A030516 -id:A030516 - cp's OEIS frontend

A280298 Numbers with 67 divisors.

73786976294838206464, 30903154382632612361920641803529, 13552527156068805425093160010874271392822265625, 59768263894155949306790119265585619217025149412430681649, 539407797827634189900210968137750826278309533633974732577186113975161

Offset: 1

Omar E. Pol, Dec 31 2016

Also, 66th powers of primes.

More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 67.

			a(1) = 2^66, a(2) = 3^66, a(3) = 5^66, a(4) = 7^66, a(5) = 11^66.

OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280299, A280301, A261700.

Array[Prime[#]^66 &, {5}] (* Michael De Vlieger, Dec 31 2016 *)

PARI
```
a(n)=prime(n)^66
```

a(n) = A000040(n)^(67-1) = A000040(n)^66.

A000005(a(n)) = 67.

A280299 Numbers with 71 divisors.

Original entry on oeis.org

1180591620717411303424, 2503155504993241601315571986085849, 8470329472543003390683225006796419620513916015625, 143503601609868434285603076356671071740077383739246066639249, 7897469567994392174328988784504809847540729881935024059662581894710332201

Offset: 1

Omar E. Pol, Dec 31 2016

Also, 70th powers of primes.

More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 71.

			a(1) = 2^70, a(2) = 3^70, a(3) = 5^70, a(4) = 7^70, a(5) = 11^70.

OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280301, A261700.

Array[Prime[#]^70 &, {5}] (* Michael De Vlieger, Dec 31 2016 *)

PARI
```
a(n)=prime(n)^70
```

a(n) = A000040(n)^(71-1) = A000040(n)^70.

A000005(a(n)) = 71.

A280301 Numbers with 73 divisors.

Original entry on oeis.org

4722366482869645213696, 22528399544939174411840147874772641, 211758236813575084767080625169910490512847900390625, 7031676478883553279994550741476882515263791803223057265323201, 955593817727321453093807642925081991552428315714137911219172409259950196321

Offset: 1

Omar E. Pol, Dec 31 2016

Also, 72nd powers of primes.

More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 73.

			a(1) = 2^72, a(2) = 3^72, a(3) = 5^72, a(4) = 7^72, a(5) = 11^72.

OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280299, A280346, A280347, A280349, A261700.

Array[Prime[#]^72 &, {5}] (* Michael De Vlieger, Dec 31 2016 *)

PARI
```
a(n)=prime(n)^72
```

a(n) = A000040(n)^(73-1) = A000040(n)^72.

A000005(a(n)) = 73.

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1369, 1521, 1681, 1849, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 4225, 4489, 4761, 5041, 5329, 6241, 6561, 6889, 7225, 7569, 7921, 8649, 9025, 9409, 10201, 10609, 11449, 11881, 12321, 12769, 13225, 14161, 14641, 15129, 15625

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

nonn

Since for numbers m^2 in the sequence the width at the diagonal of SRS(m^2) is 1, the area m of its center part is odd so that this sequence is a proper subsequence of A016754 and since SRS(m^2) has an odd number of parts it is a proper subsequence of A319529. The smallest odd square not in this sequence is 225 = 15^2.  SRS(225) is {113, 177, 113}, its center part has maximum width 2, its width at the diagonal is 1.
The k+1 parts of SRS(p^(2k)), p an odd prime and k >= 0, through the diagonal including the center part have areas (p^(2k-i) + p^i)/2 for 0 <= i <= k. They form a strictly decreasing sequence. Since p^(2k) has 2k+1 divisors and SRS(p^(2k)) has 2k+1 parts, all of width 1 (A357581), the even powers of odd primes form a proper subsequence of A244579. For the subsequence of squares of odd primes p, SRS(p^2) consists of the 3 parts { (p^2 + 1)/2, p, (p^2 + 1)/2 } see A001248, A247687 and A357581.
The areas of the parts of SRS(m^2) need not be in descending order through the diagonal as a(112) = 275^2 = 75625 with SRS(75625) = (37813, 7565, 3443, 1525, 715, 738, 275, 738, 715, 1525, 3443, 7565, 37813) demonstrates.
An equivalent description of the sequence is: The center part of SRS(m^2) has width 1, m is odd, and A249223(m^2, m-1) = 0.
Conjectures (true for all a(n) <= 10^8):
(1) The central part of SRS(a(n)) is the minimum of all parts of SRS(a(n)), 1 <= n.
(2) The terms in this sequence are the squares of the terms in A244579.

