A007955 -id:A007955 - cp's OEIS frontend

A334731 a(n) = Product_{d|n} gcd(sigma(d), pod(d)) where sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 48, 1, 4, 3, 1, 1, 36, 1, 4, 1, 4, 1, 576, 1, 2, 1, 224, 1, 5184, 1, 1, 3, 2, 1, 144, 1, 4, 1, 40, 1, 2304, 1, 16, 9, 4, 1, 2304, 1, 2, 9, 4, 1, 864, 1, 1792, 1, 2, 1, 995328, 1, 4, 1, 1, 1, 20736, 1, 4, 3, 128, 1, 5184, 1, 2

Offset: 1

Jaroslav Krizek, May 09 2020

nonn

			a(6) = gcd(sigma(1), pod(1)) * gcd(sigma(2), pod(2)) * gcd(sigma(3), pod(3)) * gcd(sigma(6), pod(6)) = gcd(1, 1) * gcd(3, 2) * gcd(4, 3) * gcd(12, 36) = 1 * 1 * 1 * 12 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(d), pod(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A306682 (gcd(sigma(n), pod(n))).

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

a[n_] := Product[GCD[DivisorSigma[1, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)

pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(sigma(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

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

A334793 a(n) = Sum_{d|n} lcm(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 3, 7, 27, 11, 45, 15, 91, 34, 113, 23, 1797, 27, 213, 917, 5211, 35, 5904, 39, 24137, 1785, 509, 47, 333637, 386, 705, 2950, 66093, 59, 811055, 63, 103515, 4385, 1193, 4925, 10085352, 75, 1485, 6117, 2584201, 83, 3113715, 87, 256085, 183194, 2165, 95

Offset: 1

Jaroslav Krizek, May 12 2020

			a(6) = lcm(tau(1), pod(1)) + lcm(tau(2), pod(2)) + lcm(tau(3), pod(3)) + lcm(tau(6), pod(6)) = lcm(1, 1) + lcm(2, 2) + lcm(2, 3) + lcm(4, 36) = 1 + 2 + 6 + 36 = 45.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A334662 (Sum_{d|n} gcd(tau(d), pod(d))), A334784 (Sum_{d|n} lcm(tau(d), sigma(d))).

Cf. A000005 (tau(n)), A007955 (pod(n)), A324528 (lcm(tau(n), pod(n))).

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

pod:= proc(n) option remember; convert(numtheory:-divisors(n),`*`) end proc:
f:= proc(n) local d; add(ilcm(numtheory:-tau(d), pod(d)),d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Jan 02 2025

a[n_] := DivisorSum[n, LCM[(d = DivisorSigma[0, #]), #^(d/2)] &]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)

a(n) = sumdiv(n, d, lcm(numdiv(d), vecprod(divisors(d)))); \\ Michel Marcus, May 12 2020

a(p) = 2p + 1 for p = odd primes (A065091).

A174896 a(n) = numbers k in increasing order such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

Complement of A174895(n). A174897(a(n)) = 0, A174898(a(n)) = 1.

More terms from Michel Marcus, Sep 18 2013

A174899 Record values of A007955(m), where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 2, 3, 8, 36, 64, 100, 1728, 5832, 8000, 331776, 810000, 10077696, 254803968, 46656000000, 139314069504, 351298031616, 531441000000, 782757789696, 1586874322944, 42998169600000000, 634562281237118976, 198359290368000000000, 634033809653760000000000

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = A007955(A034287(n)).

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A007955 (product of divisors), A034287 (record indices).

{my(m=0); for(n=i=1, 10^3, my(t=vecprod(divisors(n))); if(t>m, print1(t, ", "); m=t))} \\ Andrew Howroyd, Jan 05 2020

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020

A174933 a(n) = Sum_{d|n} A007955(d) * A000027(d) = Sum_{d|n} A007955(d) * (d), where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 5, 10, 37, 26, 230, 50, 549, 253, 1030, 122, 20998, 170, 2798, 3410, 16933, 290, 105449, 362, 161062, 9320, 10774, 530, 7984134, 3151, 17750, 19936, 617486, 842, 24304630, 962, 1065509, 36068, 39598, 42950, 362923273, 1370, 55238, 59498, 102561574

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)*1 + b(2)*2 + b(4)*4 = 1*1 + 2*2 + 8*4 = 37.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A007955.

