A007955 -id:A007955 - cp's OEIS frontend

A306671 a(n) = gcd(tau(n), pod(n)) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

1, 2, 1, 1, 1, 4, 1, 4, 3, 4, 1, 6, 1, 4, 1, 1, 1, 6, 1, 2, 1, 4, 1, 8, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 9, 1, 4, 1, 8, 1, 8, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 8, 1, 8, 1, 4, 1, 12, 1, 4, 3, 1, 1, 8, 1, 2, 1, 8, 1, 12, 1, 4, 3, 2, 1, 8, 1, 10, 1, 4, 1, 12, 1, 4

Offset: 1

Jaroslav Krizek, Mar 04 2019

nonn

Sequence of the smallest numbers k such that a(k) = n: 1, 2, 9, 6, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 120, ...

			For n=6: a(6) = gcd(tau(6), pod(6)) = gcd(4, 36) = 4.

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

Cf. A000005, A007955, A009205, A046642.

[GCD(NumberOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

a(n) = gcd(numdiv(n), vecprod(divisors(n))); \\ Michel Marcus, Mar 04 2019

a(n) = 1 for numbers in A046642.

a(n) = tau(n) for numbers in A120736.

A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 3, 4, 11, 6, 42, 8, 75, 31, 108, 12, 1778, 14, 206, 234, 1099, 18, 5901, 20, 8116, 452, 498, 24, 333618, 131, 692, 760, 22166, 30, 810372, 32, 33867, 1104, 1176, 1238, 10085333, 38, 1466, 1538, 2568180, 42, 3112382, 44, 85690, 91386, 2142, 48, 255138610

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

			For n = 4, with b(n) = A007955(n), a(4) = b(1) + b(2) + b(4) = 1 + 2 + 8 = 11.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007429, A007955, A206032, A266265.

Subsequences: A008864, A181388 \ {0}.

a[n_] := DivisorSum[n, #^(DivisorSigma[0, #]/2) &]; Array[a, 50] (* Amiram Eldar, Oct 23 2021 *)

a(n) = sumdiv(n, d, vecprod(divisors(d))); \\ Michel Marcus, Dec 09 2014 and Oct 23 2021

from math import isqrt
from sympy import divisor_count, divisors
def A175317(n): return sum(isqrt(d)**c if (c:=divisor_count(d)) & 1 else d**(c//2) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 24 2022

From Bernard Schott, Oct 26 2021: (Start)
a(1) = 1 (the only fixed point).
a(p) = p+1 for prime p only.
a(2^k) = A181388(k+1). (End)

Corrected by Jaroslav Krizek, Apr 02 2010

Edited and more terms from Michel Marcus, Dec 09 2014

A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 1, 8, 1, 8, 1, 2, 1, 12, 1, 4, 1, 1, 1, 8, 1, 2, 1, 8, 1, 3, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 1, 3, 1, 1, 8, 1, 2, 1

Offset: 1

Jaroslav Krizek, Aug 01 2020

nonn

a(n) = tau(n) for numbers n: 1, 6, 14, 22, 30, 38, 42, 46, 54, 56, 60, 62, 66, 70, 78, 86, 94, 96, 102, ...

			a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.

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

Cf. A009205 (gcd(tau(n), sigma(n))), A306671 (gcd(tau(n), pod(n))), A306682 (gcd(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).

[GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];

a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* Amiram Eldar, Aug 01 2020 *)

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ Antti Karttunen, Aug 10 2020

a(p) = 1 for p = primes (A000040).

a(n) = gcd(A007955(n), A009205(n)). - Antti Karttunen, Aug 10 2020

Data section extended up to a(105) by Antti Karttunen, Aug 10 2020

A306682 a(n) = gcd(sigma(n), pod(n)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 4, 1, 4, 3, 1, 1, 3, 1, 2, 1, 4, 1, 12, 1, 2, 1, 56, 1, 72, 1, 1, 3, 2, 1, 1, 1, 4, 1, 10, 1, 48, 1, 4, 3, 4, 1, 4, 1, 1, 9, 2, 1, 24, 1, 8, 1, 2, 1, 24, 1, 4, 1, 1, 1, 144, 1, 2, 3, 16, 1, 3, 1, 2, 1, 4, 1, 24, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

See A324527(n) = the smallest numbers k such that a(k) = n.

			For n=6: a(6) = gcd(tau(6), pod(6)) = gcd(4, 36) = 4.

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

Cf. A000005, A000203, A007955, A009205, A046642, A120736, A324526, A324527.

[GCD(SumOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

a(n) = my(d=divisors(n)); gcd(vecsum(d), vecprod(d)); \\ Michel Marcus, Mar 05 2019

a(n) = 1 for numbers in A014567.

a(n) = tau(n) for numbers in A324526.

A324528 a(n) = lcm(tau(n), pod(n)) where tau(k) = the number of divisors of k (A000005) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 2, 6, 24, 10, 36, 14, 64, 27, 100, 22, 1728, 26, 196, 900, 5120, 34, 5832, 38, 24000, 1764, 484, 46, 331776, 375, 676, 2916, 65856, 58, 810000, 62, 98304, 4356, 1156, 4900, 10077696, 74, 1444, 6084, 2560000, 82, 3111696, 86, 255552, 182250, 2116, 94

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

			For n=4: a(4) = lcm(tau(4), pod(4)) = lcm(3, 8) = 24.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A007955, A046642, A120736.

[LCM(NumberOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

Table[LCM[DivisorSigma[0,n],Times@@Divisors[n]],{n,50}] (* Harvey P. Dale, Aug 14 2019 *)

a(n) = my(d=divisors(n)); lcm(vecprod(d), #d); \\ Michel Marcus, Mar 05 2019

a(n) = pod(n) for numbers n in A120736.

a(n) = tau(n) * pod(n) for numbers n in A046642.