			The center part of SRS(a(3)) = SRS(25) has area 5, all 3 parts have width 1, and 25 with 3 divisors also belongs to A244579.
The center part of SRS(a(7)) = SRS(169) has area 13, all 3 parts have width 1, and 169 with 3 divisors also belongs to A244579.
The center part of SRS(a(10)) = SRS(441) has area 21 and width 1, but the maximum width of SRS(441) is 2. Number 441 has 9 divisors and SRS(441) has 7 parts while 21 has 4 divisors and SRS(21) has 4 parts so that 21 is in A244579 while 441 is not.

Cf. A001248, A030514, A030516, A030629, A030631, A179645 (even prime power sequences).

Cf. A003056, A016754, A237591, A237593, A244579, A247687, A249223, A319529, A341969, A357581.

(* t237591 and partsSRS compute rows in A237270 and A237591, respectively *)
(* t249223 and widthPattern are also defined in A376829 *)
row[n_] := Floor[(Sqrt[8 n+1]-1)/2]
t237591[n_] := Map[Ceiling[(n+1)/#-(#+1)/2]-Ceiling[(n+1)/(#+1)-(#+2)/2]&, Range[row[n]]]
partsSRS[n_] := Module[{widths=t249223[n], legs=t237591[n], parts, srs}, parts=widths legs; srs=Map[Apply[Plus, #]&, Select[SplitBy[Join[parts, Reverse[parts]], #!=0&], First[#]!=0&]]; srs[[Ceiling[Length[srs]/2]]]-=Last[widths]; srs]
t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, row[n]]]
widthPattern[n_] := Map[First, Split[Join[t249223[n], Reverse[t249223[n]]]]]
centerQ[n_] := Module[{pS=partsSRS[n]}, Sqrt[n]==pS[[(Length[pS]+1)/2]]]/;OddQ[n]
widthQ[n_] := Module[{wP=SplitBy[widthPattern[n], #!=0&]}, wP[[(Length[wP]+1)/2]]]=={1}/;OddQ[n]
a377654[m_, n_] := Select[Map[#^2&, Range[m, n, 2]], centerQ[#]&&widthQ[#]&]/;OddQ[m]
a377654[1, 125]

A133542 Sum of sixth powers of five consecutive primes.

Original entry on oeis.org

1905628, 6732373, 30869213, 77899469, 225817709, 818869469, 1701546341, 4243135181, 8946193541, 15119520701, 25303912709, 46580770157, 86195577389, 132965847509, 217102866629, 334423935221, 463593800381, 664500722261

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=1905628 because 2^6+3^6+5^6+7^6+11^6=1905628.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133541, A133543.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@(Partition[Prime[Range[30]],5,1]^6)  (* Harvey P. Dale, Mar 13 2011 *)

a(n) = A133528(n) + A030516(n+4). - Michel Marcus, Nov 09 2013

A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

Original entry on oeis.org

32, 72, 108, 128, 200, 243, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1024, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3125, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4096, 4232, 4500, 4563, 4608

Offset: 1

V. Raman, Sep 07 2012

nonn

Powerful numbers (A001694) that are not squares of cubefree numbers (A004709), cubes of squarefree numbers (A062838), or 6th powers of primes (A030516). - Amiram Eldar, Feb 07 2023

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

Cf. A143610, A216426.

Cf. A001694, A002117, A004709, A013661, A013664, A062838, A085966.

With[{max = 5000}, Union[Table[i^2*j^3, {j, 2, max^(1/3)}, {i, 2, Sqrt[max/j^3]}] // Flatten]] (* Amiram Eldar, Feb 07 2023 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4, 3), for(a=2, sqrtint(lim\b^3), listput(v, a^2*b^3))); Set(v) \\ Charles R Greathouse IV, Jan 03 2014

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A216427(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j, b = isqrt(x), integer_nthroot(x,6)[0]
        l, c = 0, n+x-1+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(1, b+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Sum_{n>=1} 1/a(n) = 1 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - P(6) = 0.12806919584708298724..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

A137487 Numbers with 24 divisors.

