A030629 -id:A030629 - cp's OEIS frontend

A183062 Numbers with 59 divisors.

288230376151711744, 4710128697246244834921603689, 34694469519536141888238489627838134765625, 10367793076318844190248738727596255138212949486449

Offset: 1

Omar E. Pol, Jul 31 2011

Also, 58th 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.

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.

a(n)=prime(n)^58 \\ Charles R Greathouse IV, Jul 31 2011

a(n) = A000040(n)^(59-1) = A000040(n)^58.

A000005(a(n)) = 59.

A183085 Numbers with 61 divisors.

Original entry on oeis.org

1152921504606846976, 42391158275216203514294433201, 867361737988403547205962240695953369140625, 508021860739623365322188197652216501772434524836001

Offset: 1

Omar E. Pol, Jul 31 2011

nonn

Also, 60th 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.

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.

PARI
```
a(n)=prime(n)^60
```

a(n) = A000040(n)^(61-1) = A000040(n)^60.

A000005(a(n)) = 61.

A351198 Sum of the 10th powers of the primes dividing n.

Original entry on oeis.org

0, 1024, 59049, 1024, 9765625, 60073, 282475249, 1024, 59049, 9766649, 25937424601, 60073, 137858491849, 282476273, 9824674, 1024, 2015993900449, 60073, 6131066257801, 9766649, 282534298, 25937425625, 41426511213649, 60073, 9765625, 137858492873, 59049, 282476273

Offset: 1

Wesley Ivan Hurt, Feb 04 2022

nonn

Inverse Möbius transform of n^10 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), this sequence (k=10).

Cf. A010051, A030629 (p^10).

Array[DivisorSum[#, #^10 &, PrimeQ] &, 50]
f[p_, e_] := p^10; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = vecsum(apply(x->x^10, factor(n)[, 1])); \\ Michel Marcus, Feb 05 2022

from sympy import primefactors
def A351198(n): return sum(p**10 for p in primefactors(n)) # Chai Wah Wu, Feb 04 2022

a(n) = Sum_{p|n, p prime} p^10.
G.f.: Sum_{k>=1} prime(k)^10 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022
Additive with a(p^e) = p^10. - Amiram Eldar, Jun 20 2022
a(n) = Sum_{d|n} d^10 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

Original entry on oeis.org

420, 630, 660, 780, 924, 990, 1020, 1050, 1092, 1140, 1170, 1380, 1386, 1428, 1470, 1530, 1540, 1596, 1638, 1650, 1710, 1716, 1740, 1820, 1860, 1932, 1950, 2070, 2142, 2220, 2244, 2380, 2394, 2436, 2460, 2508, 2550, 2574, 2580, 2604, 2610, 2652, 2660, 2790

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

Theorem 4 in Goldston-Graham-Pintz-Yildirim proves that a(n+1) = a(n) + 1 for infinitely many n. - Charles R Greathouse IV, Jul 17 2015, corrected by M. F. Hasler, Jul 17 2019

T. D. Noe, Table of n, a(n) for n = 1..1000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers (2008)
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,2}; Select[Range[4000],f]

is(n)=vecsort(factor(n)[,2])==[1, 1, 1, 2]~ \\ Charles R Greathouse IV, Jul 17 2015

Definition reworded by M. F. Hasler, Jul 17 2019

A261700 Numbers with 101 divisors.

Original entry on oeis.org

1267650600228229401496703205376, 515377520732011331036461129765621272702107522001, 7888609052210118054117285652827862296732064351090230047702789306640625, 3234476509624757991344647769100216810857203198904625400933895331391691459636928060001

Offset: 1

Omar E. Pol, Aug 28 2015

Also, 100th 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.

			a(1) = 2^100, a(2) = 3^100, a(3) = 5^100, a(4) = 7^100.

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.

a(n)=prime(n)^100 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(101-1) = A000040(n)^100.

A000005(a(n)) = 101.

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293

Offset: 0

Omar E. Pol, Sep 09 2018

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.

The partial sums of column k give the column k of A319076.

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

PARI
```
T(n, k) = prime(k)^n;
```

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

Original entry on oeis.org

1260, 1980, 2100, 2340, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3900, 4140, 4284, 4410, 4788, 4950, 5100, 5148, 5220, 5580, 5700, 5796, 5850, 6468, 6660, 6732, 6900, 7260, 7308, 7350, 7380, 7524, 7644, 7650, 7700, 7740, 7812, 7956, 8460, 8550, 8700

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296. - M. F. Hasler, Jul 17 2019

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,2}; Select[Range[10000], f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\60), t1=p^2; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A179691 Numbers p^5q^2r where p, q, r are 3 distinct primes.

Original entry on oeis.org

1440, 2016, 2400, 3168, 3744, 4704, 4860, 4896, 5472, 5600, 6624, 6804, 7840, 8352, 8800, 8928, 10400, 10656, 10692, 11616, 11808, 12150, 12384, 12636, 13536, 13600, 15200, 15264, 16224, 16524, 16992, 17248, 17568, 18400, 18468, 19296, 19360

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 (and A030514 .. A030629, A189982 .. A189990 etc, cf. A101296). - M. F. Hasler, Jul 17 2019

Subsequence of A175746 (numbers with 36 divisors).

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,5}; Select[Range[20000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179691(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 n+x-sum(primepi(x//(p**5*q**2)) for p in primerange(integer_nthroot(x,5)[0]+1) for q in primerange(isqrt(x//p**5)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(x//p**7) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,8)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Name improved by M. F. Hasler, Jul 17 2019

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Giovanni Teofilatto, Oct 16 2009

nonn

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009

The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..6955

Cf. A000005.

Cf. A073915. - R. J. Mathar, Oct 26 2009

[n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019

Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)

isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A280298 Numbers with 67 divisors.

Original entry on oeis.org

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.

cp's OEIS Frontend

A183062 Numbers with 59 divisors.

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A183085 Numbers with 61 divisors.

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A351198 Sum of the 10th powers of the primes dividing n.

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A189982 Numbers with prime signature (2,1,1,1), i.e., factorization p*q*r*s^2 with distinct primes p, q, r, s.

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Extensions

A261700 Numbers with 101 divisors.

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A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

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PARI

Formula

A179690 Numbers of the form p^2*q^2*r*s where p, q, r, and s are distinct primes.

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Mathematica

PARI

A179691 Numbers p^5*q^2*r where p, q, r are 3 distinct primes.

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A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

A179691 Numbers p^5q^2r where p, q, r are 3 distinct primes.