A324121 -id:A324121 - cp's OEIS frontend

A324058 a(n) = A324121(A005940(1+n)) = gcd(A324054(n), A005940(1+n)*A106737(n)).

1, 1, 2, 1, 2, 12, 1, 1, 2, 2, 12, 4, 1, 3, 4, 1, 2, 8, 4, 6, 4, 24, 6, 12, 3, 3, 2, 1, 4, 24, 1, 3, 2, 4, 12, 56, 4, 48, 2, 10, 4, 16, 24, 24, 2, 18, 120, 4, 1, 3, 6, 1, 6, 12, 1, 3, 4, 4, 24, 8, 1, 3, 2, 1, 2, 2, 4, 12, 4, 48, 6, 8, 28, 8, 24, 112, 6, 24, 8, 2, 4, 16, 24, 336, 8, 96, 12, 120, 6, 24, 4, 6, 8, 720, 6, 36, 3, 3, 2, 21, 6, 36, 3, 15, 14, 6

Offset: 0

Antti Karttunen, Feb 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000005, A000203, A005940, A038040, A106737, A324054, A324057, A324121.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A106737(n) = sum(k=0, n, (binomial(n+k, n-k)*binomial(n, k)) % 2);
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324058(n) = gcd(A324054(n), A005940(1+n)*A106737(n));
\\ Alternatively as:
A324121(n) = gcd(sigma(n),n*numdiv(n));
A324058(n) = A324121(A005940(1+n));

a(n) = A324121(A005940(1+n)) = gcd(A324054(n), A005940(1+n)*A106737(n)).

A324188 a(n) = A324121(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 12, 2, 1, 4, 3, 1, 4, 12, 2, 2, 3, 1, 24, 4, 1, 2, 3, 3, 12, 6, 24, 4, 6, 4, 8, 2, 1, 2, 3, 1, 8, 24, 4, 4, 3, 1, 12, 6, 1, 6, 3, 1, 4, 120, 18, 2, 24, 24, 16, 4, 10, 2, 48, 4, 56, 12, 4, 2, 1, 1, 12, 6, 1, 2, 1, 1, 24, 12, 48, 40, 12, 8, 16, 4, 1, 20, 3, 3, 4, 36, 6, 14, 15, 3, 36, 6, 21, 2, 3, 3, 36, 6, 720, 8, 6, 4, 24, 6, 120, 12

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A054429, A324058, A163511, A324121, A324183, A324184, A324187, A324189.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A324121(n) = gcd(sigma(n),n*numdiv(n));
A324188(n) = A324121(A163511(n));

A324183(n) = if(!n,1,n = ((3<<#binary(n\2))-n-1); my(e=0,m=1); while(n>0, if(!(n%2), m *= (1+e); e=0, e++); n >>= 1); (m*(1+e)));
A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));
A324188(n) = gcd(A324184(n), A163511(n)*A324183(n));

a(n) = A324121(A163511(n)) = gcd(A324184(n), A163511(n)*A324183(n)).

For n > 0, a(n) = A324058(A054429(n)).

A336314 a(n) = A324121(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 2, 1, 4, 3, 2, 1, 12, 3, 2, 1, 12, 1, 6, 1, 4, 1, 8, 4, 36, 1, 10, 1, 24, 3, 2, 3, 4, 24, 4, 1, 12, 1, 56, 1, 24, 1, 2, 3, 4, 1, 4, 1, 6, 9, 6, 1, 4, 8, 8, 1, 12, 9, 48, 1, 4, 1, 2, 24, 120, 5, 2, 3, 18, 7, 12, 1, 36, 2, 10, 3, 24, 1, 12, 3, 4, 3, 112

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A324121, A336315.
Cf. A336317 (gives the positions where this coincides with A323173).
Cf. also A335914.

A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A324121(n) = gcd(sigma(n),n*numdiv(n));
A336314(n) = A324121(A122111(n));

\\ Or as a standalone program:
A336314(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); gcd(s,x*d));

a(n) = A324121(A122111(n)) = gcd(A323173(n), A122111(n)*A336315(n)).

