A030516 -id:A030516 - cp's OEIS frontend

A030627 Numbers with 9 divisors.

36, 100, 196, 225, 256, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6561, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129

Offset: 1

Jeff Burch

nonn

Numbers of the form p^8 (8th row of A120458) or p^2*r^2 (A085986), where p and r are distinct primes. - R. J. Mathar, Mar 01 2010

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000005, A030515, A030516, A030626, A085986, A120458, A179645.

Select[Range[90000],DivisorSigma[0,#]==9&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==9 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, primerange
def A030627(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=isqrt(x))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(isqrt(s)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n)) = 9. - Juri-Stepan Gerasimov, Oct 10 2009

Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 + P(8) = 0.0678286..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A139572 Numbers with 37 divisors.

Original entry on oeis.org

68719476736, 150094635296999121, 14551915228366851806640625, 2651730845859653471779023381601, 30912680532870672635673352936887453361, 12646218552730347184269489080961456410641

Offset: 1

Omar E. Pol, May 07 2008

36th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139573.

A139572 := proc(n) ithprime(n)^36 ; end proc: seq(A139572(n),n=1..10) ; # R. J. Mathar, Feb 05 2010

Prime[Range[10]]^36 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^36 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(37-1) = A000040(n)^36.

More terms from R. J. Mathar, Feb 05 2010

A133528 Sum of sixth powers of four consecutive primes.

Original entry on oeis.org

134067, 1905564, 6731644, 30853588, 77781820, 224046148, 814042660, 1677408772, 4196089300, 8798157652, 14524697380, 24416409028, 44015043748, 81445473148, 126644484460, 206323651300, 312259574092, 421413266740

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=134067 because 2^6+3^6+5^6+7^6=134067.

Cf. A034963, A133524, A133525, A133526, A133527.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a, {n, 1, 100}]

a(n) = A133533(n) + A030516(n+3). - Michel Marcus, Nov 08 2013

A139574 Numbers with 43 divisors.

Original entry on oeis.org

4398046511104, 109418989131512359209, 227373675443232059478759765625, 311973482284542371301330321821976049, 54763699237492901685126120802225273763666521, 61040881526285814362156628321386486455989674569

Offset: 1

Omar E. Pol, May 09 2008

42nd powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139575.

Prime[Range[9]]^42 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^42 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(43-1)=A000040(n)^42.

More terms from R. J. Mathar, May 11 2008

A072499 Product of divisors of n which are <= n^(1/2).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 8, 1, 6, 1, 8, 3, 2, 1, 24, 5, 2, 3, 8, 1, 30, 1, 8, 3, 2, 5, 144, 1, 2, 3, 40, 1, 36, 1, 8, 15, 2, 1, 144, 7, 10, 3, 8, 1, 36, 5, 56, 3, 2, 1, 720, 1, 2, 21, 64, 5, 36, 1, 8, 3, 70, 1, 1152, 1, 2, 15, 8, 7, 36, 1, 320, 27, 2, 1, 1008, 5, 2, 3, 64, 1

Offset: 1

Amarnath Murthy, Jul 20 2002

nonn

a(1) = 1 and a(24) = 24. For each pair of primes p,q such that p < q < p^2, if n = p^3*q, then a(n) = n. There are others as well; e.g., a(40) = 40. - Don Reble, Aug 02 2002
Row products of the table in A161906. - Reinhard Zumkeller, Mar 08 2013
It appears that the fixed points belong to 3 categories: p^6 (A030516), p^3*q, or p*q*r. - Michel Marcus, May 16 2014

			a(20) = 8. The divisors of 20 are 1,2,4,5,10 and 20. a(20) = 1*2*4 = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072500, A072501, A161906.

Cf. A072504, A066839.

a072499 = product . a161906_row  -- Reinhard Zumkeller, Mar 08 2013

a[n_] := Times @@ Select[Divisors[n], #^2 <= n &]; Array[a, 100] (* Amiram Eldar, Jul 31 2022 *)

a(n) = my(d = divisors(n)); prod(i=1, #d, if (d[i]^2 <= n, d[i], 1)); \\ Michel Marcus, May 16 2014

from math import prod
from itertools import takewhile
from sympy import divisors
def A072499(n): return prod(takewhile(lambda x:x**2<=n,divisors(n))) # Chai Wah Wu, Dec 19 2023

More terms from Sascha Kurz, Feb 02 2003

A139573 Numbers with 41 divisors.

