A003958 -id:A003958 - cp's OEIS frontend

A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

1, -2, -2, -2, -2, 4, -2, -2, -4, 4, -2, 4, -2, 4, 4, -2, -2, 8, -2, 4, 4, 4, -2, 4, -8, 4, -8, 4, -2, -8, -2, -2, 4, 4, 4, 8, -2, 4, 4, 4, -2, -8, -2, 4, 8, 4, -2, 4, -12, 16, 4, 4, -2, 16, 4, 4, 4, 4, -2, -8, -2, 4, 8, -2, 4, -8, -2, 4, 4, -8, -2, 8, -2, 4, 16, 4, 4, -8, -2, 4, -16, 4, -2, -8, 4, 4, 4, 4, -2, -16

Offset: 1

Antti Karttunen, Nov 16 2021

Multiplicative because both A003958 and A063441 are.

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003959 with factor A003958. For example, convolving this with A003968 (the Möbius transform of A003959) produces A003966, the Möbius transform of A003958.

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

Cf. A003958, A003959, A003966, A003968, A063441, A349356 (Dirichlet inverse), A349357 (sum with it).

Cf. also A349382.

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 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349355(n) = sumdiv(n,d,A003958(n/d)*A063441(d));

a(n) = Sum_{d|n} A003958(n/d) * A063441(d).

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

A349620 Dirichlet convolution of A003415 with the Dirichlet inverse of A003958.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 8, 4, 2, 1, 5, 1, 2, 2, 20, 1, 7, 1, 5, 2, 2, 1, 12, 6, 2, 15, 5, 1, 3, 1, 48, 2, 2, 2, 17, 1, 2, 2, 12, 1, 3, 1, 5, 7, 2, 1, 28, 8, 11, 2, 5, 1, 24, 2, 12, 2, 2, 1, 7, 1, 2, 7, 112, 2, 3, 1, 5, 2, 3, 1, 40, 1, 2, 11, 5, 2, 3, 1, 28, 54, 2, 1, 7, 2, 2, 2, 12, 1, 10, 2, 5, 2, 2, 2, 64, 1, 15

Offset: 1

Antti Karttunen, Nov 25 2021

nonn

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

Cf. A003415, A097945.

Cf. also A349133, A349394, A349618, A349619, A349621.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, MoebiusMu[#] * EulerPhi[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A097945(n) = (moebius(n)*eulerphi(n)); \\ Also Dirichlet inverse of A003958.
A349620(n) = sumdiv(n,d,A003415(n/d)*A097945(d));

a(n) = Sum_{d|n} A003415(n/d) * A097945(d).

A351444 a(n) = n - A003958(n) + A003958(sigma(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

1, 3, 2, 9, 3, 6, 2, 15, 17, 10, 3, 16, 7, 10, 9, 45, 5, 38, 5, 28, 10, 16, 3, 30, 39, 26, 23, 28, 9, 26, 2, 55, 15, 26, 13, 104, 19, 28, 21, 52, 13, 32, 11, 46, 53, 28, 3, 76, 49, 94, 23, 76, 9, 54, 19, 58, 25, 46, 9, 64, 31, 34, 51, 189, 29, 50, 17, 76, 27, 50, 5, 164, 37, 74, 73, 82, 19, 66, 5, 136

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

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

Cf. A000203, A003958, A322582, A351442, A351445.

Cf. A351446 (fixed points), A351443 (odd terms there).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A351444(n) = (n - A003958(n) + A003958(sigma(n)));

a(n) = A322582(n) + A351442(n) = n - A003958(n) + A003958(sigma(n)).

a(n) = n + A351445(n).

A351456 a(n) = A003958(sigma(A003961(n))), where A003958 is multiplicative with a(p^e) = (p-1)^e, A003961 multiplicative with a(prime(k)^e) = prime(1+k)^e, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 2, 12, 1, 2, 2, 4, 30, 1, 6, 24, 4, 2, 2, 100, 4, 30, 2, 12, 4, 6, 8, 8, 36, 4, 24, 24, 1, 2, 18, 72, 12, 4, 2, 360, 12, 2, 8, 4, 10, 4, 2, 72, 30, 8, 8, 200, 108, 36, 8, 48, 8, 24, 6, 8, 4, 1, 30, 24, 16, 18, 60, 1092, 4, 12, 4, 48, 16, 2, 36, 120, 4, 12, 72, 24, 12, 8, 12, 100, 700, 10, 16, 48, 4, 2, 2, 24

Offset: 1

Antti Karttunen, Feb 12 2022

Cf. A000203, A003958, A003961, A003973, A151800, A339905, A351442, A351457.

Cf. also A326042.

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); };
A351456(n) = A003958(sigma(A003961(n)));

Multiplicative with a(p^e) = A003958(1 + q + ... + q^e), where q = nextPrime(p) = A151800(p).
a(n) = A003958(A003973(n)) = A351442(A003961(n)) = A003958(A000203(A003961(n))).
a(n) = A351457(n) + A339905(n).

A353792 a(n) = A003958(sigma(n)) * A064989(sigma(n)).