Original entry on oeis.org

360, 420, 480, 504, 540, 600, 630, 660, 672, 756, 780, 792, 864, 924, 936, 990, 1020, 1050, 1056, 1092, 1120, 1140, 1152, 1170, 1176, 1188, 1224, 1248, 1350, 1368, 1380, 1386, 1400, 1404, 1428, 1470, 1500, 1530, 1540, 1596, 1632, 1638, 1650, 1656, 1710

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^23, p^2*q^7, p*q^2*r^3 (like 360, 504), p*q*r^5 (like 480, 672), p*q*r*s^2 (like 420, 660), p^3*q^5 (like 864) or p*q^11, where p, q, r and s are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[3000],DivisorSigma[0,#]==24&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==24 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=24.

A138408 a(n) = prime(n)^6 - prime(n).

Original entry on oeis.org

62, 726, 15620, 117642, 1771550, 4826796, 24137552, 47045862, 148035866, 594823292, 887503650, 2565726372, 4750104200, 6321363006, 10779215282, 22164361076, 42180533582, 51520374300, 90458382102, 128100283850

Offset: 1

Artur Jasinski, Mar 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers

Cf. A000040, A030516.

[NthPrime((n))^6 - NthPrime((n)): n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^6 - p], {n, 1, 50}]; a
#^6-#&/@Prime[Range[20]] (* Harvey P. Dale, Jun 09 2013 *)

forprime(p=2,1e3,print1(p^6-p", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A030516(n) - A000040(n). - Elmo R. Oliveira, Jan 28 2023

A138448 a(n) = (prime(n)^6 - prime(n)^2)/15.

Original entry on oeis.org

4, 48, 1040, 7840, 118096, 321776, 1609152, 3136368, 9869024, 39654832, 59166848, 171048336, 316673504, 421424080, 718614208, 1477623888, 2812035344, 3434691376, 6030558512, 8540018592, 10088948064, 16205829952, 21796024432

Offset: 1

Artur Jasinski, Mar 19 2008

Cf. A030516, A001248, A138441, A030514.

A138448:=n->(ithprime(n)^6-ithprime(n)^2)/15: seq(A138448(n), n=1..40); # Wesley Ivan Hurt, Apr 14 2017

a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^2)/15], {n, 1, 50}]; a

forprime(p=2,1e3,print1((p^6-p^2)/15", ")) \\ Charles R Greathouse IV, Jun 16 2011

From Elmo R. Oliveira, Jan 19 2023: (Start)
a(n) = (A030516(n) - A001248(n))/15.
a(n) = (2 * A138441(n))/15.
a(n) = (A001248(n) * (A030514(n) - 1))/15. (End)

A259417 Even powers of the odd primes listed in increasing order.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649

Offset: 1

Hartmut F. W. Hoft, Jun 26 2015

nonn

Each of the following sequences, p^(q-1) with p >= 2 and q > 2 primes, except their respective first elements, powers of 2, is a subsequence:
A001248(p) = p^2,  A030514(p) = p^4,  A030516(p) = p^6,
A030629(p) = p^10, A030631(p) = p^12, A030635(p) = p^16,
A030637(p) = p^18, A137486(p) = p^22, A137492(p) = p^28,
A139571(p) = p^30, A139572(p) = p^36, A139573(p) = p^40,
A139574(p) = p^42, A139575(p) = p^46, A173533(p) = p^52,
A183062(p) = p^58, A183085(p) = p^60.
See also the link to the OEIS Wiki.
The sequences A053182(n)^2, A065509(n)^4, A163268(n)^6 and A240693(n)^10 are subsequences of this sequence.
The odd numbers in A023194 are a subsequence of this sequence.

			a(11) = 5^4 = 625 is followed by a(12) = 3^6 = 729 since no even power of an odd prime falls between them.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..473
OEIS Wiki, Index entries for number of divisors.

a259417[bound_] := Module[{q, h, column = {}}, For[q = Prime[2], q^2 <= bound, q = NextPrime[q], For[h = 1, q^(2*h) <= bound, h++, AppendTo[column, q^(2*h)]]]; Prepend[Sort[column], 1]]
a259417[25000] (* data *)
With[{upto=25000},Select[Union[Flatten[Table[Prime[Range[2,Floor[ Sqrt[ upto]]]]^n,{n,0,Log[2,upto],2}]]],#<=upto&]] (* Harvey P. Dale, Nov 25 2017 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (P(2*k) - 1/2^(2*k)) = 1.21835996432366585110..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

cp's OEIS Frontend

A280298 Numbers with 67 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A280299 Numbers with 71 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A280301 Numbers with 73 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A133542 Sum of sixth powers of five consecutive primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A137487 Numbers with 24 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A138408 a(n) = prime(n)^6 - prime(n).

Views