A030629 -id:A030629 - cp's OEIS frontend

A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

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

Offset: 1

V. Raman, Sep 07 2012

nonn

Terms of A216427 that are not 5th powers of squarefree numbers (A113850) and not 10th powers of primes (A030629). - Amiram Eldar, Feb 07 2023

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

Cf. A143610.
Subsequence of A216427. - Zak Seidov, Jan 03 2014
Cf. A002117, A013661, A013663, A013664, A013668, A030629, A085966, A113850.

With[{upto=4000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^3,#[[2]]^2 #[[1]]^3}& /@ Subsets[Range[2,Surd[upto,2]],{2}]]],#<=upto&]](* Harvey P. Dale, Jan 04 2014 *)
pMx = 25; mx = 2^3 pMx^2; t = Flatten[Table[If[a != b, a^2 b^3, 0], {a, 2, mx^(1/2)}, {b, 2, mx^(1/3)}]]; Union[Select[t, 0 < # <= mx &]] (* T. D. Noe, Jan 02 2014 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4,3), for(a=2, sqrtint(lim\b^3), if(a!=b, listput(v, a^2*b^3)))); Set(v) \\ Charles R Greathouse IV, Jan 02 2014

from math import isqrt
from sympy import integer_nthroot, mobius, primepi
def A216426(n):
    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, a, d = isqrt(x), integer_nthroot(x,6)[0], integer_nthroot(x,5)[0], integer_nthroot(x,10)[0]
        l, c = 0, n+x-2+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(d+1, b+1))+primepi(d)+sum(mobius(k)*(a//k**2+j//k**3) for k in range(1, d+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = sum(mobius(k)*((k2-1)//k**2) for k in range(1, isqrt(k2-1)+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) = 2 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - zeta(5)/zeta(10) - P(6) - P(10) = 0.09117811499514578262..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

Name corrected by Charles R Greathouse IV, Jan 02 2014

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236

Offset: 1

Amiram Eldar, Feb 19 2025

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

A182681 a(n) is the smallest n-digit number with exactly 11 divisors, a(n) = 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 1024, 59049, 0, 9765625, 0, 282475249, 0, 25937424601, 137858491849, 2015993900449, 41426511213649, 420707233300201, 4808584372417849, 13422659310152401, 174887470365513049, 1822837804551761449, 15516041187205853449, 110462212541120451001, 1091533853073393531649

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) is the smallest n-digit number of the form p^10 (p = prime), a(n) = 0 if no such number exists.

See A182682(n) - the largest n-digit number with exactly 11 divisors.

Cf. A030629 (numbers with 11 divisors).

a(n)=(10^n>n=nextprime(sqrtn(10^(n-1),10))^10)*n  \\ M. F. Hasler, Nov 27 2010

A000005(a(n)) = 11.

a(n) = min { 0 } union ( A030629 intersect [10^(n-1), 10^n-1] )

A182682 a(n) = the largest n-digit number with exactly 11 divisors, a(n) = 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 1024, 59049, 0, 9765625, 0, 282475249, 0, 25937424601, 137858491849, 6131066257801, 41426511213649, 819628286980801, 4808584372417849, 52599132235830049, 713342911662882601, 9468276082626847201, 73742412689492826049, 339456738992222314849, 9099059901039401398249

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) = the largest n-digit number of the form p^10 (p = prime), a(n) = 0 if no such number exists.

Cf. A030629, A182681.

A000005(a(n)) = 11.

a(n) >= A182681(n).

A236216 Sum of the tenth powers of the first n primes.

Original entry on oeis.org

1024, 60073, 9825698, 292300947, 26229725548, 164088217397, 2180082117846, 8311148375647, 49737659589296, 470444892889497, 1290073179870298, 6098657552288147, 19521316862440548, 41132799175724797, 93731931411554846, 268619401777067895, 779736155077709296

Offset: 1

Robert Price, Jan 20 2014

Robert Price, Table of n, a(n) for n = 1..1000
OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
Partial sums of A030629.

Table[Sum[Prime[k]^10, {k, n}], {n, 1000}]

a(n) = Sum_{k=1..n} prime(k)^10.

A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

Original entry on oeis.org

1, 6, 8, 12, 10, 14, 2, 15, 48, 18, 20, 3, 28, 21, 5, 7, 11, 4, 22, 24, 32, 13, 17, 9, 19, 23, 26, 29, 27, 25, 30, 33, 31, 37, 40, 34, 41, 35, 49, 43, 47, 16, 38, 42, 39, 46, 44, 53, 51, 59, 45, 54, 56, 60, 50, 61, 66, 52, 55, 57, 67, 71, 58, 62, 72, 63, 192, 65

Offset: 1

David James Sycamore, Jan 14 2022

nonn

If d(j(n)) is prime p then d(a(n+1)) must be properly divisible by p. In practice the proper divisor for computation of a(n+1) toggles between d(j(n)) and d(k).
Conjecture: This is a permutation of the positive integers. Numbers with the same number (tau) of divisors appear in their natural orders (e.g., primes, semiprimes, squares).
The plot, after the first few terms, resolves itself into points tightly packed on and around a straight line of slope 1, with exceptional points appearing as significant upward or downward "spikes".
When d(j(n)) is prime p appearing for the first time in the sequence J = {d(j(a(n)), n>=1}, then a(n+1) is the smallest number with 2p divisors, which produces a significantly large upward spike above the straight line (6, 12, 48, 192, 3072, 12288, ...).
When d(j(a(n)) is 2p, seen for the first time in J, then a(n+1) is the smallest number with p divisors, which produces a large downward spike, below the straight line (2, 4, 16, 64, 1024, 4096, ...).
The sequence of fixed points starts: 1, 46, 69, 74, 110, 140, 142, 152, 154, 178, ... apparently becoming denser as n increases.