Original entry on oeis.org

1, 4, 1, 30, 4, 4, 1, 48, 132, 16, 4, 30, 30, 4, 4, 870, 16, 528, 12, 120, 1, 16, 4, 48, 870, 120, 12, 30, 48, 16, 1, 480, 4, 64, 4, 3960, 306, 48, 30, 192, 120, 4, 70, 120, 528, 16, 4, 870, 1224, 3480, 16, 900, 64, 48, 16, 48, 12, 192, 48, 120, 870, 4, 132, 14238, 120, 16, 208, 480, 4, 16, 16, 6336, 1116, 1224, 870

Offset: 1

Antti Karttunen, May 11 2022

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. A003958, A064989, A350073, A351442, A353791, A353793, A353794 [= a(A003961(n))].
Cf. A046528 (positions of 1's).
Cf. also A353750.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353792(n) = { my(s=sigma(n)); (A003958(s)*A064989(s)); };

A353792(n) = { my(f=factor(n),s); prod(i=1, #f~, s = sigma(f[i,1]^f[i,2]); A003958(s)*A064989(s)); };

Multiplicative with a(p^e) = A003958(1 + p + ... + p^e) * A064989(1 + p + ... + p^e).
a(n) = A353791(A000203(n)).
a(n) = A351442(n) * A350073(n) = A003958(A000203(n)) * A064989(A000203(n)).

A345047 a(n) = A003958(n) / A345046(n), where A003958(n) is multiplicative with a(p^e) = (p-1)^e, and A345046(n) gives the least common multiple of the same factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 4, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 4, 1, 2, 1, 1, 4, 6, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

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

Cf. A003958, A345046.

Cf. also A345045, A353784.

A345047(n) = { my(f=factor(n)~, g=vector(#f, i, (f[1, i]-1)^f[2, i])); factorback(g)/lcm(g); };

a(n) = A003958(n) / A345046(n).

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)).

A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 8, 1, 2, 0, 18, 0, 2, 2, 41, 0, 22, 0, 22, 2, 2, 0, 98, 1, 2, 12, 26, 0, 40, 0, 172, 2, 2, 2, 148, 0, 2, 2, 130, 0, 48, 0, 34, 28, 2, 0, 426, 1, 34, 2, 38, 0, 158, 2, 162, 2, 2, 0, 278, 0, 2, 32, 645, 2, 64, 0, 46, 2, 56, 0, 706, 0, 2, 36, 50, 2, 72, 0, 590, 91, 2, 0, 350, 2, 2, 2, 226, 0, 348

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A322582 with A348507.

Question: Is a(n) >= A305809(n) for all n?

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003958, A003959, A305809, A322582, A348507, A349129.

f1[p_, e_] := (p - 1)^e; s1[1] = 0; s1[n_] := n - Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s2[1] = 0; s2[n_] := Times @@ f2 @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, s1[#]*s2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 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); };
A322582(n) = (n-A003958(n));
A348507(n) = (A003959(n)-n);
A349139(n) = sumdiv(n,d,A322582(d)*A348507(n/d));

a(n) = Sum_{d|n} A322582(d) * A348507(n/d).

A351440 Numbers k for which A003958(sigma(k)) + A064989(sigma(k)) is equal to A003958(k) + A064989(k).

Original entry on oeis.org

1, 6, 28, 496, 8128, 30240, 32760, 240408, 2178540, 6828720, 13042080, 23569920, 33550336, 42402048, 45532800, 142990848, 1379454720

Offset: 1

Antti Karttunen, Feb 12 2022

Cf. A000203, A003958, A064989, A351442.
Subsequence of A351446.
Subsequences: A000396, A336702.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
isA351440(n) = { my(s=sigma(n)); ((A003958(s)+A064989(s)) == (A003958(n)+A064989(n))); };

A351447 Numbers k for which A003958(sigma(k)) = 2*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

2, 98, 120, 136, 312, 520, 672, 888, 1080, 1120, 1464, 1480, 1752, 2440, 2520, 2808, 2912, 2920, 3420, 3768, 3848, 4632, 5880, 6048, 6280, 6344, 6552, 6648, 6664, 7512, 7592, 7720, 7992, 8181, 8288, 8892, 9528, 10104, 10968, 11080, 12464, 12520, 12984, 13176, 13664, 14712, 15288

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Numbers k such that A351442(k) = 2*A003958(k).

In contrast, numbers x for which A064989(sigma(x)) = 2*A064989(x) seem to consist just of {2} followed by A005820: 2, 120, 672, 523776, ..., etc, which (also) contains as its subsequence all the odd terms of A336702 multiplied by 2.

Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A064989, A351442.

Subsequences: A005820 (3-perfect numbers), odd terms of A336702 doubled, the terms of A351443 doubled (2, 98, 81810, ...), A351448 (odd terms in this sequence).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351447(n) = (A003958(sigma(n))==2*A003958(n));

cp's OEIS Frontend

A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

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A349620 Dirichlet convolution of A003415 with the Dirichlet inverse of A003958.

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A351444 a(n) = n - A003958(n) + A003958(sigma(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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A351456 a(n) = A003958(sigma(A003961(n))), where A003958 is multiplicative with a(p^e) = (p-1)^e, A003961 multiplicative with a(prime(k)^e) = prime(1+k)^e, and sigma is the sum of divisors function.

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A353792 a(n) = A003958(sigma(n)) * A064989(sigma(n)).

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A345047 a(n) = A003958(n) / A345046(n), where A003958(n) is multiplicative with a(p^e) = (p-1)^e, and A345046(n) gives the least common multiple of the same factors.

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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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A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

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