 is a square for some 0 < k < n/2.
(iii) If n > 1 is not among 4, 7, 60, 199, 267, then k*phi(n-k) is a square for some 0 < k < n.
We have verified part (i) of the conjecture for n up to 2*10^6.

			a(17) = 1 since phi(5)*phi(12) = 4*4 = 4^2.
a(24) = 1 since phi(4)*phi(20) = 2*8 = 4^2.
a(56) = 1 since phi(8)*phi(48) = 4*16 = 8^2.

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

Cf. A000010, A000290, A234246, A236567, A237016.

SQ[n_]:=IntegerQ[Sqrt[n]]
p[n_,k_]:=SQ[EulerPhi[k]*EulerPhi[n-k]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,100}]

A237049 Number of ordered ways to write n = i + j + k (0 < i <= j <= k) with i,j,k not all equal such that sigma(i)sigma(j)sigma(k) is a cube, where sigma(m) denotes the sum of all positive divisors of m.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 2, 1, 1, 1, 3, 4, 2, 3, 3, 1, 2, 3, 2, 3, 2, 2, 3, 2, 1, 0, 5, 3, 4, 3, 1, 2, 4, 1, 2, 3, 5, 7, 5, 6, 3, 4, 6, 7, 6, 7, 3, 8, 2, 7, 6, 4, 3, 8, 7, 6, 6, 2, 7, 5, 7, 2, 8, 4, 8, 6, 5, 7, 7, 9, 10, 5, 9, 7, 11, 3, 6, 7, 8, 8, 7, 5, 6, 5

Offset: 1

Zhi-Wei Sun, Feb 02 2014

nonn

Conjecture: For every k = 2, 3, ... there is a positive integer N(k) such that any integer n > N(k) can be written as n_1 + n_2 + ... + n_k with n_1, n_2, ..., n_k positive and distinct such that the product sigma(n_1)*sigma(n_2)*...*sigma(n_k) is a k-th power. In particular, we may take N(2) = 309, N(3) = 42, N(4) = 25, N(5) = 24, N(6) = 27 and N(7) = 32.

This is similar to the conjecture in A233386.

			 a(9) = 1 since 9 = 1 + 1 + 7 with sigma(1)*sigma(1)*sigma(7) = 1*1*8 = 2^3.
a(41) = 1 since 41 = 2 + 6 + 33 with sigma(2)*sigma(6)*sigma(33) = 3*12*48 = 12^3.
a(50) = 1 since 50 = 2 + 17 + 31 with sigma(2)*sigma(17)*sigma(31) = 3*18*32 = 12^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Cf. A000203, A000578, A233386, A236998, A237016.

sigma[n_]:=DivisorSigma[1,n]
CQ[n_]:=IntegerQ[n^(1/3)]
p[i_,j_,k_]:=CQ[sigma[i]*sigma[j]*sigma[k]]
a[n_]:=Sum[If[p[i,j,n-i-j],1,0],{i,1,(n-1)/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

A237050 Number of ways to write n = i_1 + i_2 + i_3 + i_4 + i_5 (0 < i_1 <= i_2 <= i_3 <= i_4 <= i_5) with i_1, i_2, ..., i_5 not all equal such that the product i_1i_2i_3i_4i_5 is a fifth power.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 4, 3, 3, 3, 5, 4, 5, 7, 3, 5, 3, 4, 3, 3, 4, 6, 4, 4, 4, 4, 2, 4, 3, 5, 5, 3, 5, 4, 8, 7, 7, 9, 10, 9, 12, 7, 6, 9, 10, 9, 9, 8, 8, 7, 10, 7, 10, 10, 10, 10, 5, 8, 13, 10, 9, 8, 12, 15, 10, 12, 9, 8

Offset: 1

Zhi-Wei Sun, Feb 02 2014

nonn

Conjecture: For each k = 3, 4, ... there is a positive integer M(k) such that any integer n > M(k) can be written as i_1 + i_2 + ... + i_k with i_1, i_2, ..., i_k positive and not all equal such that the product i_1*i_2*...*i_k is a k-th power. In particular, we may take M(3) = 486, M(4) = 23, M(5) = 26, M(6) = 36 and M(7) = 31.

This is motivated by the conjectures in A233386 and A237049.

			a(25) = 1 since 25 = 1 + 4 + 4 + 8 + 8 with 1*4*4*8*8 = 4^5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300
Tianxin Cai and Deyi Chen, A new variant of the Hilbert-Waring problem, Math. Comp. 82 (2013), 2333-2341.

Cf. A000584, A233386, A236998, A237016, A237049.

QQ[n_]:=IntegerQ[n^(1/5)]
a[n_]:=Sum[If[QQ[i*j*h*k*(n-i-j-h-k)],1,0],{i,1,(n-1)/5},{j,i,(n-i)/4},{h,j,(n-i-j)/3},{k,h,(n-i-j-h)/2}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A236998 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) is a square}|, where phi(.) is Euler's totient function.

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A237049 Number of ordered ways to write n = i + j + k (0 < i <= j <= k) with i,j,k not all equal such that sigma(i)sigma(j)sigma(k) is a cube, where sigma(m) denotes the sum of all positive divisors of m.

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A237050 Number of ways to write n = i_1 + i_2 + i_3 + i_4 + i_5 (0 < i_1 <= i_2 <= i_3 <= i_4 <= i_5) with i_1, i_2, ..., i_5 not all equal such that the product i_1i_2i_3i_4i_5 is a fifth power.

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cp's OEIS Frontend

A236998 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) is a square}|, where phi(.) is Euler's totient function.

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A237049 Number of ordered ways to write n = i + j + k (0 < i <= j <= k) with i,j,k not all equal such that sigma(i)*sigma(j)*sigma(k) is a cube, where sigma(m) denotes the sum of all positive divisors of m.

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Mathematica

A237050 Number of ways to write n = i_1 + i_2 + i_3 + i_4 + i_5 (0 < i_1 <= i_2 <= i_3 <= i_4 <= i_5) with i_1, i_2, ..., i_5 not all equal such that the product i_1*i_2*i_3*i_4*i_5 is a fifth power.

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A237049 Number of ordered ways to write n = i + j + k (0 < i <= j <= k) with i,j,k not all equal such that sigma(i)sigma(j)sigma(k) is a cube, where sigma(m) denotes the sum of all positive divisors of m.

A237050 Number of ways to write n = i_1 + i_2 + i_3 + i_4 + i_5 (0 < i_1 <= i_2 <= i_3 <= i_4 <= i_5) with i_1, i_2, ..., i_5 not all equal such that the product i_1i_2i_3i_4i_5 is a fifth power.