A007422 -id:A007422 - cp's OEIS frontend

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

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

A066423 Composite numbers n such that the product of proper divisors of the n does not equal n.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 24, 25, 28, 30, 32, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 121, 124, 126, 128, 130, 132

Offset: 1

Robert G. Wilson v, Dec 26 2001

nonn

A084115(a(n))>1; complement of A084116. - Reinhard Zumkeller, May 12 2003

			The fourth composite number is 9. Its proper or aliquot divisors are 1 and 3. The product of 1 and 3 equals 3 which is not equal to 9. Therefore 9 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A048741, A007422.

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Do[m = Composite[n]; If[ Apply[ Times, Drop[ Divisors[m], -1]] != m, Print[m]], {n, 1, 100} ]
Select[Range[150],CompositeQ[#]&&Times@@Most[Divisors[#]]!=#&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 18 2020 *)

is(n)=my(d=numdiv(n)); d>4 || d==3 \\ Charles R Greathouse IV, Oct 15 2015

A189974 Numbers m such that d(m-1) = d(m+1) = 4, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

7, 9, 34, 56, 86, 92, 94, 124, 142, 144, 160, 184, 186, 202, 204, 214, 216, 218, 220, 236, 248, 266, 300, 302, 304, 320, 322, 328, 340, 342, 392, 394, 412, 414, 416, 446, 452, 470, 472, 516, 518, 534, 536, 544, 552, 580, 582, 590, 634, 668, 670, 680, 686

Offset: 1

Juri-Stepan Gerasimov, May 03 2011

nonn

Numbers m such that m-1 and m+1 are both multiplicatively perfect numbers A007422.
Conjecture: all terms but the first two are even numbers. - Harvey P. Dale, Jul 21 2025
Proof of conjecture: if m is odd and > 10 then either m-1 or m+1 is divisible by 4 and > 8 as well. Let t be the number from {m-1, m+1} divisible by 4. Then t is a power of 2 that is > 8 and so has more than two divisors or it has an odd prime divisor such that it has more than 4 divisors. Both exclude the odd m > 8 from the sequence. - David A. Corneth, Aug 05 2025

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000005, A007422.

with(numtheory): A189974 := proc(n) option remember: local k: if(n=1)then return 7:else k:=procname(n-1)+1: do if(tau(k-1)=4 and tau(k+1)=4)then return k: fi: k:=k+1: od: fi: end: seq(A189974(n),n=1..60); # Nathaniel Johnston, May 04 2011

Select[Range[2, 754], DivisorSigma[0, # - 1] == DivisorSigma[0, # + 1] == 4 &]
Flatten[Position[Partition[DivisorSigma[0,Range[700]],3,1],?(#[[1]]==#[[3]]==4&),1,Heads->False]]+1 (* _Harvey P. Dale, Jul 21 2025 *)

A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 8, 16, 11, 49, 10, 81, 12, 13, 121, 14, 625, 18, 64, 17, 169, 15, 2401, 20, 729, 24, 19, 289, 21, 14641, 28, 15625, 30, 36, 23, 361, 22, 28561, 32, 117649, 40, 100, 48, 29, 529, 26, 83521, 44, 1771561, 42, 196, 80, 1024, 31, 841, 27

Offset: 1

Leroy Quet, May 31 2006

From Peter Munn, May 17 2023: (Start)
As a square array A(n,m), n, m >= 1, read by ascending antidiagonals, A(n,m) is the n-th positive integer with m+1 divisors.
Thus both formats list the numbers with m+1 divisors in their m-th column. For the corresponding sequences giving numbers with a specific number of divisors see the index entries link.
(End)

			Looking at the 4th row, 7 is the 4th positive integer with 2 divisors, 25 is the 3rd positive integer with 3 divisors, 8 is the 2nd positive integer with 4 divisors and 16 is the first positive integer with 5 divisors. So the 4th row is (7,25,8,16).
The triangle T(n,m) begins:
  n\m:    1     2     3     4     5     6     7
  ---------------------------------------------
   1 :    2
   2 :    3     4
   3 :    5     9     6
   4 :    7    25     8    16
   5 :   11    49    10    81    12
   6 :   13   121    14   625    18    64
   7 :   17   169    15  2401    20   729    24
  ...
Square array A(n,m) begins:
  n\m:     1      2      3       4      5  ...
  --------------------------------------------
   1 :     2      4      6      16     12  ...
   2 :     3      9      8      81     18  ...
   3 :     5     25     10     625     20  ...
   4 :     7     49     14    2401     28  ...
   5 :    11    121     15   14641     32  ...
  ...