			a(1)=1, so j(1)=2, d(j(1))=2, a prime, so we need the smallest unused k such that d(k) is properly divisible by 2, hence a(2)=6.
a(2)=6, j(2)=4, d(j(2))=3, a prime so we need the smallest unused k such that d(k) is properly divisible by 3, hence a(3)=8.

Michael De Vlieger, Table of n, a(n) for n = 1..10001
Michael De Vlieger, Scatterplot of a(n), n = 1..256, color coded to show primes in red, evens in blue, 2 in magenta, odd composites in black. A gold highlight indicates terms such that (a(n)+1) | a(n+1). Outlying points are annotated.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16. Terms in A3680(p) such that p is prime appear annotated in red, and those in A061286 appear in blue.

Cf. A000040, A001248, A030514, A054753, A030516, A030626, A085986, A178739, A030629, A030630, A030631, A030632, A030633, A030634, A005179, A061286, A003680.

Block[{c, d, j, k, u, nn}, nn = 120; j = u = 2; c[] = 0; c[1] = 1; Do[d[i] = DivisorSigma[0, i], {i, 2^(Ceiling@ Log2[nn] + 3)}]; {1}~Join~Reap[Do[k = u; While[Nand[Or[Divisible[d[j], d[k]], Divisible[d[k], d[j]]], d[j] != d[k], c[k] == 0], k++]; Sow[k]; c[k] = i; j = k + 1; If[k == u, While[c[u] != 0, u++]], {i, nn}] ][[-1, -1]] ] (* _Michael De Vlieger, Jan 14 2022 *)

isok(k, last, set) = if (!setsearch(set, k), my(ndk=numdiv(k), ndl=numdiv(last+1)); (ndl != ndk) && ((!(ndk % ndl)) || (!(ndl % ndk))));
lista(nn) = {my(last = 1, list = List(last), set = Set(list)); for (n=2, nn, my(k=1); while (!isok(k, last, set), k++); listput(list, k); set = Set(list); last = k;); Vec(list);} \\ Michel Marcus, Jan 15 2022

More terms from Michael De Vlieger, Jan 14 2022

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378

Offset: 1

Amiram Eldar, Feb 19 2025

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));

A096329 Prime(p^10) where p is the n-th prime.

Original entry on oeis.org

8161, 733561, 174978379, 6065997263, 679646318387, 3851884421117, 61935696080369, 195412630499981, 1402090078377899, 15243760266373817, 30260601850590289, 186278471527230527, 534126386413471121, 870547653763149821, 2166779476441061129, 7419756995015118023

Offset: 1

Cino Hilliard, Aug 02 2004

a(5)-a(7) from the Nth Prime Page.

			2^10 = 125, prime(1024) = 8161.

Andrew Booker, Nth Prime Page.

Cf. A000040, A030629.

Table[ Prime[ Prime[n]^10], {n, 7}] (* Robert G. Wilson v, Aug 07 2004 *)

a(n)=prime(prime(n)^10) \\ Charles R Greathouse IV, Nov 02 2014

a(n) ~ 10n^10 log^11 n. - Charles R Greathouse IV, Nov 02 2014

a(n) = A000040(A030629(n)). - Amiram Eldar, Jul 11 2024

Edited and extended by Robert G. Wilson v, Aug 07 2004

a(8)-a(16) from Charles R Greathouse IV, Nov 02 2014

A280350 Numbers with 97 divisors.

Original entry on oeis.org

79228162514264337593543950336, 6362685441135942358474828762538534230890216321, 12621774483536188886587657044524579674771302961744368076324462890625, 1347137238494276547832006567721872890819326613454654477690085519113574118965817601, 9412343651268540526001186511911506574868063110469548823950876000379062365652829504091329792873336961

Offset: 1

Omar E. Pol, Jan 02 2017

Also, 96th 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 = 97.

			a(1) = 2^96, a(2) = 3^96, a(3) = 5^96, a(4) = 7^96, a(5) = 11^96.

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, A280301, A280346, A280347, A280349, A261700.

[NthPrime(n)^96: n in [1..5]]; // Vincenzo Librandi, Jan 06 2017

With[{p = 25}, Table[Prime[n]^(Prime[p] - 1), {n, 5}]] (* Michael De Vlieger, Jan 02 2017 *)

PARI
```
a(n)=prime(n)^96
```

a(n) = A000040(n)^(97-1) = A000040(n)^96.

A000005(a(n)) = 97.

cp's OEIS Frontend

A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

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A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

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A182681 a(n) is the smallest n-digit number with exactly 11 divisors, a(n) = 0 if no such number exists.

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A182682 a(n) = the largest n-digit number with exactly 11 divisors, a(n) = 0 if no such number exists.

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A236216 Sum of the tenth powers of the first n primes.

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A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

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A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

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A096329 Prime(p^10) where p is the n-th prime.

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