A065119 -id:A065119 - cp's OEIS frontend

A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

3, 6, 12, 18, 24, 36, 48, 72, 96, 108

Offset: 1

Vladimir Shevelev, May 13 2016

Theorem. Ramanujan's largely composite numbers (A067128) having the greatest prime divisor p_k = prime(k) do not exceed Product_{2 <= p <= p_k} p^((2*ceiling(log_p(p_(k + 1)) - 1).
Proof. Let N be in A067128 with prime power factorization 2^l_1 * 3^l_2 * ... * p_k^l_k.
First let us show that l_1 <= 2x_1-1 such that 2^x_1 > p_(k+1).
Indeed, consider N_1 = 2^(l_1-x_1)*3^l_2*...*p_k^l_k*p_(k+1).
Since 2^x_1 > p_(k+1) then N_1But d(N_1) > d(N) if l_1 >= 2*x_1, so l_1 <= 2x_1-1.
Analogously we find l_i <= 2x_i-1 if p_i^x_i > p_(k+1), i <= k.
Therefore N <= 2^(2*x_1-1)*3^(2*x_2-1)*...* p_k^(2*x_k-1) and the theorem easily follows.
QED
The inequality of the theorem gives a way to find the full sequence for every p_k. In particular, in case p_k = 2 we have the sequence {2, 4, 8}. For other cases see A273016, A273018.

Cf. A067128, A065119 (the intersection of these two sequences is the present sequence). Cf. also A003586, A273016, A273018.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Select[a, FactorInteger[#][[-1, 1]] == 3 &] (* Michael De Vlieger, May 13 2016 *)

A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 2, 1, 7, 2, 8, 3, 3, 5, 9, 2, 3, 6, 2, 4, 10, 2, 11, 1, 4, 7, 3, 2, 12, 8, 5, 3, 13, 3, 14, 5, 2, 9, 15, 2, 4, 3, 6, 6, 16, 2, 3, 4, 7, 10, 17, 2, 18, 11, 3, 1, 4, 4, 19, 7, 8, 3, 20, 2, 21, 12, 2, 8, 4, 5, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 14 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Prepending 1 to the positions of 1's gives A000079.
Positions of first appearances are A008578.
Positions of 2's are A065119.
The non-augmented version is A286470, also A355526.
The non-augmented minimal version is A355524, also A355525.
The minimal version is A355531.
Row maxima of A355534, which has Heinz number A325351.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A086761 Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.

Original entry on oeis.org

5, 10, 20, 25, 40, 50, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 640, 800, 1000, 1250, 1280, 1600, 2000, 2500, 2560, 3125, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 15625, 16000, 20000, 20480, 25000, 25600, 31250, 32000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 02 2003

nonn

A206787(a(n)) = 6. - Reinhard Zumkeller, Feb 12 2012
All terms have the form 2^a 5^b with a >= 0 and b > 0. - T. D. Noe, Feb 13 2012
If the above holds for all terms then this sequence is 5 * A003592. - David A. Corneth, Jul 04 2018

Cf. A003592, A051664, A065119.

Select[Range[1000], Count[CoefficientList[Cyclotomic[#, x], x], 0] == EulerPhi[#] - 4 &] (* T. D. Noe, Feb 13 2012 *)

is(n) = v = Vec(polcyclo(n)); sum(i=1,#v,v[i]!=0) == 5 \\ David A. Corneth, Jul 04 2018

More terms from T. D. Noe, Feb 13 2012

A301461 Number of integers less than or equal to n whose largest prime factor is 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Ralph-Joseph Tatt, Mar 21 2018

nonn

a(n) increases when n has the form 2^a*3^b, with a >= 0 and b > 0.

A distinct sequence can be generated for each prime number; this sequence is for the prime number 3. For an example using another prime number see A301506.

			a(12) = a(2^2 * 3^1); 3 is the largest prime factor, so a(12) exceeds the previous term by 1. For a(13), 13 is a prime, so there is no increase from the previous term.

Cf. A003586, A065119, A071521.