[&+[&*Divisors(d)*d:d in Divisors(n)]:n in [1..40]]; // Marius A. Burtea, Jan 05 2020

a(n)={sumdiv(n, d, vecprod(divisors(d))*d)} \\ Andrew Howroyd, Jan 05 2020

from math import isqrt
from sympy import divisor_count, divisors
def A174933(n): return sum(isqrt(d)**(c+2) if (c:=divisor_count(d)) & 1 else d**(c//2+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 25 2022

Terms a(31) and beyond from Andrew Howroyd, Jan 05 2020

A174939 a(n) = Sum_{k<=n} A007955(k) * A007955(k) = Sum_{k<=n} A007955(k)^2, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 5, 14, 78, 103, 1399, 1448, 5544, 6273, 16273, 16394, 3002378, 3002547, 3040963, 3091588, 4140164, 4140453, 38152677, 38153038, 102153038, 102347519, 102581775, 102582304, 110177896480, 110177912105, 110178369081, 110178900522, 110660790826, 110660791667

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)^2 + b(2)^2 + b(3)^2 + b(4)^2 = 1^2 + 2^2 + 3^2 + 8^2 = 78.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007955, A062758.

Accumulate@ Array[#^DivisorSigma[0, #] &, 29] (* Michael De Vlieger, May 03 2022 *)

a(n) = sum(k=1, n, k^numdiv(k)); \\ Michel Marcus, May 03 2022

from sympy import divisor_count
from itertools import count, islice
def agen():
    an = 1
    for k in count(2):
        yield an
        an += k**divisor_count(k)
print(list(islice(agen(), 29))) # Michael S. Branicky, May 03 2022

a(n) = Sum_{k=1..n} A062758(k). - Michel Marcus, May 03 2022

a(27) and beyond from Michael S. Branicky, May 03 2022

A324555 a(n) = the smallest number m such that gcd(tau(m), pod(m)) = n where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 9, 6, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 120, 18939904, 180, 94633984, 240, 35721, 11264, 2218786816, 360, 10000, 53248, 900, 1344, 225754218496, 720, 1031865892864, 840, 7144929, 1114112, 1960000, 1260, 94076963651584, 4980736

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

a(n) = the smallest number m such that A306671(m) = n.

If a(17) exists, it must be bigger than 10^7.

			For n=3; a(3) = 9 because gcd(tau(9), pod(9)) = gcd (3, 27) = 3 and 9 is the smallest.

Cf. A000005, A007955, A306671.

[Min([n: n in[1..10^6] | GCD(NumberOfDivisors(n), &*[d: d in Divisors(n)]) eq k]): k in [1..16]]

Array[Block[{m = 1}, While[GCD[DivisorSigma[0, m], Times @@ Divisors@ m] != #, m++]; m] &, 16] (* Michael De Vlieger, Mar 24 2019 *)

a(n) = {my(k=1, vk = divisors(k)); while(gcd(#vk, vecprod(vk)) != n, k++; vk = divisors(k)); k;} \\ Michel Marcus, Mar 06 2019

a(17)-a(38) from Jon E. Schoenfield, Mar 07 2019

A324982 a(n) = numerator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 2, 5, 14, 7, 25, 9, 62, 23, 59, 13, 1819, 15, 109, 245, 3382, 19, 1987, 21, 2731, 465, 257, 25, 250747, 271, 355, 775, 22295, 31, 405385, 33, 28434, 1121, 599, 1253, 6726169, 39, 745, 1557, 642763, 43, 1556549, 45, 28657, 61031, 1085, 49, 765671783, 713

Offset: 1

Jaroslav Krizek, Mar 22 2019

Sum_{d|n} (pod(d)/tau(d)) > 1 for all n > 1.

			Sum_{d|n} (pod(d)/tau(d)) for n >= 1: 1, 2, 5/2, 14/3, 7/2, 25/2, 9/2, 62/3, 23/2, 59/2, ...
For n=4; Sum_{d|4} (pod(d)/tau(d)) = pod(1)/tau(1) + pod(2)/tau(2) + pod(4)/tau(4) = 1/1 + 2/2 + 8/3 = 14/3;  a(4) = 14.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A007955, A324983 (denominators).

