A003959 -id:A003959 - cp's OEIS frontend

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A349140.
Dirichlet convolution of sigma (A000203) with A349141.
Dirichlet convolution of A060640 with A348971.

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

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.

Cf. also A349123, A348983.

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
a(n) = Sum_{d|n} d * A349140(n/d).
a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
For all n >= 1, a(n) >= A349123(n) >= A348983(n).

A349170 a(n) = Sum_{d|n} d * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

1, 5, 7, 19, 11, 35, 15, 65, 37, 55, 23, 133, 27, 75, 77, 211, 35, 185, 39, 209, 105, 115, 47, 455, 91, 135, 175, 285, 59, 385, 63, 665, 161, 175, 165, 703, 75, 195, 189, 715, 83, 525, 87, 437, 407, 235, 95, 1477, 169, 455, 245, 513, 107, 875, 253, 975, 273, 295, 119, 1463, 123, 315, 555, 2059, 297, 805, 135, 665

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003959 with the identity function, A000027.

Dirichlet convolution of sigma (A000203) with A003968.

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

Cf. A000027, A000203, A003959, A003968, A038040, A349129, A349130, A349140, A349171 (Möbius transform), A349172, A349173.

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

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349170(n) = sumdiv(n,d,d*A003959(n/d));

a(n) = Sum_{d|n} d * A003959(n/d).
a(n) = Sum_{d|n} A349171(d).
a(n) = Sum_{d|n} A000203(d) * A003968(n/d).
a(n) = A038040(n) + A349140(n).
For all n >= 1, a(n) >= A349129(n) >= A349130(n).
Multiplicative with a(p^e) = (p+1)^(e+1) - p^(e+1). - Amiram Eldar, Nov 09 2021

A349171 a(n) = Sum_{d|n} phi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and phi is Euler totient function.

Original entry on oeis.org

1, 4, 6, 14, 10, 24, 14, 46, 30, 40, 22, 84, 26, 56, 60, 146, 34, 120, 38, 140, 84, 88, 46, 276, 80, 104, 138, 196, 58, 240, 62, 454, 132, 136, 140, 420, 74, 152, 156, 460, 82, 336, 86, 308, 300, 184, 94, 876, 154, 320, 204, 364, 106, 552, 220, 644, 228, 232, 118, 840, 122, 248, 420, 1394, 260, 528, 134, 476, 276

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003959 with Euler totient function phi, A000010.

Möbius transform of A349170.

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

Cf. A000010, A003959, A018804, A349141, A349170 (inverse Möbius transform), A349172, A349131.

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

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349171(n) = sumdiv(n,d,eulerphi(d)*A003959(n/d));

a(n) = Sum_{d|n} A000010(d) * A003959(n/d).
a(n) = Sum_{d|n} A008683(d) * A349170(n/d).
a(n) = Sum_{k=1..n} A003959(gcd(n, k)).
a(n) = A018804(n) + A349141(n).
For all n >= 1, a(n) >= A349131(n).
Multiplicative with a(p^e) = p*(p+1)^e - (p-1)*p^e. - Amiram Eldar, Nov 09 2021

A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.

Original entry on oeis.org

1, 6, 8, 24, 12, 48, 16, 84, 44, 72, 24, 192, 28, 96, 96, 276, 36, 264, 40, 288, 128, 144, 48, 672, 102, 168, 212, 384, 60, 576, 64, 876, 192, 216, 192, 1056, 76, 240, 224, 1008, 84, 768, 88, 576, 528, 288, 96, 2208, 184, 612, 288, 672, 108, 1272, 288, 1344, 320, 360, 120, 2304, 124, 384, 704, 2724, 336, 1152, 136

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A001615 with A003959.

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

Cf. A001615, A003959, A327251, A349142, A349170, A349171, A349172, A349173.

f[p_, e_] := (p + 2)*(p + 1)^e - (p + 1)*p^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)));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349172(n) = sumdiv(n,d,A001615(d)*A003959(n/d));

a(n) = Sum_{d|n} A001615(d) * A003959(n/d).
a(n) = A327251(n) + A349142(n).
For all n >= 1, a(n) >= A349132(n).
Multiplicative with a(p^e) = (p+2)*(p+1)^e - (p+1)*p^e. - Amiram Eldar, Nov 09 2021

A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 2, 56, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 6, 72, 24, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96, 144, 72, 3, 74, 114, 124, 140, 96, 168, 80, 6, 1, 126

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 1 <> 91 = 7*13 = a(4)*a(9).

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

Cf. A000203, A003959, A034448, A348944, A348945, A348947, A348948.

Cf. also A344695, A348047, A348503, A348733.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), (Times @@ f2 @@@ f + Times @@ f3 @@@ f)/2]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348946(n) = gcd(sigma(n), ((1/2)*(A003959(n)+A034448(n))));

a(n) = gcd(A000203(n), A348944(n)).
a(n) = gcd(A000203(n), A348945(n)) = gcd(A348944(n), A348945(n));
a(n) = A348944(n) / A348947(n) = A000203(n) / A348948(n).

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.

Original entry on oeis.org

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)).

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

A348048 a(n) = sigma(n) / gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 5, 7, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 7, 1, 57, 13

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Not multiplicative. For example, a(196) = 133 != a(4) * a(49).

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

Cf. A000203, A003959, A348029, A348047, A348049.

Cf. also A344696.

f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := (s = DivisorSigma[1, n]) / GCD[s, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348048(n) = { my(u=sigma(n)); (u/gcd(u, A003959(n))); };

a(n) = A000203(n) / A348047(n) = A000203(n) / gcd(A000203(n), A003959(n)).

A348944 a(n) = (1/2) * (A003959(n)+A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 18, 13, 18, 12, 28, 14, 24, 24, 49, 18, 39, 20, 42, 32, 36, 24, 72, 31, 42, 46, 56, 30, 72, 32, 138, 48, 54, 48, 97, 38, 60, 56, 108, 42, 96, 44, 84, 78, 72, 48, 196, 57, 93, 72, 98, 54, 138, 72, 144, 80, 90, 60, 168, 62, 96, 104, 397, 84, 144, 68, 126, 96, 144, 72, 261, 74, 114, 124, 140, 96

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 97 != 91 = 7*13 = a(4)*a(9).

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

Arithmetic mean of A003959 and A034448.
Cf. A348732, A348733, A348945.
Cf. also A325973.

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f) / 2; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

cp's OEIS Frontend

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A349170 a(n) = Sum_{d|n} d * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1).

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A349171 a(n) = Sum_{d|n} phi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and phi is Euler totient function.

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A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.

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A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

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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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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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