Cf. A301506, A071521, A070939.

clear;clc;
prime = 3;
limit = 10000;
largest_divisor = ones(1,limit+1);
for k = 0:limit
    f = factor(k);
    largest_divisor(k+1) = f(end);
end
for i = 1:limit+1
    FQN(i) = sum(largest_divisor(1:i)==prime);
end
output = [0:limit;FQN]'

Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 3] &, 80, 0] (* Michael De Vlieger, Apr 21 2018 *)

gpf(n) = if (n<=1, n, vecmax(factor(n)[,1]));
a(n) = sum(k=1, n, gpf(k)==3); \\ Michel Marcus, Mar 27 2018

From David A. Corneth, Mar 27 2018: (Start)
a(n) - a(n - 1) = 1 if and only if n is in 3 * A003586. If n isn't in that sequence then a(n) = a(n - 1).
a(3 * n + b) = A071521(n), n > 0, 0 <= b < 3. (End)
a(n) = A071521(n) - A070939(n). - Ridouane Oudra, Mar 24 2025

A382487 The number of divisors of n whose largest prime factor is 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 5, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 3, 0, 0, 1

Offset: 1

Amiram Eldar, Mar 29 2025

The number of 3-smooth divisors of n that are not powers of 2.

The number of terms of A065119 that divide n.

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

Cf. A001511, A001651, A065119, A072078, A007949, A051064, A169611, A301461, A306771.

a[n_] := (IntegerExponent[n, 2] + 1) * IntegerExponent[n, 3]; Array[a, 100]

a(n) = (valuation(n, 2) + 1) * valuation(n, 3);

a(n) = A072078(n) - A001511(n).
a(n) = A001511(n) * A007949(n).
a(n) = 0 if and only if n is in A001651.
a(n) = 1 if and only if n is in A306771.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.
In general, the asymptotic mean of the number prime(k+1)-smooth divisors of n that are not prime(k)-smooth, for k >= 1, is (1/(prime(k+1)-1)) * Product_{i=1..k} (prime(i)/(prime(i)-1)).
Dirichlet g.f.: (zeta(s)/(1-1/2^s))*(1/(1-1/3^s) - 1).

A085459 Numbers k such that k-th cyclotomic polynomial has exactly 3 positive coefficients.

Original entry on oeis.org

3, 9, 10, 20, 27, 40, 50, 80, 81, 100, 160, 200, 243, 250, 320, 400, 500, 640, 729, 800, 1000, 1250, 1280, 1600, 2000, 2187, 2500, 2560, 3200, 4000, 5000, 5120, 6250, 6400, 6561, 8000, 10000, 10240, 12500, 12800, 16000, 19683, 20000, 20480, 25000, 25600

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 14 2003

Sequence appears to consist of 3^i, i > 0; and 2^i*5^j, i, j > 0. Are there any other terms? - David Wasserman, Feb 01 2005

			9 is a member because the 9th cyclotomic polynomial is P(x) = x^6+x^3+1.

Cf. A065119, A086765.

Select[Range@ 5000, Count[CoefficientList[Cyclotomic[#, x], x], ?(# > 0 &)] == 3 &] (* _Michael De Vlieger, Oct 26 2017 *)

n = 0; while (1, n++; p = polcyclo(n, x); d = poldegree(p); c = 0; i = 0; while (c < 4 && i <= d, if (polcoeff(p, i) > 0, c++); i++); if (c == 3, print(n))); \\ David Wasserman, Feb 01 2005

More terms from David Wasserman, Feb 01 2005

A280990 Least prime p such that n divides phi(p*n).

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 31, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 31, 31, 2, 67, 17, 71, 3, 37, 19, 13, 5, 41, 7, 43, 11, 31, 23, 47, 3, 7, 5, 103, 13, 53, 3, 11, 7, 19, 29, 59, 31, 61, 31, 7, 2, 131, 67, 67, 17, 139, 71, 71, 3, 73, 37, 31, 19, 463

Offset: 1

Altug Alkan, Jan 12 2017

nonn

n*a(n) are 2, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 465, 32, 289, ...

a(n) <= A034694(A007947(n)). If n is in A050384 then a(n) = A034694(n). - Robert Israel, Jan 12 2017

			a(15) = 31 because 15 does not divide phi(p*15) for p < 31 where p is prime and phi(31*15) = 2*4*30 is divisible by 15.

