A003958 -id:A003958 - cp's OEIS frontend

A349131 a(n) = Sum_{d|n} phi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and phi is Euler totient function.

1, 2, 4, 4, 8, 8, 12, 8, 14, 16, 20, 16, 24, 24, 32, 16, 32, 28, 36, 32, 48, 40, 44, 32, 52, 48, 46, 48, 56, 64, 60, 32, 80, 64, 96, 56, 72, 72, 96, 64, 80, 96, 84, 80, 112, 88, 92, 64, 114, 104, 128, 96, 104, 92, 160, 96, 144, 112, 116, 128, 120, 120, 168, 64, 192, 160, 132, 128, 176, 192, 140, 112, 144, 144, 208

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with Euler totient function phi, A000010.

Möbius transform of A349130.

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

Cf. A000010, A003958, A018804, A348981, A349130 (inverse Möbius transform), A349132, A349171.

f[p_, e_] := (p - 1)*p^e - (p - 2)*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349131(n) = sumdiv(n,d,eulerphi(d)*A003958(n/d));

a(n) = Sum_{d|n} A000010(d) * A003958(n/d).
a(n) = Sum_{d|n} A008683(d) * A349130(n/d).
a(n) = Sum_{k=1..n} A003958(gcd(n, k)).
a(n) = A018804(n) - A348981(n).
For all n >= 1, a(n) <= A349171(n).
Multiplicative with a(p^e) = (p-1)*p^e - (p-2)*(p-1)^e. - Amiram Eldar, Nov 09 2021
Dirichlet g.f.: (zeta(s-1)/zeta(s)) / Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Dec 24 2023

A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 22, 24, 40, 22, 60, 26, 56, 60, 46, 34, 96, 38, 100, 84, 88, 46, 132, 70, 104, 84, 140, 58, 240, 62, 94, 132, 136, 140, 240, 74, 152, 156, 220, 82, 336, 86, 220, 240, 184, 94, 276, 140, 280, 204, 260, 106, 336, 220, 308, 228, 232, 118, 600, 122, 248, 336, 190, 260, 528, 134, 340, 276, 560

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with Dedekind psi function, A001615.

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

Cf. A001615, A003958, A327251, A348982, A349130, A349131, A349133, A349172.

f[p_, e_] := (p + 1)*p^e - p*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349132(n) = sumdiv(n,d,A001615(d)*A003958(n/d));

a(n) = Sum_{d|n} A001615(d) * A003958(n/d).
a(n) = A327251(n) - A348982(n).
For all n >= 1, a(n) <= A349172(n).
Multiplicative with a(p^e) = (p+1)*p^e - p*(p-1)^e. - Amiram Eldar, Nov 09 2021

A351445 a(n) = A003958(sigma(n)) - A003958(n), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

0, 1, -1, 5, -2, 0, -5, 7, 8, 0, -8, 4, -6, -4, -6, 29, -12, 20, -14, 8, -11, -6, -20, 6, 14, 0, -4, 0, -20, -4, -29, 23, -18, -8, -22, 68, -18, -10, -18, 12, -28, -10, -32, 2, 8, -18, -44, 28, 0, 44, -28, 24, -44, 0, -36, 2, -32, -12, -50, 4, -30, -28, -12, 125, -36, -16, -50, 8, -42, -20, -66, 92, -36, 0

Offset: 1

Antti Karttunen, Feb 12 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A351442, A351444, A351457 [= a(A003961(n))].
Cf. A351446 (positions of zeros), A351443 (odd terms there).
Cf. also A348736.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351445(n) = (A003958(sigma(n)) - A003958(n));

a(n) = A351442(n) - A003958(n) = A351444(n) - n.

A353794 a(n) = A353791(sigma(A003961(n))), where A353791(n) = A003958(n) * A064989(n).

Original entry on oeis.org

1, 1, 4, 132, 1, 4, 4, 12, 870, 1, 30, 528, 16, 4, 4, 4900, 12, 870, 4, 132, 16, 30, 48, 48, 1224, 16, 528, 528, 1, 4, 306, 3960, 120, 12, 4, 114840, 120, 4, 64, 12, 70, 16, 4, 3960, 870, 48, 64, 19600, 9180, 1224, 48, 2112, 48, 528, 30, 48, 16, 1, 870, 528, 208, 306, 3480, 1191372, 16, 120, 16, 1584, 192, 4, 1116

Offset: 1

Antti Karttunen, May 11 2022

It is conjectured that a(n) is not a multiple of A353793(n) on any other n except on n=1. See also A353795.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A003961, A003973, A064989, A326042, A351456, A353791, A353792, A353793, A353795 [numbers k such that k divides a(k)].

Cf. also A353790.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353794(n) = { my(s=sigma(A003961(n))); (A003958(s)*A064989(s)); };

Multiplicative with a(p^e) = A003958(1 + q + ... + q^e) * A064989(1 + q + ... + q^e), where q is the least prime larger than p.
a(n) = A353791(A003973(n)) = A353792(A003961(n)).
a(n) = A326042(n) * A351456(n) = A064989(A003973(n)) * A003958(A003973(n)).

A340081 a(n) = gcd(n-1, A003958(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 8, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 12, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 16, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4, 1, 2, 1, 96, 1, 2, 1, 100

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340082, A340083.

Cf. also A049559, A340071.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340081(n) = gcd(n-1, A003958(n));

a(n) = gcd(n-1, A003958(n)).
a(n) = A003958(n) / A340082(n).
For n > 1, a(n) = (n-1) / A340083(n).

