A003959 -id:A003959 - cp's OEIS frontend

A348945 a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 0, 0, 75, 0, 0, 0, 6, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 18, 0, 24, 0, 0, 0, 0, 0, 0, 0, 270, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 108, 48, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 300, 0, 0, 0, 10

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

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

Cf. A000203, A003959, A034448, A048146, A348029, A348944, A348946.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 0; a[n_] := (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2 - Times @@ f1 @@@ f; 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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348945(n) = (A348944(n)-sigma(n));

a(n) = A348944(n) - A000203(n) = ((1/2) * (A003959(n)+A034448(n))) - A000203(n).

a(n) = (1/2) * (A348029(n)-A048146(n)).

A348949 a(n) = A003959(A276086(n)), where 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, 3, 4, 12, 16, 48, 6, 18, 24, 72, 96, 288, 36, 108, 144, 432, 576, 1728, 216, 648, 864, 2592, 3456, 10368, 1296, 3888, 5184, 15552, 20736, 62208, 8, 24, 32, 96, 128, 384, 48, 144, 192, 576, 768, 2304, 288, 864, 1152, 3456, 4608, 13824, 1728, 5184, 6912, 20736, 27648, 82944, 10368, 31104, 41472, 124416, 165888, 497664, 64

Offset: 0

Antti Karttunen, Nov 07 2021

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

Cf. A003959, A276086, A348950, A348997, A348999.

Cf. also A324653, A346470, A348996.

A348949(n) = { my(m=1, p=2); while(n, m *= ((1+p)^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A003959(A276086(n)).

a(n) = A276086(n) + A348950(n).

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.

Original entry on oeis.org

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

A349621 Dirichlet convolution of A003415 with the Dirichlet inverse of A003959.

Original entry on oeis.org

0, 1, 1, 1, 1, -2, 1, 0, 2, -2, 1, -3, 1, -2, -2, -4, 1, -5, 1, -3, -2, -2, 1, -4, 4, -2, 3, -3, 1, 3, 1, -16, -2, -2, -2, -7, 1, -2, -2, -4, 1, 3, 1, -3, -5, -2, 1, -4, 6, -9, -2, -3, 1, -12, -2, -4, -2, -2, 1, 5, 1, -2, -5, -48, -2, 3, 1, -3, -2, 3, 1, -8, 1, -2, -9, -3, -2, 3, 1, -4, 0, -2, 1, 5, -2, -2, -2, -4

Offset: 1

Antti Karttunen, Nov 25 2021

sign

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

Cf. A003415, A003959, A063441.

Cf. also A349173, A349394, A349618, A349619, A349620.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, MoebiusMu[#] * DivisorSigma[1, #] * 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]));
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349621(n) = sumdiv(n,d,A003415(n/d)*A063441(d));

a(n) = Sum_{d|n} A003415(n/d) * A063441(d).

A348049 a(n) = A003959(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, 9, 1, 1, 1, 9, 16, 1, 1, 9, 1, 1, 1, 81, 1, 16, 1, 9, 1, 1, 1, 9, 36, 1, 8, 9, 1, 1, 1, 27, 1, 1, 1, 144, 1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 81, 64, 36, 1, 9, 1, 8, 1, 9, 1, 1, 1, 9, 1, 1, 16, 729, 1, 1, 1, 9, 1, 1, 1, 144, 1, 1, 36, 9, 1, 1, 1, 81, 256, 1, 1, 9, 1, 1, 1, 9, 1, 16, 1, 9, 1, 1, 1, 27, 1, 64

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Not multiplicative. For example, a(196) = 192 != 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, A005117 (positions of 1's), A348029, A348047, A348048.

Cf. also A344697.

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

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348049(n) = { my(u=A003959(n)); (u/gcd(u, sigma(n))); };

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

A348929 a(n) = gcd(n, A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 36, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 2, 3, 4, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Nov 07 2021

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

Cf. A003959, A085731, A126865, A323166, A348507, A348928, A348999 [= a(A276086(n))].

Differs from similar A126795 for the first time at n=36, where a(36) = 36, while A126795(36) = 12.