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

Cf. A000010, A007694, A007947, A034694, A050384, A109395.

Cf. A000079, A065119, A086761: for those n such that a(n)=2,3,5. - Michel Marcus, Jan 20 2017

f:= proc(n) local p;
    p:= 2;
    while numtheory:-phi(p*n) mod n <> 0 do p:= nextprime(p) od:
    p
end proc:
map(f, [$1..100]); # Robert Israel, Jan 12 2017

lpp[n_]:=Module[{p=2},While[Mod[EulerPhi[p*n],n]!=0,p=NextPrime[p]];p]; Array[lpp,80] (* Harvey P. Dale, Sep 26 2020 *)

a(n)=my(k = 1); while (eulerphi(prime(k)*n) % n != 0, k++); prime(k);

a(n)=my(t=n/gcd(eulerphi(n),n)); if(t==1, return(2)); forstep(p=if(t%2,2*t,t)+1, if(isprime(t), t, oo),lcm(t,2), if(isprime(p), return(p))); t \\ Charles R Greathouse IV, Jan 20 2017

a(p^k) = p for all primes p and k >= 1. - Robert Israel, Jan 12 2017

a(n) << n^5 by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Jan 20 2017

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Amiram Eldar, Jan 16 2024

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

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

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]

gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

A384184 Order of the permutation of {0,...,n-1} formed by successively swapping elements at i and 2*i mod n, for i = 0,...,n-1.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 2, 8, 3, 4, 5, 4, 6, 4, 6, 16, 4, 6, 9, 8, 4, 10, 28, 8, 10, 12, 9, 8, 14, 12, 12, 32, 5, 8, 70, 12, 18, 18, 24, 16, 10, 8, 7, 20, 210, 56, 126, 16, 110, 20, 60, 24, 26, 18, 120, 16, 9, 28, 29, 24, 30, 24, 60, 64, 6, 10, 33, 16

Offset: 1

Mia Boudreau, May 29 2025

nonn

a(2*n) = 2*a(n) since the cycle lengths of the permutation with size 2*n is effectively that of size n twice, doubled. Thus, the LCM/order is doubled.

			For n = 11, the permutation is {0,3,4,7,8,1,2,9,10,5,6} and it has order a(11) = 5.

Mia Boudreau, Table of n, a(n) for n = 1..10000
Mia Boudreau, Java program, GitHub

Cf. A003418, A000027, A051732, A004626, A065119, A001122, A155072, A001133, A001134, A001135, A225759.

from sympy.combinatorics import Permutation
def a(n):
   L = list(range(n))
   for i in range(n):
       if (j:= (i << 1) % n) != i:
           L[i],L[j] = L[j],L[i]
   return Permutation(L).order() # Darío Clavijo, Jun 05 2025

a(2*n) = 2*a(n).
a(2^n) = 2^n.
Conjecture: a(2^n + 2^x) = 2^n * (x-n) if x > n.
a(2^n - 1) = A003418(n-1).
s(2^n + 1) = A000027(n).
a(2*n - 1) = A051732(n).
a(A004626(n)) % 2 = 1.
a(A065119(n)) = n/3.
a(A001122(n)) = (n-1) / 2.
a(A155072(n)) = (n-1) / 4.
a(A001133(n)) = (n-1) / 6.
a(A001134(n)) = (n-1) / 8.
a(A001135(n)) = (n-1) / 10.
a(A225759(n)) = (n-1) / 16.

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15

Offset: 0

Gus Wiseman, Nov 19 2023

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]

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cp's OEIS Frontend

A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

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A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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Mathematica

A086761 Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.

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Mathematica

PARI

Extensions

A301461 Number of integers less than or equal to n whose largest prime factor is 3.

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MATLAB

Mathematica

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A382487 The number of divisors of n whose largest prime factor is 3.

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Mathematica

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A085459 Numbers k such that k-th cyclotomic polynomial has exactly 3 positive coefficients.

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Mathematica

PARI

Extensions

A280990 Least prime p such that n divides phi(p*n).

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Maple

Mathematica

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A369209 Numbers whose number of divisors has the largest prime factor 3.

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