[Numerator(&+[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];

Array[Numerator@ DivisorSum[#, Apply[Times, Divisors@ #]/DivisorSigma[0, #] &] &, 49] (* Michael De Vlieger, Mar 24 2019 *)

a(n) = numerator(sumdiv(n, d, vecprod(divisors(d))/numdiv(d))); \\ Michel Marcus, Mar 23 2019

a(p) = p + 2 for p = odd primes.

A324983 a(n) = denominator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 6, 2, 2, 4, 15, 2, 2, 2, 2, 4, 2, 2, 6, 6, 2, 4, 6, 2, 4, 2, 5, 4, 2, 4, 6, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 30, 6, 2, 4, 6, 2, 4, 4, 6, 4, 2, 2, 4, 2, 2, 4, 35, 4, 4, 2, 2, 4, 4, 2, 6, 2, 2, 12, 6, 4, 4, 2, 10, 20, 2, 2, 12, 4

Offset: 1

Jaroslav Krizek, Mar 22 2019

Sum_{d|n} (pod(d)/tau(d)) > 1 for all n > 1.

			Sum_{d|n} (pod(d)/tau(d)) for n >= 1: 1, 2, 5/2, 14/3, 7/2, 25/2, 9/2, 62/3, 23/2, 59/2, ...
For n=4; Sum_{d|4} (pod(d)/tau(d)) = pod(1)/tau(1) + pod(2)/tau(2) + pod(4)/tau(4) = 1/1 + 2/2 + 8/3 = 14/3;  a(4) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A007955, A324982 (numerators).

[Denominator(&+[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]

Array[Denominator@ DivisorSum[#, Apply[Times, Divisors@ #]/DivisorSigma[0, #] &] &, 85] (* Michael De Vlieger, Mar 24 2019 *)

a(n) = denominator(sumdiv(n, d, vecprod(divisors(d))/numdiv(d))); \\ Michel Marcus, Mar 23 2019

a(p) = 2 for p = odd primes.

A325030 a(n) = Product_{d|n} (sigma(d)*pod(d)) 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, 6, 12, 336, 30, 31104, 56, 322560, 4212, 324000, 132, 84276412416, 182, 1580544, 1944000, 10239344640, 306, 2483164449792, 380, 6096384000000, 9483264, 13799808, 552, 1610547321930095001600, 116250, 31004064, 122821920, 108806975520768, 870

Offset: 1

Jaroslav Krizek, Apr 25 2019

nonn

n divides a(n) for all n.

			a(6) = (sigma(1)*pod(1)) * (sigma(2)*pod(2)) * (sigma(3)*pod(3)) * (sigma(6)*pod(6)) = (1*1) * (3*2) * (4*3) * (12*36) = 31104.

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

Cf. A000203, A007955, A325029.

[&*[&+ [c: c in Divisors(d)] * &*[c: c in Divisors(d)]: d in Divisors(n)]: n in [1..100]]

Table[Times@@(DivisorSigma[1,#]Times@@Divisors[#]&/@Divisors[n]),{n,30}] (* Harvey P. Dale, Dec 10 2024 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, my(dd=divisors(d[k])); vecsum(dd)*vecprod(dd)); \\ Michel Marcus, Apr 25 2019

a(n) = Product_{d|n} sigma(d) * Product_{d|n} pod(d) = A206032(n) * A266265(n).

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

cp's OEIS Frontend

A334731 a(n) = Product_{d|n} gcd(sigma(d), pod(d)) where 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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A334793 a(n) = Sum_{d|n} lcm(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

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A174896 a(n) = numbers k in increasing order such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m.

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A174899 Record values of A007955(m), where A007955(m) = product of divisors of m.

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A174933 a(n) = Sum_{d|n} A007955(d) * A000027(d) = Sum_{d|n} A007955(d) * (d), where A007955(m) = product of divisors of m.

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A174939 a(n) = Sum_{k<=n} A007955(k) * A007955(k) = Sum_{k<=n} A007955(k)^2, where A007955(m) = product of divisors of m.

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A324555 a(n) = the smallest number m such that gcd(tau(m), pod(m)) = 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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A324982 a(n) = numerator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).