A167344 -id:A167344 - cp's OEIS frontend

A306709 For n > 1, a(n) = gcd(A001414(n), A167344(n)) where A001414(n) is the sum of primes p dividing n (with repetition) and A167344(n) = b(n) is the totally multiplicative sequence with b(p) = (p-1)*(p+1) = p^2 - 1; a(1) = 0.

0, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 9, 8, 1, 1, 8, 1, 9, 2, 1, 1, 9, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 12, 2, 1, 3, 16, 1, 1, 12, 1, 15, 1, 1, 1, 1, 2, 12, 4, 1, 1, 1, 16, 1, 2, 1, 1, 12, 1, 3, 1, 3, 18, 16, 1, 3, 2, 2, 1, 12, 1, 3, 1, 1, 18, 18, 1, 1, 4, 1, 1, 2, 2, 9, 32, 1, 1, 1, 4, 27

Offset: 1

Juri-Stepan Gerasimov, May 20 2019

Positions of records: 0, 1, 8, 14, 35, 39, 65, 87, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A003958, A003959, A167344.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A167344(n) = { my(f=factor(n)); for(i=1,#f~,f[i,1] = (f[i,1]^2)-1); factorback(f); };
A306709(n) = if(1==n, 0, gcd(A001414(n), A167344(n))); \\ Antti Karttunen, Jan 03 2021

a(n) = gcd(sopfr(n), A003958(n)*A003959(n)) for n > 1; a(1) = 0.

a(p) = 1 for all primes p. - Antti Karttunen, Jan 03 2021

A335915 Fully multiplicative with a(2) = 1, and a(p) = A000265(p-1)*A000265(p+1) = A000265(p^2 - 1), for odd primes p.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 15, 1, 21, 3, 3, 1, 9, 1, 45, 3, 3, 15, 33, 1, 9, 21, 1, 3, 105, 3, 15, 1, 15, 9, 9, 1, 171, 45, 21, 3, 105, 3, 231, 15, 3, 33, 69, 1, 9, 9, 9, 21, 351, 1, 45, 3, 45, 105, 435, 3, 465, 15, 3, 1, 63, 15, 561, 9, 33, 9, 315, 1, 333, 171, 9, 45, 45, 21, 195, 3, 1, 105, 861, 3, 27, 231, 105, 15, 495, 3, 63, 33, 15, 69

Offset: 1

Antti Karttunen, Jul 09 2020

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A336118(i) = A336118(j).

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

Cf. A000265.

Cf. also A167344, A324400, A335904, A336118, A336455, A336456, A336466, A336467.

A000265(n) = (n>>valuation(n,2));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };

Completely multiplicative with a(2) = 1, and for odd primes p, a(p) = A000265(p-1)*A000265(p+1).
For all n >= 1, A335904(a(n)) = A336118(n).
For all n >= 0, a(2^n) = a(3^n) = 1, a(5^n) = a(7^n) = 3^n.
a(n) = A336466(n) * A336467(n). - Antti Karttunen, Jan 31 2021

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A340368 Multiplicative with a(p^e) = (p - 1) * (p + 1)^(e-1).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 9, 8, 4, 10, 6, 12, 6, 8, 27, 16, 8, 18, 12, 12, 10, 22, 18, 24, 12, 32, 18, 28, 8, 30, 81, 20, 16, 24, 24, 36, 18, 24, 36, 40, 12, 42, 30, 32, 22, 46, 54, 48, 24, 32, 36, 52, 32, 40, 54, 36, 28, 58, 24, 60, 30, 48, 243, 48, 20, 66, 48, 44, 24, 70, 72, 72, 36, 48, 54, 60, 24, 78, 108, 128, 40

Offset: 1

Antti Karttunen, Jan 06 2021

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A167344, A173557, A327564, A340323.

f[p_, e_] := (p - 1)*(p + 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 12 2022 *)

A340368(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]-1)*((f[i,1]+1)^(f[i,2]-1))));

a(n) = A167344(n) / A340323(n).
a(n) = A173557(n) * A327564(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 3/p^2 + 2/p^4) /  Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.4313799748... . - Amiram Eldar, Nov 12 2022

cp's OEIS Frontend

A306709 For n > 1, a(n) = gcd(A001414(n), A167344(n)) where A001414(n) is the sum of primes p dividing n (with repetition) and A167344(n) = b(n) is the totally multiplicative sequence with b(p) = (p-1)*(p+1) = p^2 - 1; a(1) = 0.

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A335915 Fully multiplicative with a(2) = 1, and a(p) = A000265(p-1)*A000265(p+1) = A000265(p^2 - 1), for odd primes p.

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A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

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A340368 Multiplicative with a(p^e) = (p - 1) * (p + 1)^(e-1).

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