A340082 a(n) = A003958(n) / gcd(n-1, A003958(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 4, 1, 1, 4, 1, 4, 3, 10, 1, 2, 2, 12, 4, 2, 1, 8, 1, 1, 5, 16, 12, 4, 1, 18, 12, 4, 1, 12, 1, 10, 4, 22, 1, 2, 3, 16, 16, 4, 1, 8, 20, 6, 9, 28, 1, 8, 1, 30, 12, 1, 3, 4, 1, 16, 11, 8, 1, 4, 1, 36, 16, 6, 15, 24, 1, 4, 1, 40, 1, 12, 16, 42, 28, 10, 1, 16, 4, 22, 15, 46, 36, 2, 1, 36

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340081, A340083, A340085 (gives the odd part).

Cf. also A160595, A340072.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340082(n) = { my(u=A003958(n)); u/gcd(n-1, u); };

a(n) = A003958(n) / A340081(n) = A003958(n) / gcd(n-1, A003958(n)).

A349129 a(n) = Sum_{d|n} A003958(d) * A003959(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

1, 4, 6, 13, 10, 24, 14, 40, 28, 40, 22, 78, 26, 56, 60, 121, 34, 112, 38, 130, 84, 88, 46, 240, 76, 104, 120, 182, 58, 240, 62, 364, 132, 136, 140, 364, 74, 152, 156, 400, 82, 336, 86, 286, 280, 184, 94, 726, 148, 304, 204, 338, 106, 480, 220, 560, 228, 232, 118, 780, 122, 248, 392, 1093, 260, 528, 134, 442, 276

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with A003959.

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

Cf. A003958, A003959, A168066, A349139.

Cf. also A349130, A349131, A349132, A349133 and A349170, A349171, A349172, A349173.

f[p_, e_] := ((p + 1)^(e + 1) - (p - 1)^(e + 1))/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349129(n) = sumdiv(n,d,A003958(d)*A003959(n/d));

Multiplicative with a(p^e) = ((p+1)^(e+1) - (p-1)^(e+1))/2. - Amiram Eldar, Nov 09 2021

For all n >= 1, A349130(n) <= a(n) <= A349170(n).

A349356 Dirichlet convolution of A003959 with A097945 (Dirichlet inverse of A003958), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 18, 8, 4, 2, 12, 2, 4, 4, 54, 2, 16, 2, 12, 4, 4, 2, 36, 12, 4, 32, 12, 2, 8, 2, 162, 4, 4, 4, 48, 2, 4, 4, 36, 2, 8, 2, 12, 16, 4, 2, 108, 16, 24, 4, 12, 2, 64, 4, 36, 4, 4, 2, 24, 2, 4, 16, 486, 4, 8, 2, 12, 4, 8, 2, 144, 2, 4, 24, 12, 4, 8, 2, 108, 128, 4, 2, 24, 4, 4, 4, 36, 2, 32, 4, 12, 4

Offset: 1

Antti Karttunen, Nov 16 2021

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003958 with factor A003959. For example, convolving this with A349133 produces A349173.

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

Cf. A003958, A003959, A097945, A349355 (Dirichlet inverse), A349357 (sum with it).

Cf. also A349133, A349173, A349381.

f[p_, e_] := 2*(p + 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A097945(n) = (moebius(n)*eulerphi(n)); \\ Also Dirichlet inverse of A003958.
A349356(n) = sumdiv(n,d,A003959(n/d)*A097945(d));

a(n) = Sum_{d|n} A003959(n/d) * A097945(d).

Multiplicative with a(p^e) = 2*(p+1)^(e-1). - Amiram Eldar, Nov 16 2021

A351443 Odd numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 49, 40905, 106353, 140211, 275301, 302697, 499041, 597213, 1094913, 1284417, 1578933, 2004345, 2266137, 2560653, 3247857, 3444201, 3738717, 4425921, 5014953, 5123817, 5211297, 5407641, 5505813, 5996673, 6193017, 6870339, 7174737, 8156457, 8941833, 9432693, 9825381, 9923553

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Odd numbers k for which A351442(k) = A003958(k), or equally, for which k = A351444(k) = A322582(k) + A351442(k).

The 13th term, 2004345, is one of the rare abundant numbers (A005101, A005231) in this sequence.

Odd terms in A351446.
Cf. A000203, A003958, A005101, A005231, A322582, A351442, A351444, A351445.
These terms doubled form a subsequence of A351447.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351442(n) = A003958(sigma(n));
isA351443(n) = ((n%2) && (A351442(n)==A003958(n)));

A340083 a(n) = (n-1) / gcd(n-1, A003958(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 2, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 4, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 63, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89, 5, 91, 23

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340081, A340082.

Cf. also A160596, A340073.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340083(n) = ((n-1)/gcd(n-1, A003958(n)));

a(n) = (n-1) / A340081(n) = (n-1) / gcd(n-1, A003958(n)).

cp's OEIS Frontend

A349131 a(n) = Sum_{d|n} phi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and phi is Euler totient function.

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A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

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A351445 a(n) = A003958(sigma(n)) - A003958(n), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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A353794 a(n) = A353791(sigma(A003961(n))), where A353791(n) = A003958(n) * A064989(n).

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A340081 a(n) = gcd(n-1, A003958(n)).

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A340082 a(n) = A003958(n) / gcd(n-1, A003958(n)).

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A349129 a(n) = Sum_{d|n} A003958(d) * A003959(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and A003959 is fully multiplicative with a(p) = (p+1).

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A349356 Dirichlet convolution of A003959 with A097945 (Dirichlet inverse of A003958), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

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