A048943 -id:A048943 - cp's OEIS frontend

A258455 Divisorial primes: primes p of the form p = 1 + Product_{d|k} d for some k.

2, 3, 37, 101, 197, 677, 5477, 8837, 17957, 21317, 42437, 98597, 106277, 148997, 217157, 331777, 401957, 454277, 1196837, 1378277, 1674437, 1705637, 1833317, 1865957, 2390117, 2735717, 3118757, 3147077, 3587237, 3865157, 4104677, 4519877, 4726277, 5410277, 6728837

Offset: 1

Jaroslav Krizek, May 30 2015

nonn

Primes p of the form p = A007955(k) + 1 for some k.
This sequence is a sorted version of A118370.
Corresponding values of k are in A118369.
Conjectures:
(1) if 1+ Product_{d|k} d for k > 2 is a prime p, then p-1 is a square.
(2) except for n = 2, a(n) - 1 are squares.
(3) subsequence of A062459 (primes of form x^2 + mu(x)).
From Robert Israel, Jun 08 2015: (Start)
The first n > 4 for which a(n) does not end in 7 is a(918) = 34188010001.
Statements (1) and (2) are true.
Note that if k = p_1^(a_1) ... p_m^(a_m) is the prime factorization of k, then A007955(k) = p_1^(a_1*M/2) ... p_m^(a_m*M/2) where M = (a_1+1)*...*(a_m+1).  Now if M has any odd factor r > 1, A007955(k) = x^r for some x > 1 and then p = A007955(k)+1 is divisible by x+1.  So for p to be prime, M must be a power of 2.
  Now if A007955(k) is not a square, we need M/2 to be odd, so M = 2.  That can only happen if m=1 and a_1=1.  For p to be odd we need k to be even, so this means p_1 = 1, and then k=2. (End)
Union of prime 3 (where A007955(3-1) is not a square), A258896 (primes p such that p-1 = A007955(sqrt(p-1))) and A258897 (primes p such that p-1 = A007955(k) for some k < sqrt(p-1)). - Jaroslav Krizek, Jun 14 2015
Contrary to the above, this is not a subsequence of A062459: 24^4+1 = 331777 is in this sequence but not A062459. - Charles R Greathouse IV, Sep 22 2015

			The prime 37 is in sequence because there is n = 6 with divisors 1, 2, 3, 6 such that 6*3*2*1 + 1 = 37.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007955, A048943, A118369, A118370, A174895.

Set(Sort([&*(Divisors(n))+1: n in [1..1000000] | IsPrime(&*(Divisors(n))+1)]));

N:= 10^8: # to get all terms <= N
K:= floor(sqrt(N)):
sort(convert(select(t -> t <= N and isprime(t),{2,seq(convert(numtheory:-divisors(k),`*`)+1,k=2..K,2)}),list)); # Robert Israel, Jun 08 2015

terms = 35; n0 = 1000; Clear[f]; f[nmax_] := f[nmax] = Reap[For[n = 1, n <= nmax, n++, If[PrimeQ[p = Times @@ Divisors[n] + 1], Sow[p]]]][[2, 1]] // Sort // Take[#, terms]&; f[n0]; f[nmax = 2*n0]; While[f[nmax] != f[nmax/2], Print[nmax]; nmax = 2*nmax]; f[nmax] (* Jean-François Alcover, May 31 2015 *)
Take[Sort[Select[Table[Times@@Divisors[n]+1,{n,3000}],PrimeQ]],40] (* Harvey P. Dale, Apr 18 2018 *)

list(lim)=my(v=List()); lim\=1; for(n=1,sqrtint(lim-1), my(d=divisors(n), t=prod(i=2,#d,d[i])+1); if(t<=lim && isprime(t), listput(v, t))); Set(v) \\ Charles R Greathouse IV, Jun 08 2015

A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 24, 26, 30, 33, 34, 35, 38, 39, 40, 42, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Chris Boyd, Sep 15 2013

nonn

Subsequence of A048943. According to Gerard P. Michon, one of the criteria for N to belong to A048943 is that it has at least two prime factors with odd multiplicities. By definition, the composite factor c in any term of A229153 conforms to this criterion.

From a(1) to a(63), identical to the given terms of A119847, except for the single term a(55) = 120.

Chris Boyd, Table of n, a(n) for n = 1..10000

Complement of A265640.

Cf. A048943, A229125.

iscomposite(n)={if(!isprime(n)&&n!=1,return(1));}
test(n)={if(iscomposite(core(n)),return(1));}
for(n=1,200,if(test(n)==1,print1(n",")))

lista(nn) = {for(n=1,nn, if(!ispseudoprime(core(n)) && !issquare(n), print1(n, ", ")));} \\ Altug Alkan, Feb 04 2016

list(lim)=my(v=List()); forsquarefree(c=6,lim\=1, if(#c[2]~ > 1, for(m=1,sqrtint(lim\c[1]), listput(v, c[1]*m^2)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2022

from math import isqrt
from sympy import primepi, mobius
def A229153(n):
    def f(x):
        c = n+x+(a:=isqrt(x))
        for y in range(1,a+1):
            m = x//y**2
            c += primepi(m)-sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 30 2025

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

A259023 Numbers n such that Product_{d|n} d = k^2 for some k > n and simultaneously number k^2 + 1 is a divisorial prime (A258455).

