A007955 -id:A007955 - cp's OEIS frontend

A337323 a(n) = gcd(n, 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).

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

Offset: 1

Jaroslav Krizek, Aug 23 2020

GCD(n, tau(n), sigma(n), pod(n)) = GCD(n, tau(n), sigma(n)). - David A. Corneth, Aug 24 2020

			a(6) = gcd(6, tau(6), sigma(6), pod(6)) = gcd(6, 4, 12, 36) = 2.

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

Cf. A336722 (gcd(tau(n), sigma(n), pod(n))).
Cf. A337324 (least m such that gcd(m, tau(m), sigma(m), pod(m)) = n).
Cf. A336723 (lcm(tau(n), sigma(n), pod(n))) = (lcm(n, tau(n), sigma(n), pod(n))).
Cf. A000005, A000203, A007955.

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

f:= proc(n) uses numtheory; igcd(n, tau(n), sigma(n)) end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2020

a[n_] := GCD @@ {n, DivisorSigma[0, n], DivisorSigma[1, n]}; Array[a, 100] (* Amiram Eldar, Aug 24 2020 *)

a(n) = my(f=factor(n)); gcd([n, sigma(f), numdiv(f)]); \\ Michel Marcus, Apr 01 2021

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

a(n) = 1 for n = p^k, p prime, k >= 0 (A000961). - Bernard Schott, Apr 01 2021

A174897 a(n) = characteristic function of numbers k such that A007955(m) = k has solution for some m, where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = characteristic function of numbers from A174895(n).

a(n) = 1 if A007955(m) = n for any m, else 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Cf. A007955, A174895, A174898.

Block[{nn = 105, t}, t = ConstantArray[0, nn]; ReplacePart[t, Map[# -> 1 &, TakeWhile[Sort@ Array[Times @@ Divisors@ # &, nn], # <= 105 &]]]] (* Michael De Vlieger, Oct 20 2017 *)

up_to = 65537;
v174897 = vector(up_to);
A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ This function from Charles R Greathouse IV, Feb 11 2011
for(k=1, up_to, t=A007955(k); if(t<=up_to, v174897[t] = 1));
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,v174897,"b174897_upto65537.txt");
\\ Antti Karttunen, Oct 20 2017

a(n) = 1 - A174898(n).

Name edited and more terms added by Antti Karttunen, Oct 20 2017

A174898 a(n) = characteristic function of numbers k such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = characteristic function of numbers from A174896(n).

a(n) = 1 if A007955(m) not equal to n for any m, else 0.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (computed from the b-file of A174897)
Index entries for characteristic functions

Cf. A007955, A174896, A174897.

a(n) = 1 - A174897(n).

More terms from Antti Karttunen, Oct 20 2017

A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

Original entry on oeis.org

2, 37, 101, 197, 677, 5477, 8837, 17957, 21317, 42437, 98597, 106277, 148997, 217157, 401957, 454277, 1196837, 1378277, 1674437, 1705637, 1833317, 1865957, 2390117, 2735717, 3118757, 3147077, 3587237, 3865157, 4104677, 4519877, 4726277, 5410277, 6728837, 7043717

Offset: 1

Jaroslav Krizek, Jun 20 2015

nonn

Sequence lists divisorial primes p from A258455 such that p-1 = A007955(sqrt(p-1)).
If 1 + Product_{d|k} d for some k > 1 is a prime p other than 3, then p-1 is a square and p is either of the form k^2 + 1 or h^2 + 1 where h>k. In this sequence are divisorial primes of the first kind. Divisorial primes of the second kind are in A258897.
With number 3, complement of A258897 with respect to A258455.
All terms > 2 are of the form 4*q^2 + 1 where q = prime (see A052292).
Subsequence of A002496 (primes of the form k^2 + 1), and the corresponding k are a subsequence of A007422. - Michel Marcus, Jul 09 2015

			Number 101 is in sequence because 100 is the product of divisors of 10; 101 - 1 = 100 = A007955(sqrt(101 - 1)).

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A007955, A052292, A258455, A258897, A259199.

[n: n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

lista(nn) = {forprime(p=2, nn, if (issquare(pp=(p-1)) && (k=sqrtint(pp)) && (d=divisors(k)) && (1+prod(j=1, #d, d[j])==p), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

For n>1; a(n) = 4*(A052291(n))^2 + 1 = A052292(n).