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

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348929(n) = gcd(n, A003959(n));

a(n) = gcd(n, A003959(n)) = gcd(n, A348507(n)) = gcd(A003959(n), A348507(n)).

A348947 a(n) = A348944(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, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 23, 1, 1, 1, 1, 46, 1, 1, 1, 97, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 23, 1, 6, 1, 1, 1, 1, 1, 1, 1, 397, 1, 1, 1, 1, 1, 1, 1, 87, 1, 1, 1, 1, 1, 1, 1, 49, 169, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 46, 1, 1, 1, 227

Offset: 1

Antti Karttunen, Nov 05 2021

Numerator of ratio A348944(n) / A000203(n).

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 <> 1 = a(4)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348948 (denominators).

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_] := (s = (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2) / GCD[Times @@ f1 @@@ f, s]; 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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348947(n) = { my(u=A348944(n)); (u/gcd(sigma(n),u)); };

a(n) = A348944(n) / A348946(n) = A348944(n) / gcd(A000203(n), A348944(n)).

A348948 a(n) = sigma(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, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 20, 1, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 20, 1, 5, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 31, 121, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 217

Offset: 1

Antti Karttunen, Nov 05 2021

Denominator of ratio A348944(n) / A000203(n).

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) = 91 <> 1 = a(4)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348947 (numerators).

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_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, (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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348948(n) = { my(s=sigma(n)); (s/gcd(s,A348944(n))); };

a(n) = A000203(n) / A348946(n) = A000203(n) / gcd(A000203(n), A348944(n)).

A348950 a(n) = A348507(A276086(n)), where A348507(n) = A003959(n) - 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

0, 1, 1, 6, 7, 30, 1, 8, 9, 42, 51, 198, 11, 58, 69, 282, 351, 1278, 91, 398, 489, 1842, 2331, 8118, 671, 2638, 3309, 11802, 15111, 50958, 1, 10, 11, 54, 65, 258, 13, 74, 87, 366, 453, 1674, 113, 514, 627, 2406, 3033, 10674, 853, 3434, 4287, 15486, 19773, 67194, 5993, 22354, 28347, 98166, 126513, 418914, 15, 94, 109

Offset: 0

Antti Karttunen, Nov 06 2021

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

Cf. A003959, A276086, A348507, A348949, A348999.

Cf. also A327860.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A348507(n) = (A003959(n) - n);
A348950(n) = A348507(A276086(n));

A348950(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)); (m2-m1); };

a(n) = A348949(n) - A276086(n) = A348507(A276086(n)).

A348972 a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 3, 4, 1, 6, 1, 8, 1, 1, 1, 12, 4, 14, 1, 1, 3, 18, 3, 20, 2, 1, 1, 24, 4, 1, 1, 2, 12, 30, 1, 32, 1, 1, 1, 1, 48, 38, 1, 1, 54, 42, 1, 44, 4, 12, 1, 48, 4, 1, 1, 1, 18, 54, 3, 1, 4, 1, 1, 60, 8, 62, 1, 2, 1, 1, 1, 68, 2, 1, 3, 72, 12, 74, 1, 2, 12, 1, 1, 80, 2, 1, 1, 84, 16, 1, 1, 1, 12, 90, 3, 1, 4, 1, 1, 1, 4, 98

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Cf. A003415, A003959, A129283, A348970, A348973, A348974.

Cf. also A343226.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n])), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348972(n) = gcd(A003959(n),(n+A003415(n)));

a(n) = gcd(A003959(n), A129283(n)) = gcd(A003959(n), n+A003415(n)).
a(n) = gcd(A003959(n), A348970(n)) = gcd(A129283(n), A348970(n)).
a(n) = A129283(n) / A348973(n) = A003959(n) / A348974(n).

cp's OEIS Frontend

A348945 a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

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A348949 a(n) = A003959(A276086(n)), where 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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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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A349621 Dirichlet convolution of A003415 with the Dirichlet inverse of A003959.

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A348049 a(n) = A003959(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.

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A348929 a(n) = gcd(n, A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A348947 a(n) = A348944(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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A348948 a(n) = sigma(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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