A143610 -id:A143610 - cp's OEIS frontend

A377846 Powerful numbers that are not divisible by the cubes of more than one distinct prime.

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Amiram Eldar, Nov 09 2024

Subsequence of A377821 and first differs from it at n = 33: A377821(33) = 432 = 2^4 * 3^3 is not a term of this sequence.

Numbers whose prime factorization has exponents that are all larger than 1 and no more than one exponent is larger than 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Complement of A376936 within A001694.
Subsequence of A377821.
Subsequences: A143610, A377847.
Cf. A082020.

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, Length[e] == 1 || e[[-2]] == 2]; With[{max = 1300}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || q[#] &]]

is(k) = if(k == 1, 1, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && (#e == 1 || e[#e - 1] == 2));

Sum_{n>=1} 1/a(n) = (15/Pi^2) * (1 + Sum_{p prime} 1/((p-1)*(p^2+1))) = 1.92240214785252516795... .

A166988 Products n of a square of a prime and a cube of a prime such that n-1 and n+1 are semiprimes.

Original entry on oeis.org

392, 14792, 19652, 48668, 55112, 197192, 291848, 783752, 908552, 963272, 1203052, 1541768, 1670792, 5081672, 5903048, 8193532, 9732872, 10089032, 10285412, 12241352, 13333448, 13960328, 14087432, 14818568, 15882248, 16290632

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 26 2009

nonn

Intersection of A143610 and A124936.

			392 = 7^2*2^3; 391 = 17*23 and 393 = 3*131 are semiprimes, hence 392 is in the sequence.
14792 = 2^3*43^2 is in the sequence because 14791=7*2113 and 14793=3*4931 are semiprimes.

Vincenzo Librandi, Table of n, a(n) for n = 1..112

Cf. A001248 (squares of primes), A030078 (cubes of primes), A001358 (semiprimes).

f2[n_]:=Last/@FactorInteger[n]=={2,3}||Last/@FactorInteger[n]=={3,2}; f1[n_]:=Plus@@Last/@FactorInteger[n]==2; lst={};Do[If[f2[n],If[f1[n-1]&&f1[n+1],AppendTo[lst,n]]],{n,10!}];lst
With[{prs=Prime[Range[300]]},Union[Select[Times@@@Tuples[{prs^2, prs^3}], PrimeOmega[#-1] == PrimeOmega[#+1]==2&]]] (* Harvey P. Dale, Aug 13 2013 *)

{m=17000000; v=[]; forprime(j=2, sqrtint(m\8), a=j^2; g=sqrtn(m\a, 3); forprime(k=2, g, n=a*k^3; if(nKlaus Brockhaus, Oct 29 2009

Edited by Klaus Brockhaus and R. J. Mathar, Oct 28 2009

A203663 Achilles number whose double is also an Achilles number.

Original entry on oeis.org

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136

Offset: 1

Antonio Roldán, Jan 04 2012

nonn

Every term is a multiple of 4.

			15552 is in the sequence because 15552 = 2^6*3^5 (Achilles number) and 15552*2 = 2^7*3^5 is also an Achilles number.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..4148 from Robert Israel)

Cf. A052486, A143610, A203662.

filter:= proc(n) local e2,F;
  e2:= padic:-ordp(n,2);
  if e2 < 2 then return false fi;
  F:= map(t -> t[2], ifactors(n/2^e2)[2]);
  min(F) > 1 and igcd(e2,op(F))=1 and igcd(e2+1,op(F))=1
end proc:
select(filter, [seq(i,i=4..10^5,4)]); # Robert Israel, Jan 16 2018

achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD@@ee == 1];
Select[Range[4, 10^5, 4], achillesQ[#] && achillesQ[2#]&] (* Jean-François Alcover, Sep 25 2020 *)

achilles(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n) } \\ M. F. Hasler, 2010
{ for (n=1, 10^6, if (achilles(n)==1 && achilles(2*n)==1, print1(n,", "))); } \\ Antonio Roldán, Oct 07 2012

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint
def A203663_gen(): # generator of terms
    return map(lambda x:x[0],filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1,map(lambda x: (x,factorint(x<<1).values()),(A052486(i) for i in count(1)))))
A203663_list = list(islice(A203663_gen(),30)) # Chai Wah Wu, Sep 10 2024

A355462 Powerful numbers divisible by exactly 2 distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Amiram Eldar, Jul 03 2022

nonn

First differs from A286708 at n = 25.
Number of the form p^i * q^j, where p != q are primes and i,j > 1.
Numbers k such that A001221(k) = 2 and A051904(k) >= 2.
The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).
675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

			36 is a term since 36 = 2^2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Intersection of A001694 and A007774.
Subsequence of A286708.
Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.
Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .

A367781 Numbers with prime signature p^3 * q^2 whose difference from the next such number is a record low.

Original entry on oeis.org

72, 1323, 1352, 49923, 27725061981125

Offset: 1

Jon E. Schoenfield, Nov 29 2023

If it exists, a(6) > 10^20.

Do there exist two numbers with prime signature p^3 * q^2 that differ by less than 3? If not, this sequence terminates at a(5) = 27725061981125.

			A143610, the sequence of numbers with prime signature p^3 * q^2, is 72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, ...; the differences between its successive terms are 36, 92, 192, 108, 175, 293, 157, 198, 29, 20, 940, ..., and the first five record lows among those differences are 36, 29, 20, 5, 3, ...
------------------------------------------------  ----  -  --------------
  Two successive numbers of the form p^3 * q^2    diff  n            a(n)
------------------------------------------------  ----  -  --------------
2^3 *      3^2 =    72 | 3^3 *       2^2 =   108    36  1              72
3^3 *      7^2 =  1323 | 2^3 *      13^2 =  1352    29  2            1323
2^3 *     13^2 =  1352 | 7^3 *       2^2 =  1372    20  3            1352
3^3 *     43^2 = 49923 | 2^3 *      79^2 = 49928     5  4           49923
5^3 * 470957^2 =   ... | 2^3 * 1861621^2 =   ...     3  5  27725061981125

Cf. A143610.

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Original entry on oeis.org

72, 108, 200, 392, 432, 500, 648, 675, 968, 1125, 1323, 1352, 1372, 1800, 2000, 2312, 2592, 2700, 2888, 3087, 3267, 3528, 3888, 4232, 4500, 4563, 5000, 5292, 5324, 5400, 5488, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10125, 10584, 10952, 11979

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

			72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

Cf. A051903, A051904, A072777.
Subsequence of A001694.
Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

Original entry on oeis.org

4, 9, 25, 49, 72, 108, 121, 144, 169, 200, 288, 289, 324, 361, 392, 400, 500, 529, 576, 675, 784, 800, 841, 961, 968, 972, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1600, 1681, 1849, 1936, 2025, 2209, 2304, 2312, 2500, 2704, 2809, 2888, 2916, 3087, 3136, 3200

Offset: 1

Amiram Eldar, Feb 19 2025

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).

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

Subsequence of A001694, A377816 and A377818.
Subsequences: A001248, A143610, A179646, A179689, A179699, A189988, A189990, A190106, A190115, A190470, A190471.
Cf. A036966, A065483.

With[{max = 3200}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], 2] == 1 &]]

isok(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x==2), e) == 1);

Sum_{n>=1} 1/a(n) = Sum_{p prime}((p-1)/(p^3-p^2+1)) * Product_{p prime} (1 + 1/(p^2*(p-1))) = 0.53045141423939736076... .

A216417 Numbers of the form p^2*q^3 where p, q are (not necessarily distinct) primes.

Original entry on oeis.org

32, 72, 108, 200, 243, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3125, 3267, 4232, 4563, 5324, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 16807, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436, 27848, 29768

Offset: 1

V. Raman, Sep 07 2012

nonn

Union of A143610 (where p != q) and A050997 (where p = q).

With[{nn=50},Take[Union[First[#]^2 Last[#]^3&/@ Tuples[ Prime[ Range[ nn]],2]], nn]] (* Harvey P. Dale, Jan 17 2014 *)

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A377846 Powerful numbers that are not divisible by the cubes of more than one distinct prime.

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A166988 Products n of a square of a prime and a cube of a prime such that n-1 and n+1 are semiprimes.

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A203663 Achilles number whose double is also an Achilles number.

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A355462 Powerful numbers divisible by exactly 2 distinct primes.

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A367781 Numbers with prime signature p^3 * q^2 whose difference from the next such number is a record low.

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A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

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A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

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A216417 Numbers of the form p^2*q^3 where p, q are (not necessarily distinct) primes.