Index entries for sequences related to divisors of numbers

Columns: A000040, A001248, A007422, A030514, A030515, A030516, A030626, A030627, A030628, ... (see the index entries link for more).
Cf. A073915.
Diagonals (equivalently, rows of the square array) start: A005179\{1}, A161574.
Cf. A091538.

t[n_, m_] := Block[{c = 0, k = 1}, While[c < n + 1 - m, k++; If[DivisorSigma[0, k] == m + 1, c++ ]]; k]; Table[ t[n, m], {n, 11}, {m, n}] // Flatten (* Robert G. Wilson v, Jun 07 2006 *)

More terms from Robert G. Wilson v, Jun 07 2006

A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

Original entry on oeis.org

0, 1, 1, -1, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 2, -1, 1, 3, 1, 3, 2, 2, 1, 4, -1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, -1, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, -1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, -1, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, -1, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, -1

Offset: 1

Juri-Stepan Gerasimov, Sep 13 2017

sign

If the number of divisors (nd) of n > 1 is odd, then a(n) = -1, else a(n) = nd/2. - Michel Marcus, Sep 14 2017
First occurrence of k beginning with -1 is A293570(r). - Robert G. Wilson v, Oct 10 2017
Records occur for A293570(r): 4, 6, 12, 24, 48, 60, 192, 240, 3072, 12288, 196608, 786432, 12582912, 805306368, etc. - Robert G. Wilson v, Oct 10 2017

			a(10) = 2 because divisors of 10 are 1,2,5,10 with product 100 = 10^2.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Numbers n such that the product of divisors of n is n^k: A000040 (k = 1), A007422 (k = 2), A162947 (k = 3), A111398 (k = 4), A030628 (k = 5), A030630 (k = 6).

Cf. A000037, A000290, A056924, A007955, A293570.

Table[Boole[n == 1] + If[OddQ@ #, -1, #/2] &@ DivisorSigma[0, n], {n, 100}] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = if (n==1, 0, my(nd = numdiv(n)); if (nd % 2, -1, nd/2)); \\ Michel Marcus, Sep 14 2017

a(n)=my(k=numdiv(n)); if(k%2, if(n>1, -1, 0), k/2) \\ Charles R Greathouse IV, Sep 19 2017

a(1) = 0, a(A000290(n+1)) = -1, a(A000037(n+1)) = A056924(A000037(n+1)), where A000290 = the squares and A000037 = the nonsquares.

Definition corrected by Charles R Greathouse IV, Sep 13 2017

A338483 a(n) is the smallest number having n smaller numbers with the same number of divisors.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Ivan N. Ianakiev, Oct 30 2020

nonn

Inspired by A047983.

Are there prime terms greater than 31?

			The smallest number having two smaller numbers (2 and 3) with the same number of divisors is 5, so a(2) is 5.

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

Cf. A000005, A007422, A030513, A047983.

N:= 500: # for terms before the first term > N
T:= map(numtheory:-tau, [$1..N]):
M:= max(T):
V:= Vector(M):
for n from 1 to N do
  v:= T[n];
  V[v]:= V[v]+1;
  if not assigned(R[V[v]]) then R[V[v]]:= n fi
od:
for nn from 1 while assigned(R[nn]) do od:
seq(R[i],i=2..nn-1); # Robert Israel, Oct 30 2020

f[n_]:=With[{tau=DivisorSigma[0,n]},Length[Select[Range[n-1],DivisorSigma[0,#]==tau&]]];t=Table[f[n],{n,1,300}]; a[n_]:=FirstPosition[t,n]; Rest[a/@Range[0,65]]//Flatten (* f(n) by Jean-François Alcover at A047983 *)

f(n) = {my(d=numdiv(n)); sum(k=1, n-1, (numdiv(k)==d))} \\ A047983
a(n) = my(k=1); while (f(k)!= n, k++); k; \\ Michel Marcus, Oct 30 2020

A047983(a(n)) = n. - Rémy Sigrist, Dec 06 2020

A385350 Numbers j such that the product of odd proper divisors of j is j.