A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

1, 6, 10, 14, 26, 74, 94, 134, 146, 206, 314, 326, 386, 466, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2974, 2986, 3046, 3106, 3134, 3214, 3254, 3274, 3314, 3326, 3334, 3446

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

First deviation from A259020 is at a(15).
With number 2 complement of A259023 with respect to A118369.
1 together with squarefree semiprimes (A006881) k such that k^2 + 1 is prime. Without the squarefree restriction there will be only one more term, 4. - Amiram Eldar, Sep 25 2022

			The number 10 is in sequence because 10^2 = 1*2*5*10 = 100 and simultaneously 101 is prime.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Union of {1} and (intersection of A005574 and A006881).
Subsequence of A007422, A048943, A259020, A118369.
Cf. A007955, A007422, A048943, A052291, A118369, A258455, A259020, A259023, A258897.

[Floor(Sqrt(n-1)): n in [1..10000000] | IsPrime(n) and n-1 eq (&*(Divisors(Floor(Sqrt(n-1)))))];

Prepend[2*Select[Prime[Range[2, 300]], PrimeQ[4 #^2 + 1] &], 1] (* Amiram Eldar, Sep 25 2022 *)

a = [n for n in range(1,100000) if is_prime(n^2+1) and n^2==prod(list(divisors(n)))] # Danny Rorabaugh, Sep 21 2015

a(n) = 2*A052291(n) for n > 1. - Amiram Eldar, Sep 25 2022

A334662 a(n) = Sum_{d|n} gcd(tau(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 3, 2, 4, 2, 8, 2, 8, 5, 8, 2, 15, 2, 8, 4, 9, 2, 17, 2, 11, 4, 8, 2, 27, 3, 8, 6, 11, 2, 22, 2, 11, 4, 8, 4, 33, 2, 8, 4, 23, 2, 22, 2, 11, 10, 8, 2, 30, 3, 11, 4, 11, 2, 26, 4, 23, 4, 8, 2, 43, 2, 8, 10, 12, 4, 22, 2, 11, 4, 22, 2, 57, 2, 8, 8, 11, 4, 22

Offset: 1

Jaroslav Krizek, May 07 2020

nonn

Inverse Möbius transform of A306671. - Antti Karttunen, May 19 2020

			a(6) = gcd(tau(1), pod(1)) + gcd(tau(2), pod(2)) + gcd(tau(3), pod(3)) + gcd(tau(6), pod(6)) = gcd(1, 1) + gcd(2, 2) + gcd(2, 3) + gcd(4, 36) = 1 + 2 + 1 + 4 = 8.

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

Cf. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(d), pod(d))).

Cf. A000005 (tau(n)), A007955 (pod(n)), A306671 (gcd(tau(n), pod(n))).

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

a(n) = sumdiv(n, d, gcd(numdiv(d), vecprod(divisors(d)))); \\ Michel Marcus, May 08 2020

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

A334663 a(n) = Sum_{d|n} gcd(sigma(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 2, 3, 2, 15, 2, 4, 3, 5, 2, 20, 2, 7, 6, 5, 2, 19, 2, 8, 4, 7, 2, 33, 3, 5, 4, 64, 2, 93, 2, 6, 6, 5, 4, 25, 2, 7, 4, 19, 2, 69, 2, 12, 10, 7, 2, 38, 3, 7, 12, 8, 2, 44, 4, 73, 4, 5, 2, 124, 2, 7, 6, 7, 4, 167, 2, 8, 6, 27, 2, 41, 2, 5, 8, 12, 4, 43, 2

Offset: 1

Jaroslav Krizek, May 07 2020

nonn

Inverse Möbius transform of A306682. - Antti Karttunen, May 09 2020

			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 = 15.

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

Cf. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334662 (Sum_{d|n} gcd(tau(d), pod(d))).
Cf. A000203 (sigma(n)), A007955 (pod(n)), A306682 (gcd(sigma(n), pod(n))).
Cf. A334731 (product instead of sum).

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

a(n) = sumdiv(n, d, gcd(sigma(d), vecprod(divisors(d)))); \\ Michel Marcus, May 08 2020

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

A083267 Product of related numbers (counted in A073757) belonging to n; related = {divisor-set, RRS}: a(n) = A007955(n)*A001783(n).

Original entry on oeis.org

1, 2, 6, 24, 120, 180, 5040, 6720, 60480, 18900, 39916800, 665280, 6227020800, 3783780, 201801600, 2075673600, 355687428096000, 496215720, 121645100408832000, 69837768000, 20858213376000, 604969665300, 25852016738884976640000, 12336143339520, 5170403347776995328000

Offset: 1

Labos Elemer, May 13 2003

nonn

			For n = 10: related terms = {1,2,5,10,3,7,9}, product = 1*2*5*10*1*3*7*9 = 18900 = a(10).

