A176297 -id:A176297 - cp's OEIS frontend

A162870 Primes p such that p-1 and p+1 each contain at least one cubed prime in their prime factorization.

919, 1999, 2647, 2663, 2969, 3511, 3833, 3943, 4751, 6857, 9127, 10313, 11287, 11719, 12041, 12583, 13033, 13337, 13879, 14249, 14633, 15497, 15607, 16903, 18089, 18199, 18251, 18521, 19751, 20249, 20359, 20681, 21751, 21977, 22409

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

nonn

The selection criterion is that p-1 and p+1 are in the subsequence 8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, 54=2*3^3,... of cubeful numbers (A046099) which actually display at least one cube in their standard prime factorization (A176297).

So at least one of the e_i in p-1=product p_i^e_i, and at least one of the e_j in p+1=product p_j^e_j must equal 3. This is more restrictive than being cubeful, so the sequence becomes a subsequence of A086708.

			271 is not in the sequence although 271 - 1 = 2*3^3*5 contains a third cube in the prime factorization, because 271 + 1 = 2^4*17 does not.
919 is in the sequence because 919 - 1 = 2*3^3*17 contains a third cube in the prime factorization and so does 919 + 1 = 2^3*5*23.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A089189, A089194.

isA162870 := proc(n)
    if isprime(n) then
        isA176297(n-1) and isA176297(n+1) ;
    else
        false;
    end if;
end proc:
for n from 1 to 40000 do
    if isA162870(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015
N:= 10^6: # to get all terms < N, where N is even
V:= Vector(N/2):
for i from 1 do
  p:= ithprime(i);
  if p^3 > N+1 then break fi;
  if p = 2 then inds:= 4*[seq(i,i=1..floor(N/8),2)]
  else inds:= p^3*select(t -> t mod p <> 0, [$1..floor(N/2/p^3)])
  fi;
  V[inds]:= 1;
od:
select(t -> V[(t-1)/2] = 1 and V[(t+1)/2] = 1 and isprime(t), [seq(t,t=3..N,2)]); # Robert Israel, Dec 08 2015

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]==3,a=1],{m,Length[FactorInteger[n]]}]; a]; lst={};Do[p=Prime[n];If[f[p-1]==1&&f[p+1]==1,AppendTo[lst,p]], {n,7!}];lst

Role of cubefree numbers clarified by R. J. Mathar, Jul 31 2007

A386799 Numbers without an exponent 3 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Amiram Eldar, Aug 02 2025

First differs from its subsequence A336592 at n = 116: a(116) = 128 = 2^7 is not a term of A336592.
Numbers k such that A295883(k) = 0.
These numbers were named semi-3-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.90470892696874750603... (Suryanarayana, 1971).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
Index entries for sequences computed from indices in prime factorization.

Complement of A176297.	
A336592 is a subsequence.
Cf. A295883.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), this sequence (k=3), A386803 (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: this sequence (m=0), A386800 (m=1), A386801 (m=2), A386802 (m=3).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 3] &]

isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 0;

A176313 First of two consecutive numbers with at least one 3 in their prime signature.

Original entry on oeis.org

135, 296, 343, 375, 999, 1160, 1431, 1592, 1624, 2295, 2375, 2456, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4887, 4913, 5048, 5144, 5319, 5480, 5831, 6183, 6344, 6375, 6615, 6776, 6858, 6859, 7479, 7624, 7640, 7749, 7911, 8072, 8375, 8775, 8936, 9125, 9207, 9368, 9624, 10071, 10232, 10375, 10503, 10632, 10664, 10984, 11124, 11319, 11367, 11528, 11624, 11799, 11960

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17.

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

A068140 lists the smallest of two consecutive numbers such that each is divisible by a cube greater than 1. See also A000578, A001235, A176297, A176350.

f[n_]:=MemberQ[Last/@FactorInteger[n], 3]; Select[Range[8!],f[#]&&f[#+1]&]

Edited by Matthew Vandermast, Dec 09 2010

A176350 First of three consecutive numbers with at least one 3 in their prime signature.

Original entry on oeis.org

6858, 22625, 28375, 40472, 48248, 49624, 58374, 59750, 94471, 102248, 103624, 107702, 112374, 122823, 129623, 133623, 136214, 136375, 153063, 164295, 187623, 190375, 197910, 199624, 210248, 211624, 220374, 221750, 238248, 246616, 260874, 264248, 275750, 280231, 298375, 300806, 312471, 329623, 336824, 346086, 349623, 352375, 356375

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

			6858=2*3^3*127,6859=19^3,6860=2^2*5*7^3,..

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000578, A001235, A176297, A176313.

f[n_]:=MemberQ[Last/@FactorInteger[n], 3]; Select[Range[9!],f[#]&&f[#+1]&&f[#+2]&]
SequencePosition[Table[If[MemberQ[FactorInteger[n][[All,2]],3],1,0],{n,360000}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 09 2019 *)

Edited by Matthew Vandermast, Dec 09 2010

A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600

Offset: 1

Amiram Eldar, Jul 29 2024

Subsequence of A176297 and first differs from it at n = 41: A176297(41) = 432 = 2^4 * 3^3 is not a term of this sequence.
Numbers whose prime factorization has least one exponent that equals 3 and no higher exponent.
Numbers k such that A051903(k) = 3.
The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) = A215267 - A088453 = 0.0920310303408826983406... .

Intersection of A046100 and A176297.
Cf. A004709, A051903, A067259.
Cf. A002117, A013662, A088453, A215267.

