A348929 -id:A348929 - cp's OEIS frontend

A348999 a(n) = A348929(A276086(n)), where A348929(n) = gcd(n, A003959(n)), A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

1, 1, 1, 6, 1, 6, 1, 2, 3, 6, 3, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 1, 6, 1, 6, 1, 2, 3, 6, 3, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 1, 6, 1, 6, 1, 2, 3, 6, 3, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 1, 6, 1, 6, 1, 2, 3, 6, 3, 18

Offset: 0

Antti Karttunen, Nov 07 2021

After each primorial number (A002110), the apparent periodicity grows more complex.

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A002110, A003959, A276086, A348929, A348949, A348950, A348998, A349000.

A348999(n) = { my(m1=1, m2=1, p=2); while(n, m1 *= (p^(n%p)); m2 *= ((1+p)^(n%p)); n = n\p; p = nextprime(1+p)); gcd(m1,m2); };

a(n) = A348929(A276086(n)).

a(n) = gcd(A276086(n), A348949(n)) = gcd(A276086(n), A348950(n)).

A348507 a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 1, 19, 7, 8, 1, 24, 1, 10, 9, 65, 1, 30, 1, 34, 11, 14, 1, 84, 11, 16, 37, 44, 1, 42, 1, 211, 15, 20, 13, 108, 1, 22, 17, 122, 1, 54, 1, 64, 51, 26, 1, 276, 15, 58, 21, 74, 1, 138, 17, 160, 23, 32, 1, 156, 1, 34, 65, 665, 19, 78, 1, 94, 27, 74, 1, 360, 1, 40, 69, 104, 19, 90, 1, 406, 175, 44, 1, 204

Offset: 1

Antti Karttunen, Oct 30 2021

nonn

a(p*(n/p)) - (n/p) = (p+1)*a(n/p) holds for all prime divisors p of n, which can be seen by expanding the left hand side as (A003959(p*(n/p)) - (p*(n/p))) - (n/p) = (p+1)*A003959(n/p)-((p+1)*(n/p)) = (p+1)*(A003959(n/p)-(n/p)) = (p+1)*a(n/p). This implies that a(n) >= A003415(n) for all n. (See also comments in A348970). - Antti Karttunen, Nov 06 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A003415, A003959, A020639, A032742, A065488, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970.
Cf. A348971 (Möbius transform) and A349139, A349140, A349141, A349142, A349143 (other Dirichlet convolutions).
Cf. also A168065 (the arithmetic mean of this and A322582), A168066.

f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);

A020639(n) = if(1==n,n,(factor(n)[1, 1]));
A348507(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (1+spf)); (s); }; \\ (Compare this with similar programs given in A003415 and in A322582) - Antti Karttunen, Nov 06 2021

a(n) = A003959(n) - n.
a(n) = A348508(n) + n.
a(n) = A001065(n) + A348029(n).
From Antti Karttunen, Nov 06 2021: (Start)
a(n) = Sum_{d|n} A348971(d).
a(n) = A003415(n) + A348970(n).
For all n >= 1, A322582(n) <= A003415(n) <= a(n).
For n > 1, a(n) = a(A032742(n))*(1+A020639(n)) + A032742(n). [See the comments above, and compare this with Reinhard Zumkeller's May 09 2011 recursive formula for A003415] (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065488 - 1. - Amiram Eldar, Jun 01 2025

A348928 a(n) = gcd(n, A003958(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 5, 2, 3, 2, 1, 4, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Nov 07 2021

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

Cf. A003958, A085731, A126865, A322582, A348929, A348998 [= a(A276086(n))].

Differs from similar A126864 for the first time at n=36, where a(36) = 4, while A126864(36) = 2.

f[p_, e_] := (p - 1)^e; a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 07 2021 *)

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A348928(n) = gcd(n, A003958(n));

a(n) = gcd(n, A003958(n)) = gcd(n, A322582(n)) = gcd(A003958(n), A322582(n)).

cp's OEIS Frontend

A348999 a(n) = A348929(A276086(n)), where A348929(n) = gcd(n, A003959(n)), A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

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A348507 a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A348928 a(n) = gcd(n, A003958(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e.

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