A336723 a(n) = lcm(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 6, 12, 168, 30, 36, 56, 960, 351, 900, 132, 12096, 182, 1176, 1800, 158720, 306, 75816, 380, 168000, 14112, 4356, 552, 1658880, 11625, 14196, 29160, 65856, 870, 810000, 992, 2064384, 17424, 31212, 58800, 917070336, 1406, 21660, 85176, 23040000, 1722, 6223392

Offset: 1

Jaroslav Krizek, Aug 01 2020

nonn

a(n) = pod(n) for numbers n: 1, 6, 30, 66, 84, 102, 120, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, ...

			a(6) = lcm(tau(6), sigma(6), pod(6)) = lcm(4, 12, 36) = 36.

Cf. A009278 (lcm(tau(n), sigma(n))), A324528 (lcm(tau(n), pod(n))), A324529 (lcm(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336722 (gcd(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = pod(k) and simultaneously A336722(k) = tau(k)).

[LCM([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];

a[n_] := LCM @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 50] (* Amiram Eldar, Aug 01 2020 *)

a(n) = my(d=divisors(n)); lcm([#d, vecsum(d), vecprod(d)]); \\ Michel Marcus, Aug 12 2020

a(p) = p^2 + p for p = primes (A000040).

A174895 a(n) = possible values of A007955(m) in increasing order, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 100, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 225, 227, 229, 233, 239

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = all values of A007955(m) in increasing order; all terms of sequence A007955 occur only once. Complement of A174896(n). A174897(a(n)) = 1, A174898(a(n)) = 0.
For every prime p, p and p^3 occur, as does the square of every semiprime pq with p and q distinct. - T. D. Noe, Oct 22 2010
For every prime p, every power p^t occurs, where t is a triangular number.

T. D. Noe, Table of n, a(n) for n = 1..1000

nn=1000; Reap[Do[prod=Times@@Divisors[n]; If[prod<=nn, Sow[prod]], {n,nn}]][[2,1]] (* T. D. Noe, Oct 22 2010 *)

list(lim)=my(v=List(primes([2,lim]))); for(k=1,sqrtint(lim\=1), listput(v,factorback(divisors(k)))); forprime(p=2,sqrtnint(lim,3), listput(v, p^3)); select(k->k<=lim, Set(v)) \\ Charles R Greathouse IV, Sep 22 2015

Corrected by Jaroslav Krizek, Apr 02 2010

Corrected and extended by T. D. Noe, Oct 22 2010

A184390 a(n) = sum of numbers from 1 to pi(n), where pi(n) = A007955(n).

Original entry on oeis.org

1, 3, 6, 36, 15, 666, 28, 2080, 378, 5050, 66, 1493856, 91, 19306, 25425, 524800, 153, 17009028, 190, 32004000, 97461, 117370, 276, 55037822976, 7875, 228826, 266085, 240956128, 435, 328050405000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

			For n = 6; pi(6) = 36; a(n) = (1/2)*36*37 = 666.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A184387, A184388, A184389, A184391, A130674.

# (#+1)/2&/@Array[Times@@Divisors[#]&,40] (* Harvey P. Dale, Oct 05 2012 *)

from math import isqrt
from sympy import divisor_count
def A184390(n): return (m:=((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)))*(m+1)//2 # Chai Wah Wu, Jun 25 2022

a(n) = Sum_{i = 1..pi(n)} i = A000217(A007955(n)) = (1/2)*A007955(n)*(A007955(n)+1).

A184391 a(n) = product of numbers from 1 to A007955(n).

Original entry on oeis.org

1, 2, 6, 40320, 120, 371993326789901217467999448150835200000000, 5040, 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000, 10888869450418352160768000000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Robert P. P. McKone, Table of n, a(n) for n = 1..11

Cf. A184387, A184388, A184389, A184390, A130674.

A184391[n_] := Product[i, {i, 1, n^(DivisorSigma[0, n]/2)}]; Table[A184391[n], {n, 1, 9}] (* Robert P. P. McKone, Feb 04 2021 *)

from math import isqrt, factorial
from sympy import divisor_count
def A184391(n): return factorial((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(n)! = A000142(A007955(n)).

a(9) from Robert P. P. McKone, Feb 04 2021

A324529 a(n) = lcm(sigma(n), pod(n)) where sigma(k) = the sum of divisors of k (A000203) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 6, 12, 56, 30, 36, 56, 960, 351, 900, 132, 12096, 182, 1176, 1800, 31744, 306, 75816, 380, 168000, 14112, 4356, 552, 1658880, 3875, 14196, 29160, 21952, 870, 810000, 992, 2064384, 17424, 31212, 58800, 917070336, 1406, 21660, 85176, 23040000, 1722, 6223392

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

			For n=4: a(4) = lcm(sigma(4), pod(4)) = lcm(7, 8) = 56.

Cf. A000005, A000203, A014567, A145551.

[LCM(SumOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 1000]]

a(n) = my(d=divisors(n)); lcm(vecsum(d), vecprod(d)); \\ Michel Marcus, Mar 05 2019

a(n) = pod(n) for numbers n in A145551.

a(n) = sigma(n) * pod(n) for numbers n in A014567.

cp's OEIS Frontend

A306671 a(n) = gcd(tau(n), pod(n)) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

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A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

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A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

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A306682 a(n) = gcd(sigma(n), pod(n)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

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A324528 a(n) = lcm(tau(n), pod(n)) where tau(k) = the number of divisors of k (A000005) and pod(n) = the product of divisors of k (A007955).

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A336723 a(n) = lcm(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

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A174895 a(n) = possible values of A007955(m) in increasing order, where A007955(m) = product of divisors of m.

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