Select[Range[600], Max[FactorInteger[#][[;; , 2]]] == 3 &]

is(k) = k > 1 && vecmax(factor(k)[,2]) == 3;

from sympy import mobius, integer_nthroot
def A375072(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**4-x//k**3) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**3) for k in range(integer_nthroot(x,4)[0]+1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A109399 Numbers with at least two 3s in their prime signature.

Original entry on oeis.org

216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000, 17064, 17576, 17928, 18360, 18522, 19000, 19224, 19656, 20520, 20952, 21000, 21816, 22248, 23000, 23112, 23544, 23625, 24408, 24696, 24840, 25704, 26136, 27000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least two values of i.
Does not include all numbers with at least two unitary prime power divisors that are cubes (see example section).
The asymptotic density of this sequence is 1 - (1 + Sum_{p prime} ((p-1)/(p^4-p+1))) * Product_{p prime} (1-1/p^3+1/p^4) = 0.0024593812036570543518... . - Amiram Eldar, Jul 22 2024

			216 = 2^3*3^3, 1000 = 2^3*5^3, 1080 = 2^3*3^3*5, ...
On the other hand, 1728 = 2^6*3^3, 8000 = 2^6*5^3 and 21952 = 2^6*7^3 are not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000578, A001235, A176297, A176313, A176350.

A176359 is a subsequence.

f[n_]:=Count[Last/@FactorInteger[n],3]>1; Select[Range[8!],f]

is(n)=#select(e->e==3, factor(n)[,2])>1 \\ Charles R Greathouse IV, Oct 19 2015

Edited by Matthew Vandermast, Dec 07 2010

A176359 Numbers with at least three 3s in their prime signature.

Original entry on oeis.org

27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000, 1704024, 1809000, 1852200, 1917000, 1971000, 2012472, 2079000, 2133000, 2148552

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least three values of i.

The asymptotic density of this sequence is 1 - (1 + s(1) + s(1)^2/2 - s(2)/2) * Product_{p prime} (1-1/p^3+1/p^4) = 0.000018992895371889141564..., where s(k) = Sum_{p prime} ((p-1)/(p^4-p+1))^k. - Amiram Eldar, Jul 22 2024

			27000 is a term since 27000 = 2^3 * 3^3 * 5^3.
74088 is a term since 74088 = 2^3 * 5^3 * 7^3.

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

Cf. A000578, A001235, A176297, A176313, A176350.

Subsequence of A109399.

f[n_]:=Count[Last/@FactorInteger[n],3]>2; Select[Range[10!],f]

is(n) = #select(x -> x == 3, factor(n)[, 2]) > 2; \\ Amiram Eldar, Jul 22 2024

Edited by Matthew Vandermast, Dec 09 2010

A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 128, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 384, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that at least one of the exponents in their prime factorization is of the form 4*m + 3.
The asymptotic density of this sequence is 1 - zeta(4)/zeta(3) = 0.0996073223... (Cohen, 1963).
The number of divisors of all the terms is divisible by 4.

			8 is a term since 8 = 2^3 and 3 is of the form 4*m + 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

Complement of A336592.
Complement of A336594 within A252849.
Cf. A002117, A008835, A013662, A036966.
A176297 is a subsequence.

Select[Range[540], Max[Mod[FactorInteger[#][[;; , 2]], 4]] == 3 &]

A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A176297 and A375072, and first differs from them at n = 20: A176297(20) = A375072(20) = 216 = 2^3 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p+p^2+p^3) = 0.089602607198058453295... .

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

Subsequence of A046100, A176297, A375072 and A375145.
Subsequences: A030078, A048109, A065036, A143610, A163569, A179670, A179695, A179700, A189975, A189984, A190109, A190378, A190382.
Cf. A002117.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, MemberQ[e, 3] && Count[e, _?(# < 3 &)] == Length[e] - 1]; Select[Range[600], q]

isok(k) = {my(e = factor(k)[, 2]~); select(x -> x > 2, e) == [3];}

A362841 Numbers with at least one 5 in their prime signature.

Original entry on oeis.org

32, 96, 160, 224, 243, 288, 352, 416, 480, 486, 544, 608, 672, 736, 800, 864, 928, 972, 992, 1056, 1120, 1184, 1215, 1248, 1312, 1376, 1440, 1504, 1568, 1632, 1696, 1701, 1760, 1824, 1888, 1944, 1952, 2016, 2080, 2144, 2208, 2272, 2336, 2400, 2430, 2464, 2528, 2592, 2656, 2673, 2720, 2784, 2848, 2912, 2976

Offset: 1

R. J. Mathar, May 05 2023

nonn

Contains all odd multiples of 2^5: Each second term of A174312 is in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^5 + 1/p^6) = 0.01863624892... . - Amiram Eldar, May 05 2023

			Contains 2^5, 2^5*3, 2^5*5, 2^5*7, 3^5, 2^5*3^2, 2^5*11, 2^5*13, 2^5*3*5, 2*3^5, etc.

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

Cf. A038109 (at least one 2), A176297 (at least one 3), A050997 (subsequence), A178740 (subsequence), A179646 (subsequence), A179667 (subsequence), A179671 (subsequence), A174312.

Select[Range[3000], MemberQ[FactorInteger[#][[;;, 2]], 5] &] (* Amiram Eldar, May 05 2023 *)

cp's OEIS Frontend

A162870 Primes p such that p-1 and p+1 each contain at least one cubed prime in their prime factorization.

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A386799 Numbers without an exponent 3 in their prime factorization.

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A176313 First of two consecutive numbers with at least one 3 in their prime signature.

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A176350 First of three consecutive numbers with at least one 3 in their prime signature.

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A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

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A109399 Numbers with at least two 3s in their prime signature.

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A176359 Numbers with at least three 3s in their prime signature.

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A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

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