Original entry on oeis.org

24, 54, 56, 88, 154, 174, 238, 248, 266, 296, 328, 374, 378, 430, 442, 472, 488, 494, 498, 510, 568, 582, 584, 680, 710, 730, 742, 786, 856, 874, 894, 918, 962, 986, 1038, 1246, 1270, 1406, 1434, 1442, 1446, 1542, 1558, 1586, 1598

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

Product_{d|n} d is the product of divisors of n (A007955).
If 1+ Product_{d|k} d for k > 2 is a prime p, then p-1 is a square.
With number 2 complement of A259021 with respect to A118369.
See A258897 - divisorial primes of the form 1 + Product_{d|a(n)} d.

			The number 24 is in sequence because A007955(24) = 331776 = 576^2 and simultaneously 331777 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A048943 (product of divisors of n is a square) and A118369 (numbers n such that Prod_{d|n} d + 1 is prime).

Cf. A007955, A258455, A259020, A259021, A258897.

[n: n in [1..2000] | &*(Divisors(n)) ne n^2 and IsSquare(&*(Divisors(n))) and IsPrime(&*(Divisors(n))+1)];

A007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2))
is(n)=my(t=A007955(n)); t>n^2 && issquare(t) && isprime(t+1) \\ Charles R Greathouse IV, Sep 01 2015

A259020 Numbers k such that 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, 576, 634, 674, 1094, 1174, 1294, 1306, 1354, 1366, 1546, 1654, 1766, 1774, 1894, 1966, 2026, 2126, 2174, 2326, 2594, 2654, 2746, 2916, 2974, 2986, 3046, 3106, 3134, 3136, 3214, 3254, 3274, 3314, 3326

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

The divisorial primes are primes of the form p = 1 + Product_{d|k} d = 1 + A007955(k) for some k.

Supersequence of A259021. Subsequence of A005574. First deviation from A259021 is at a(15).

			The number 6 is in sequence because prime 37 = 6^2 + 1 is prime of the form p = 1 + Product_{d|k} d = 1 + A007955(k) for k = 6.

OEIS wiki, Divisorial primes

Cf. A005574, A007955, A048943, A118369, A258455, A258897, A259021, A259023.

Set(Sort([1] cat [Floor(Sqrt(&*(Divisors(n)))): n in [3..10000] | IsPrime(&*(Divisors(n))+1)]));

A258456 Product of divisors of n is not a square.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 25, 28, 29, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 112, 113, 116, 117, 121, 124, 127, 131, 137

Offset: 1

Jaroslav Krizek, May 30 2015

Numbers n such that A007955(n) is not a square.
Complement of A048943.
2 is only number n from this sequence such that 1 + Product_{d|n} d is a prime.
If 1 + Product_{d|n} d for n > 2 is a prime p, then Product_{d|n} d is a square (see A258455).
m is a term if and only if m is not a fourth power and the number of divisors of m is not a multiple of 4. - Chai Wah Wu, Mar 09 2016

			9 is in sequence because product of divisors of 9 = 1*3*9 = 27 is not square.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A007955, A118369, A118370, A258455.

[n: n in [1..200] | not IsSquare(&*(Divisors(n)))];

Select[Range@ 137, ! IntegerQ@ Sqrt[Times @@ Divisors@ #] &] (* Michael De Vlieger, Jun 02 2015 *)

for(n=1,100,d=divisors(n);p=prod(i=1,#d,d[i]);if(!issquare(p),print1(n,", "))) \\ Derek Orr, Jun 12 2015

from gmpy2 import iroot
from sympy import divisor_count
A258456_list = [i for i in range(1,10**3) if not iroot(i,4)[1] and divisor_count(i) % 4] # Chai Wah Wu, Mar 10 2016

A364053 Zumkeller numbers whose divisors can be partitioned into two disjoint sets with equal products.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 78, 84, 88, 90, 96, 102, 104, 108, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 186, 198, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 258, 260, 264, 270, 276, 280, 282, 294, 306, 308, 312, 318, 330, 336, 340, 342, 348, 350, 352

Offset: 1

Ivan N. Ianakiev, Jul 04 2023

nonn

Intersection of A083207 and A048943.

			The divisors of 24 are {1,2,3,4,6,8,12,24}. They can be partitioned into two disjoint sets with equal sums, namely {4,6,8,12} and {1,2,3,24}, and two disjoint sets with equal products, namely {1,2,12,24} and {3,4,6,8}. So, 24 is a term and also a term of A347063.

Cf. A083207, A048943, A347063.

zQ[n_]:=Module[{d=Divisors[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]]; fQ[n_]:=IntegerQ[Sqrt[Times@@Divisors[n]]];
Select[Range[1000],And[fQ[#],zQ[#]]&] (*zQ and fQ by T. D. Noe at A083207 and A048943*)

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A258455 Divisorial primes: primes p of the form p = 1 + Product_{d|k} d for some k.

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A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree.

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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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A259023 Numbers n such that Product_{d|n} d = k^2 for some k > n and simultaneously number k^2 + 1 is a divisorial prime (A258455).

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A259020 Numbers k such that k^2 + 1 is a divisorial prime (A258455).

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A258456 Product of divisors of n is not a square.

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A364053 Zumkeller numbers whose divisors can be partitioned into two disjoint sets with equal products.

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