Amiram Eldar, Table of n, a(n) for n = 1..450

Cf. A073757 (count), A083266 (sum), A083268 (LCM), A083267 (product), A001783, A007955.

a[n_] := n^(DivisorSigma[0, n]/2) * Times@@ Select[Range[n], CoprimeQ[n, #] &]; Array[a, 30] (* Amiram Eldar, Jun 20 2024 *)

More terms from Amiram Eldar, Jun 20 2024

A283995 Least number with same prime signature as the n-th divisorial: a(n) = A046523(A007955(n)).

Original entry on oeis.org

1, 2, 2, 8, 2, 36, 2, 64, 8, 36, 2, 1728, 2, 36, 36, 1024, 2, 1728, 2, 1728, 36, 36, 2, 331776, 8, 36, 64, 1728, 2, 810000, 2, 32768, 36, 36, 36, 10077696, 2, 36, 36, 331776, 2, 810000, 2, 1728, 1728, 36, 2, 254803968, 8, 1728, 36, 1728, 2, 331776, 36, 331776, 36, 36, 2, 46656000000, 2, 36, 1728, 2097152, 36, 810000, 2, 1728, 36, 810000, 2, 139314069504, 2, 36

Offset: 1

Antti Karttunen, Mar 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1025

Cf. A007955, A046523, A069264, A283990.

Table[If[n == 1, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, FactorInteger[Times @@ Divisors@ n][[All, -1]]]], {n, 74}] (* Michael De Vlieger, Mar 22 2017 *)

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From Charles R Greathouse IV, Feb 11 2011
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From Charles R Greathouse IV, Aug 17 2011
A283995(n) = A046523(A007955(n));

from math import prod, isqrt
from sympy import prime, factorint, divisor_count
def A283995(n): return (lambda n:prod(prime(i+1)**e for i, e in enumerate(sorted(factorint(n).values(), reverse=True))))((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022

(define (A283995 n) (A046523 (A007955 n)))

a(n) = A046523(A007955(n)).

A322671 a(n) = Sum_{d|n} (pod(d)/d), where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 2, 4, 2, 9, 2, 12, 5, 13, 2, 155, 2, 17, 18, 76, 2, 336, 2, 415, 24, 25, 2, 13987, 7, 29, 32, 803, 2, 27035, 2, 1100, 36, 37, 38, 280418, 2, 41, 42, 64423, 2, 74133, 2, 1963, 2046, 49, 2, 5322467, 9, 2518, 54, 2735, 2, 157827, 58, 176427, 60, 61, 2

Offset: 1

Jaroslav Krizek, Dec 23 2018

nonn

			For n = 6; a(6) = pod(1)/1 + pod(2)/2 + pod(3)/3 + pod(6)/6 = 1/1 + 2/2 + 3/3 + 36/6 = 9.

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

Cf. A007955, A322672.

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

pod:= proc(n) convert(numtheory:-divisors(n),`*`) end proc:
f:= proc(n) local d; add(pod(d)/d, d = numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Dec 23 2018

Array[Sum[Apply[Times, Divisors@ d]/d, {d, Divisors@ #}] &, 59] (* Michael De Vlieger, Jan 19 2019 *)

a(n) = sumdiv(n, d, vecprod(divisors(d))/d); \\ Michel Marcus, Dec 23 2018

from math import isqrt
from sympy import divisor_count, divisors
def A322671(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

a(n) = n for n = 1, 2 and 4.
a(n) = n + (tau(n) - 1) = n + 3 for squarefree semiprimes (A006881).
a(n) = 2 if n is prime. - Robert Israel, Dec 23 2018

cp's OEIS Frontend

A337323 a(n) = gcd(n, 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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A174897 a(n) = characteristic function of numbers k such that A007955(m) = k has solution for some m, where A007955(m) = product of divisors of m.

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A174898 a(n) = characteristic function of numbers k such that A007955(m) = k has no solution for any m, where A007955(m) = product of divisors of m.

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A258896 Divisorial primes p of the form p = 1 + k^2 where k^2 = Product_{d|k} d= A007955(k) for some k.

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A259021 Numbers k such that k^2 = Product_{d|k} d (= A007955(k)) and simultaneously k^2 + 1 is a divisorial prime (A258455).

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A334662 a(n) = Sum_{d|n} gcd(tau(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

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A334663 a(n) = Sum_{d|n} gcd(sigma(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

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