A342663 -id:A342663 - cp's OEIS frontend

A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

1, 2, 9, 4, 20, 18, 42, 8, 63, 40, 88, 36, 156, 84, 180, 16, 238, 126, 342, 80, 378, 176, 460, 72, 325, 312, 405, 168, 696, 360, 930, 32, 792, 476, 840, 252, 1184, 684, 1404, 160, 1558, 756, 1806, 352, 1260, 920, 2068, 144, 1519, 650, 2142, 624, 2544, 810, 1760, 336, 3078, 1392, 3186, 720, 3660, 1860, 2646, 64, 3120

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A151799, A326041, A341528, A341529 [= a(A003961(n))], A342662, A342663, A342664.

f[p_, e_] := If[p == 2, 2^e, Module[{q = NextPrime[p, -1]}, p^e*(q^(e + 1) - 1)/(q - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));

Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
a(n) = n * A326041(n) = n * A000203(A064989(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/9) * Product_{p prime > 2} (p^3/((p+1)*(p^2-prevprime(p)))) = 0.1815217..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

A342662 a(n) = sigma(n) * A064989(n), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 3, 8, 7, 18, 24, 40, 15, 52, 54, 84, 56, 154, 120, 144, 31, 234, 156, 340, 126, 320, 252, 456, 120, 279, 462, 320, 280, 690, 432, 928, 63, 672, 702, 720, 364, 1178, 1020, 1232, 270, 1554, 960, 1804, 588, 936, 1368, 2064, 248, 1425, 837, 1872, 1078, 2538, 960, 1512, 600, 2720, 2070, 3180, 1008, 3658, 2784, 2080, 127

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A151799, A341528 [= a(A003961(n))], A341529, A342661, A342663, A342664.

f[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]*(p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A342662(n) = (sigma(n)*A064989(n));

Multiplicative with a(p^e) = q^e * (p^(e+1)-1)/(p-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
a(n) = A000203(n) * A064989(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (16/63) * Product_{p prime > 2} p^4*(p-1)/((p^3-prevprime(p))*(p^2-prevprime(p))) = 0.1935405..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

A342664 Denominator of ratio A342661(n)/A342662(n).

Original entry on oeis.org

1, 3, 8, 7, 9, 4, 20, 15, 52, 27, 21, 14, 77, 10, 4, 31, 117, 26, 170, 63, 160, 63, 114, 5, 279, 77, 64, 5, 115, 6, 464, 63, 28, 351, 6, 13, 589, 85, 308, 27, 777, 80, 902, 147, 26, 171, 516, 31, 1425, 837, 104, 539, 423, 32, 189, 25, 1360, 345, 530, 7, 1829, 232, 1040, 127, 231, 14, 2074, 117, 304, 9, 1206, 65, 2627

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A326041, A341526 [= a(A003961(n))], A341527, A342661, A342662, A342663 (numerators), A342667 [largest prime factor of a(A003961(n))], A342670.

A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));
A342662(n) = (sigma(n)*A064989(n));
A342664(n) = { my(v=A342662(n)); (v/gcd(A342661(n),v)); };
\\ Alternatively as:
A342664(n) = denominator(A342661(n)/A342662(n));

a(n) = A342662(n) / A342670(n) = A342662(n) / gcd(A342661(n), A342662(n)).

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 2, 1, 1, 2, 4, 4, 2, 12, 36, 1, 2, 6, 2, 2, 2, 4, 4, 24, 1, 6, 5, 56, 6, 72, 2, 1, 24, 2, 120, 28, 2, 12, 4, 10, 2, 12, 2, 4, 36, 8, 4, 8, 1, 1, 18, 2, 6, 30, 8, 24, 2, 6, 6, 144, 2, 12, 2, 1, 12, 144, 2, 14, 12, 240, 4, 12, 2, 2, 9, 4, 336, 24, 2, 2, 1, 2, 4, 56, 4, 12, 24, 4, 6, 72, 56, 8, 2, 8, 360

Offset: 1

Antti Karttunen, Mar 24 2021

nonn

Cf. A000203, A064989, A326041, A341530 [ = a(A003961(n))], A342661, A342662, A342663, A342664.

A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));
A342662(n) = (sigma(n)*A064989(n));
A342670(n) = gcd(A342661(n), A342662(n));

a(n) = gcd(A342661(n), A342662(n)).

a(n) = gcd(n*A000203(A064989(n)), A000203(n)*A064989(n)).

cp's OEIS Frontend

A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

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A342662 a(n) = sigma(n) * A064989(n), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma is the sum of the divisors of n.

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PARI

Formula

A342664 Denominator of ratio A342661(n)/A342662(n).

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PARI

Formula

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342662 a(n) = sigma(n) * A064989(n), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma is the sum of the divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342664 Denominator of ratio A342661(n)/A342662(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A342670 a(n) = gcd(n*sigma(A064989(n)), sigma(n)*A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A342670 a(n) = gcd(nsigma(A064989(n)), sigma(n)A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.