Original entry on oeis.org

1, 15, 21, 27, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299

Offset: 1

Ilya Gutkovskiy, Jun 26 2025

Fixed points of A385349.
Odd terms in A007422.
Also 1 with odd numbers with exactly 4 divisors. - David A. Corneth, Jun 26 2025

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A007422, A030513, A073582, A385349.

q:= n-> n=1 or n::odd and numtheory[tau](n)=4:
select(q, [$1..500])[];  # Alois P. Heinz, Jun 26 2025

A385349[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Select[Range[300], A385349[#] == # &]

isok(k) = vecprod(select((x->((x%2)==1) && (xMichel Marcus, Jun 26 2025

is(n) = (n == 1) || (bitand(n, 1) && numdiv(n) == 4) \\ David A. Corneth, Jun 26 2025

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A385350(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jun 27 2025

A189986 Numbers of the form 4k+1 having exactly 4 divisors.

Original entry on oeis.org

21, 33, 57, 65, 69, 77, 85, 93, 125, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 301, 305, 309, 321, 329, 341, 365, 377, 381, 393, 413, 417, 437, 445, 453, 469, 473, 481, 485, 489, 493, 497, 501, 505, 517, 533, 537, 545

Offset: 1

Juri-Stepan Gerasimov, May 03 2011

nonn

Intersection of A016813 and A030513; subsequence of A007422.

Numbers p*q with p == q (mod 4) together with p^3 with p == 1 (mod 4), p and q distinct primes. - Charles R Greathouse IV, May 03 2011

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A007422, A016813, A030513.

[ n: n in [1..600 by 4] | #Divisors(n) eq 4 ]; // Klaus Brockhaus, May 04 2011

Select[4Range[200]+1,DivisorSigma[0,#]==4&] (* Harvey P. Dale, May 11 2011 *)

Corrected (497 inserted) by Klaus Brockhaus, May 04 2011

A277169 Product of squares of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 36, 1, 64, 9, 100, 1, 20736, 1, 196, 225, 4096, 1, 104976, 1, 160000, 441, 484, 1, 191102976, 25, 676, 729, 614656, 1, 729000000, 1, 1048576, 1089, 1156, 1225, 78364164096, 1, 1444, 1521, 4096000000, 1, 5489031744, 1, 3748096, 4100625, 2116, 1, 28179280429056, 49, 6250000

Offset: 1

Ilya Gutkovskiy, Oct 19 2016

			a(6) = 36 because 6 has 3 proper divisors {1,2,3} and 1^2*2^2*3^2 = 36.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product
Eric Weisstein's World of Mathematics, Proper divisors

Cf. A000290, A001248, A007422, A007955, A007956, A008578, A062758.

seq(n^(numtheory:-tau(n)-2), n=1..50); # Robert Israel, Nov 13 2016

Table[n^(DivisorSigma[0, n] - 2), {n, 1, 50}]

a(n) = n^(sigma_0(n)-2).
a(n) = n^A000005(n)/A000290(n).
a(n) = A000290(A007956(n))/A000290(n).
a(n) = A000290(A007955(n)/n)/A000290(n).
a(n) = A062758(n)/A000290(n).
a(n) = 1 if n is prime or n = 1 (A008578).
a(n) = n if n is square of prime (A001248).
a(n) = n^2 if n is multiplicatively perfect number (A007422).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A066423 Composite numbers n such that the product of proper divisors of the n does not equal n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A189974 Numbers m such that d(m-1) = d(m+1) = 4, where d(k) is the number of divisors of k (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A338483 a(n) is the smallest number having n smaller numbers with the same number of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs