A342662 -id:A342662 - cp's OEIS frontend

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

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)).

A342663 Numerator of ratio A342661(n)/A342662(n): a(n) = A342661(n) / gcd(A342661(n), A342662(n)).

Original entry on oeis.org

1, 2, 9, 4, 10, 3, 21, 8, 63, 20, 22, 9, 78, 7, 5, 16, 119, 21, 171, 40, 189, 44, 115, 3, 325, 52, 81, 3, 116, 5, 465, 32, 33, 238, 7, 9, 592, 57, 351, 16, 779, 63, 903, 88, 35, 115, 517, 18, 1519, 650, 119, 312, 424, 27, 220, 14, 1539, 232, 531, 5, 1830, 155, 1323, 64, 260, 11, 2077, 68, 345, 7, 1207, 42, 2628, 1184

Offset: 1

Antti Karttunen, Mar 23 2021

Let r(row,col) = A341605(row,col)/A341606(row,col) and d(n) = A342661(n)/A342661(n) = A342663(n)/A342664(n). Then for row > 1, r(row-1,col) = d(A246278(row,col)) * r(row,col).

Cf. A000203, A064989, A246278, A326041, A341526, A341527 [= a(A003961(n))], A341605, A341606, A342661, A342662, A342664 (denominators), 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));
A342663(n) = { my(u=A342661(n)); (u/gcd(u, A342662(n))); };
\\ Alternatively as:
A342663(n) = numerator(A342661(n)/A342662(n));

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

A341528 a(n) = n * sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 8, 18, 52, 40, 144, 84, 320, 279, 320, 154, 936, 234, 672, 720, 1936, 340, 2232, 456, 2080, 1512, 1232, 690, 5760, 1425, 1872, 4212, 4368, 928, 5760, 1178, 11648, 2772, 2720, 3360, 14508, 1554, 3648, 4212, 12800, 1804, 12096, 2064, 8008, 11160, 5520, 2538, 34848, 6517, 11400, 6120, 12168, 3180, 33696, 6160, 26880

Offset: 1

Antti Karttunen, Feb 16 2021

Cf. A000203, A003961, A003973, A016754 (positions of the odd terms), A151800, A341512, A341526, A341527, A341529, A341530, A342661, A342662, A342673.

Array[#1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 56] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A003973(n) = sigma(A003961(n));
A341528(n) = (n*A003973(n));

Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1) where q = nextPrime(p).
a(n) = n * A003973(n) = n * A000203(A003961(n)).
From Antti Karttunen, Mar 29 2021: (Start)
a(n) <= A341529(n).
a(n) = A341529(n) - A341512(n).
a(n) = A342662(A003961(n)).
(End)
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 2.26342530..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022

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.

Original entry on oeis.org

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

A342667 Largest prime in the numerator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

Original entry on oeis.org

1, 2, 3, 13, 5, 2, 7, 2, 31, 5, 11, 13, 13, 7, 3, 11, 17, 31, 19, 13, 7, 11, 23, 2, 19, 13, 13, 13, 29, 2, 31, 13, 11, 17, 5, 31, 37, 19, 13, 5, 41, 7, 43, 11, 31, 23, 47, 11, 7, 19, 17, 13, 53, 13, 11, 7, 19, 29, 59, 13, 61, 31, 31, 1093, 13, 11, 67, 17, 23, 5, 71, 31, 73, 37, 19, 19, 7, 13, 79, 11, 71, 41, 83, 13

Offset: 1

Antti Karttunen, Mar 23 2021

nonn

Equally, largest prime in the denominator of ratio A342661(A003961(n)) / A342662(A003961(n)).

Cf. A000203, A003961, A006530, A064989, A341526, A341528, A341529, A326041, A342661, A342662, A342664, A342668.

Cf. also A341607.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341526(n) = { my(s=A003961(n)); numerator((sigma(s)*n)/(sigma(n)*s)); };
A342667(n) = A006530(A341526(n));

a(n) = A006530(A341526(n)).

a(n) = A006530(A342664(A003961(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

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A342663 Numerator of ratio A342661(n)/A342662(n): a(n) = A342661(n) / gcd(A342661(n), A342662(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A341528 a(n) = n * sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

A342667 Largest prime in the numerator of ratio A341528(n)/A341529(n) (when presented in its lowest terms).

Views

Author

Keywords

Comments

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.