A323166 -id:A323166 - cp's OEIS frontend

A349000 a(n) = A323166(A276086(n)), where A323166(n) = gcd(n, usigma(n)), usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

1, 1, 1, 6, 1, 6, 1, 2, 3, 6, 15, 90, 1, 2, 1, 6, 5, 30, 1, 2, 3, 6, 45, 90, 1, 2, 1, 6, 5, 30, 1, 2, 1, 6, 1, 6, 1, 2, 3, 6, 15, 90, 1, 2, 1, 6, 5, 30, 7, 14, 21, 42, 315, 630, 1, 2, 1, 6, 5, 30, 1, 2, 1, 6, 1, 6, 5, 10, 15, 30, 15, 90, 25, 50, 25, 150, 25, 150, 175, 350, 525, 1050, 7875, 15750, 25, 50, 25, 150, 125, 750

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. A034448, A276086, A323166, A348996, A348997, A348999.

A349000(n) = { my(m1=1, m2=1, p=2, u); while(n, if(n%p, u = p^(n%p); m1 *= u; m2 *= (1+u)); n = n\p; p = nextprime(1+p)); gcd(m1,m2); };

a(n) = A323166(A276086(n)) = gcd(A276086(n), A348996(n)).

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 12, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 7, 3, 6, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 12, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 21, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 23 2019

nonn

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

Cf. A034460, A323166, A325812, A325814.

Cf. also A325385.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));

a(n) = gcd(A034460(n), A325814(n)).

A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 6, 60, 90, 87360

Offset: 1

Antti Karttunen, Aug 28 2019

10^13 < a(6) <= 146361946186458562560000. - Giovanni Resta, Aug 29 2019

Peter Hagis, Jr., Lower bounds for unitary multiperfect numbers, Fibonacci Quarterly, Vol. 22, No. 2 (1984), pp. 140-143.

Fixed points of A323166, positions of zeros in A327164.
Cf. A002827 (a subsequence), A034448, A327163.
Cf. also A007691.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA327158(n) = (gcd(n,A034448(n))==n);

A323409 Greatest common divisor of Product (p_i^e_i)-1 and n, when n = Product (p_i^e_i); a(n) = gcd(n, A047994(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 4, 1, 2, 5, 14, 3, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 2, 1, 6, 1, 20, 1, 2, 1, 12, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Records 1, 2, 6, 12, 14, 20, 24, 84, 120, 168, 240, 468, 720, 1008, 1240, 1488, 1632, 7440, 9360, 14880, 32640, ... occur at n = 1, 6, 12, 36, 56, 80, 144, 168, 240, 504, 720, 1404, 3600, 4032, 4960, 8928, 13056, 14880, 28080, 44640, 65280, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A047994, A323410.

Cf. also A009195, A323166, A323406.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A323409(n) = gcd(n, A047994(n));

a(n) = gcd(n, A047994(n)), where A047994 is unitary phi.

A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 3, 4, 5, 6, 2, 8, 9, 10, 9, 12, 5, 14, 12, 8, 17, 18, 5, 20, 3, 32, 18, 24, 3, 26, 21, 28, 10, 30, 12, 32, 33, 16, 27, 48, 25, 38, 30, 56, 27, 42, 16, 44, 15, 4, 36, 48, 17, 50, 39, 24, 35, 54, 14, 72, 9, 80, 45, 60, 2, 62, 48, 80, 65, 84, 24, 68, 45, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of unitary divisors of n.

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

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A017665, A028235, A028236, A034448, A077610, A306633, A319676, A323166, A332880, A332883 (denominators).

Cf. also A348734, A348735.

a:= n-> numer(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Numerator

a(n) = numerator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = numerator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = numerator of usigma(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A034448(n) / A323166(n). - Antti Karttunen, Nov 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332883(k) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Nov 21 2022

A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 3, 19, 2, 21, 11, 23, 2, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 18, 37, 19, 39, 20, 41, 7, 43, 11, 3, 23, 47, 12, 49, 25, 17, 26, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of unitary divisors of n.

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

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A007947, A017666, A034448, A077610, A319677, A323166, A327158 (positions of 1's), A332881, A332882 (numerators).

a:= n-> denom(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Denominator

a(n) = denominator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = denominator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = denominator of usigma(n)/n.
a(p) = p, where p is prime.
a(n) = n / A323166(n). - Antti Karttunen, Nov 13 2021

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)).

A323160 a(n) = gcd(n, A323159(n)) = gcd(n, A034448(n), A048250(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

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

Cf. A034448, A048250, A323159, A323163, A323166.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A323160(n) = gcd(n, gcd(A034448(n), A048250(n)));

a(n) = gcd(n, A323159(n)) = gcd(A048250(n), A323166(n)).

a(n) = gcd(n, A034448(n), A048250(n)).

A327160 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,usigma(x)), where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 2, 4, 2, 4, 2, 5, 4, 2, 2, 6, 2, 5, 3, 6, 2, 5, 2, 4, 2, 5, 2, 4, 2, 2, 5, 6, 3, 6, 2, 7, 3, 6, 2, 4, 2, 4, 5, 5, 2, 7, 2, 7, 5, 8, 2, 4, 3, 4, 3, 4, 2, 1, 2, 7, 3, 2, 3, 4, 2, 7, 4, 8, 2, 7, 2, 7, 3, 6, 3, 3, 2, 7, 2, 6, 2, 7, 3, 6, 6, 7, 2, 1, 5, 6, 4, 8, 4, 9, 2, 9, 5, 9, 2, 4, 2, 7, 7

Offset: 1

Antti Karttunen, Aug 25 2019

nonn

Question: Is this sequence well defined for every n ? If A318882 is not well defined in whole N, then neither this can be.

			From n = 30 we can reach any of the following strictly positive numbers: 30 (e.g., with an empty sequence of transitions), 42 (as A034460(30) = 42), 54 (as A034460(42) = 54; note that A034460(54) = 30 again) and 6 as A323166(30) = A323166(42) = A323166(54) = 6 = A323166(6) = A034460(6), thus a(30) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A034460, A318882, A323166.

Cf. also A326198, A327161.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A327160aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=A034448(n)-n, b=gcd(A034448(n),n)); xs = A327160aux(a,xs); if((a==b),xs, A327160aux(b,xs))));
A327160(n) = length(A327160aux(n,Set([])));

A327164 Number of iterations of x -> gcd(usigma(x),x) needed to reach a fixed point, where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 0, 2, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A323166, A327158 (positions of zeros).

Cf. also A326194.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323166(n) = gcd(n, A034448(n));
A327164(n) = { my(u=A323166(n)); if(u==n,0,1+A327164(u)); }

cp's OEIS Frontend

A349000 a(n) = A323166(A276086(n)), where A323166(n) = gcd(n, usigma(n)), usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

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A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

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A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

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A323409 Greatest common divisor of Product (p_i^e_i)-1 and n, when n = Product (p_i^e_i); a(n) = gcd(n, A047994(n)).

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A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

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A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

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Maple

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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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A323160 a(n) = gcd(n, A323159(n)) = gcd(n, A034448(n), A048250(n)).

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