A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 6, 17, 14, 22, 10, 44, 12, 32, 36, 49, 16, 69, 18, 78, 52, 52, 22, 132, 44, 62, 68, 112, 28, 168, 30, 129, 84, 82, 92, 233, 36, 92, 100, 230, 40, 240, 42, 180, 192, 112, 46, 356, 90, 207, 132, 214, 52, 312, 148, 328, 148, 142, 58, 552, 60

Offset: 1

Labos Elemer, May 28 2004

Not all values arise and some arise more than once.
Row sums of triangle A134866. - Gary W. Adamson, Nov 14 2007
Sum of the largest parts of the partitions of n into two parts such that the smaller part divides the larger. - Wesley Ivan Hurt, Dec 21 2017
a(n) is also the sum of all parts minus the total number of parts of all partitions of n into equal parts (an interpretation of the Torlach Rush's formula). - Omar E. Pol, Nov 30 2019
If and only if sigma(n) divides a(n), then n is one of Ore's Harmonic numbers, A001599. - Antti Karttunen, Jul 18 2020

			q^2 + 2*q^3 + 5*q^4 + 4*q^5 + 12*q^6 + 6*q^7 + 17*q^8 + 14*q^9 + ...
For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 - 7 = 5. - _Omar E. Pol_, Nov 30 2019

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 30.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A001599, A038040, A134866, A152211, A244051, A324121 (= gcd(a(n), sigma(n))).

Cf. A088827 (positions of odd terms).

using AbstractAlgebra
function A094471(n)
    sum(k for k in 0:n if is_divisible_by(n, n - k))
end
[A094471(n) for n in 1:61] |> println  # Peter Luschny, Nov 14 2023

with(numtheory); A094471:=n->n*tau(n)-sigma(n); seq(A094471(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013
divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
a := n -> local k; add(`if`(divides(n - k, n), k, 0), k = 0..n):
seq(a(n), n = 1..61);  # Peter Luschny, Nov 14 2023

Table[n*DivisorSigma[0, n] - DivisorSigma[1, n], {n, 1, 100}]

{a(n) = n*numdiv(n) - sigma(n)} /* Michael Somos, Jan 25 2008 */

from math import prod
from sympy import factorint
def A094471(n):
    f = factorint(n).items()
    return n*prod(e+1 for p,e in f)-prod((p**(e+1)-1)//(p-1) for p,e in f)
# Chai Wah Wu, Nov 14 2023

def A094471(n): return sum(k for k in (0..n) if (n-k).divides(n))
print([A094471(n) for n in range(1, 62)])  # Peter Luschny, Nov 14 2023

a(n) = n*tau(n) - sigma(n) = n*A000005(n) - A000203(n). [Previous name.]
If p is prime, then a(p) = p*tau(p)-sigma(p) = 2p-(p+1) = p-1 = phi(p).
If n>1, then a(n)>0.
a(n) = Sum_{d|n} (n-d). - Amarnath Murthy, Jul 31 2005
G.f.: Sum_{k>=1} k*x^(2*k)/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 24 2018
a(n) = A038040(n) - A000203(n). - Torlach Rush, Feb 02 2019

Simpler name by Peter Luschny, Nov 14 2023

A336846 a(n) = gcd(sigma(A003961(n)), A000005(n)*A003961(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 12, 2, 4, 1, 4, 2, 6, 2, 12, 4, 1, 2, 2, 2, 2, 4, 4, 2, 120, 3, 12, 4, 6, 2, 24, 2, 2, 4, 4, 4, 1, 2, 12, 4, 8, 2, 24, 2, 26, 2, 12, 2, 6, 1, 6, 20, 18, 2, 24, 28, 24, 4, 4, 2, 12, 2, 4, 6, 1, 4, 24, 2, 2, 20, 24, 2, 20, 2, 12, 6, 6, 4, 24, 2, 2, 1, 4, 2, 36, 4, 12, 4, 8, 2, 4, 4, 6, 4, 12, 4, 12, 2, 2, 2, 3

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
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. A000005, A000203, A000290 (positions of odd terms), A003961, A003973, A324121, A336476, A336841, A336845, A336847, A336848.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336846(n) = { my(u=A003961(n),s=sigma(u)); gcd(s, numdiv(n)*u); };

a(n) = gcd(A003973(n), A336845(n)) = gcd(A003973(n), A336841(n)).
a(n) = gcd(A000203(A003961(n)), A000005(n)*A003961(n)).
a(n) = A324121(A003961(n)).

A324122 a(n) = sigma(n) - gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

0, 2, 2, 6, 4, 0, 6, 14, 12, 16, 10, 24, 12, 16, 12, 30, 16, 36, 18, 36, 28, 32, 22, 48, 30, 40, 36, 0, 28, 48, 30, 60, 36, 52, 44, 90, 36, 56, 52, 80, 40, 48, 42, 72, 72, 64, 46, 120, 54, 90, 60, 96, 52, 96, 68, 112, 76, 88, 58, 144, 60, 88, 102, 126, 80, 96, 66, 120, 84, 128, 70, 192, 72, 112, 122, 136, 92, 144, 78, 184, 120, 124, 82

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..117800

Cf. A000005, A000203, A038040, A106315, A106316, A324045, A324046, A324047, A324121.

Cf. A001599 (positions of zeros).

A324122(n) = (sigma(n) - gcd(sigma(n),n*numdiv(n)));

a(n) = A000203(n) - A324121(n) = A000203(n) - gcd(A000203(n), A038040(n)).

A336476 a(n) = gcd(A000593(n), A336475(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 12, 1, 2, 1, 2, 2, 4, 2, 2, 2, 1, 2, 4, 2, 2, 12, 2, 1, 12, 2, 4, 1, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 1, 12, 2, 2, 4, 4, 2, 4, 2, 2, 12, 2, 2, 2, 1, 4, 12, 2, 2, 12, 4, 2, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 12, 2, 2, 6, 28, 2, 4, 2, 20, 2, 2, 3, 6, 1, 2, 12, 2, 2, 24

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

All odd terms k in A001599 (Ore's Harmonic numbers) satisfy a(k) = A336475(k).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A000593, A001227, A001599, A324121, A336475.

Cf. also A324058, A336320.

A000593(n) = sigma(n>>valuation(n, 2));
A336475(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,(1+f[i,2]) * (f[i,1]^f[i,2]))); };
A336476(n) = gcd(A000593(n), A336475(n));

a(n) = gcd(A000593(n), A336475(n)).

a(n) = A324121(A000265(n)).

A336320 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324058(i) = A324058(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 3, 4, 1, 5, 4, 1, 2, 6, 4, 7, 4, 8, 7, 3, 5, 5, 2, 1, 4, 8, 1, 5, 2, 4, 3, 9, 4, 10, 2, 11, 4, 12, 8, 8, 2, 13, 14, 4, 1, 5, 7, 1, 7, 3, 1, 5, 4, 4, 8, 6, 1, 5, 2, 1, 2, 2, 4, 3, 4, 10, 7, 6, 15, 6, 8, 16, 7, 8, 6, 2, 4, 12, 8, 17, 6, 18, 3, 14

Offset: 0

Antti Karttunen, Jul 19 2020

nonn

Restricted growth sequence transform of A324058.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A005940, A324121, A324058.

Cf. also A286622.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324121(n) = gcd(sigma(n),n*numdiv(n));
A324058(n) = A324121(A005940(1+n));
v336320 = rgs_transform(vector(1+up_to,n,A324058(n-1)));
A336320(n) = v336320[1+n];

A330606 Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 36, 64, 81, 100, 121, 128, 144, 225, 256, 289, 324, 400, 484, 512, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1250, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761

Offset: 1

Amiram Eldar, Dec 20 2019

nonn

If p is prime and p == 2 (mod 3) then p^2 is in the sequence.
Let E(x) = #{n | a(n) <= x} be the number of terms of this sequence up to x. Kanold proved that there are two constants 0 < c1 < c2 and a positive number x_0 such that c1 < E(x)/sqrt(x/log(x)) < c2 for x > x_0. De Koninck and Kátai proved that there is a positive constant c such that E(x) = c * (1 + o(1)) * sqrt(x/log(x)).
Apparently most of the terms are squares or powers of 2. Terms that are not included 1250, 4802, 31250, 57122, ...
Numbers k such that A099377(k) = A038040(k) and A099378(k) = A000203(k). - Amiram Eldar, Nov 02 2021

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 75.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Imre Kátai, On an estimate of Kanold, Int. J. Math. Anal., Vol. 5, No. 8 (2007), pp. 1-12.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl II, Mathematische Annalen, Vol. 134, No. 3 (1958), pp. 225-231.

Cf. A000005, A000203, A038040, A099377, A099378, A324121.

Subsequence of A014567 and A046678.

[k:k in [1..5000]| Gcd(k*NumberOfDivisors(k),DivisorSigma(1,k)) eq 1]; // Marius A. Burtea, Dec 20 2019

Select[Range[10^4], CoprimeQ[# * DivisorSigma[0, #], DivisorSigma[1, #]] &]

cp's OEIS Frontend

A324058 a(n) = A324121(A005940(1+n)) = gcd(A324054(n), A005940(1+n)*A106737(n)).

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A324188 a(n) = A324121(A163511(n)).

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A336314 a(n) = A324121(A122111(n)).

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A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

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A336846 a(n) = gcd(sigma(A003961(n)), A000005(n)*A003961(n)).

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A324122 a(n) = sigma(n) - gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

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A336476 a(n) = gcd(A000593(n), A336475(n)).

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A336320 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324058(i) = A324058(j) for all i, j >= 0.

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