Original entry on oeis.org

1099511627776, 12157665459056928801, 9094947017729282379150390625, 6366805760909027985741435139224001, 452592555681759518058893560348969204658401

Offset: 1

Omar E. Pol, May 07 2008

40th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572.

Prime[Range[9]]^40 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^40 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(41-1)=A000040(n)^40.

More terms from Jon E. Schoenfield, May 18 2010

A139575 Numbers with 47 divisors.

Original entry on oeis.org

70368744177664, 8862938119652501095929, 142108547152020037174224853515625, 749048330965186233494494102694564493649, 801795320536133573571931534665380233173841533961

Offset: 1

Omar E. Pol, May 09 2008

46th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574.

Prime[Range[9]]^46 (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

a(n)=prime(n)^46 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(47-1)=A000040(n)^46.

More terms from R. J. Mathar, May 11 2008

A067004 Number of numbers <= n with same number of divisors as n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 5, 1, 6, 4, 5, 1, 7, 2, 8, 3, 6, 7, 9, 1, 3, 8, 9, 4, 10, 2, 11, 5, 10, 11, 12, 1, 12, 13, 14, 3, 13, 4, 14, 6, 7, 15, 15, 1, 4, 8, 16, 9, 16, 5, 17, 6, 18, 19, 17, 1, 18, 20, 10, 1, 21, 7, 19, 11, 22, 8, 20, 2, 21, 23, 12, 13, 24, 9, 22, 2, 2, 25, 23, 3, 26, 27

Offset: 1

Henry Bottomley, Dec 21 2001

nonn

			a(10)=3 since 6,8,10 each have four divisors. a(11)=5 since 2,3,5,7,11 each have two divisors.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000005, A008479, A058933, A067003. Inverse of A000040, A001248, A030513, A030514, A030515, A030516, A030626, A030627, A030628, A030629, A030630, A030631, A030632, A030633, A030634, A030635, A030636, A030637, A030638 etc.
Cf. also A079788, A138009.
A047983(n) = a(n)-1.

N:= 1000: # to get a(1) to a(N)
R:= Vector(N):
for n from 1 to N do
  v:= numtheory:-tau(n);
  R[v]:= R[v]+1;
  A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, May 04 2015

b[_] = 0;
a[n_] := a[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

a(n)=my(d=numdiv(n)); sum(k=1,n,numdiv(k)==d) \\ Charles R Greathouse IV, Sep 02 2015

Ordinal transform of A000005. - Franklin T. Adams-Watters, Aug 28 2006

a(A000040(n)^(p-1)) = n if p is prime. - Robert Israel, May 04 2015

A173533 Numbers with 53 divisors.

Original entry on oeis.org

4503599627370496, 6461081889226673298932241, 2220446049250313080847263336181640625, 88124787089723195184393736687912818113311201, 1420429319844313329730664601483335671261683881745483121, 8415003868347247618489696679505181495471801448798649088081

Offset: 1

Omar E. Pol, Oct 16 2010

52nd powers of primes.

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575.

Prime[Range[9]]^52 (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

a(n)=prime(n)^52 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(53-1) = A000040(n)^52.

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

Original entry on oeis.org

2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23

Offset: 1

Clark Kimberling, Dec 14 2010

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
The monotonic ordering of this sequence, prefixed by 1, is A000961.
The joint-rank array of this sequence is A182869.
Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
A319075 extends the array with 0th powers.
Subtable of A242378, A284457, A329332.

TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

From Peter Munn, Dec 29 2019: (Start)
A(i,j) = A182945(j,i) = A319075(j,i).
A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
A(i,i) = A062457(i).
(End)

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022

cp's OEIS Frontend

A030627 Numbers with 9 divisors.

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A139572 Numbers with 37 divisors.

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A133528 Sum of sixth powers of four consecutive primes.

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A139574 Numbers with 43 divisors.

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A072499 Product of divisors of n which are <= n^(1/2).

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A139573 Numbers with 41 divisors.

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A139575 Numbers with 47 divisors.

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