A033845 -id:A033845 - cp's OEIS frontend

A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664, 12288, 13122, 13824, 15552, 17496, 18432, 20736, 23328

Offset: 1

Enrique Pérez Herrero, Jan 05 2013

nonn

This sequence is closed under multiplication.
Also 1 and the numbers where psi(n)/n = 2, or n/phi(n)=3, or psi(n)/phi(n)=6.
Also 1 and the numbers of the form 2^i*3^j with i, j >= 1 (A033845).
If M(n) is the n X n matrix whose elements m(i,j) = 2^i*3^j, with i, j >= 1, then det(M(n))=0.
Numbers n such that Product_{i=1..q} (1 + 1/d(i)) is an integer where q is the number of the distinct prime divisors d(i) of n. - Michel Lagneau, Jun 17 2016

			psi(48) = 96 and 96/48 = 2 so 48 is in this sequence.

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..191 from Vincenzo Librandi)
R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.
E. Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Smooth Number
Wikipedia, Closure

Cf. A003586, A001615, A007694, A033950, A074946, A075592.

[6*n: n in [1..3000] | PrimeDivisors(n) subset [2, 3]]; // Vincenzo Librandi, Jan 11 2019

Select[Range[10^4],#/EulerPhi[#]==3 || #==1&]
Join[{1}, 6 Select[Range@4000, Last@Map[First, FactorInteger@#]<=3 &]] (* Vincenzo Librandi, Jan 11 2019 *)

dedekindpsi(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n]);
k=0; n=0; while(k<10000,n++; if( dedekindpsi(n) % n== 0, k++; print1(n, ", ")));

For n > 1, a(n) = 6 * A003586(n).

Sum_{n>0} 1/a(n)^k = 1 + Sum_{i>0} Sum_{j>0} 1/(2^i * 3^j)^k = 1 + 1/((2^k-1)*(3^k-1)).

A215142 Numbers n such that the difference between the greatest prime divisor of n and the sum of the other distinct prime divisors equals 1.

Original entry on oeis.org

6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 231, 288, 324, 330, 384, 432, 455, 486, 546, 576, 648, 660, 663, 693, 768, 864, 935, 972, 990, 1092, 1122, 1152, 1235, 1296, 1311, 1320, 1458, 1463, 1482, 1536, 1617, 1638, 1650, 1728, 1944, 1955

Offset: 1

Michel Lagneau, Aug 04 2012

nonn

A033845 is included in this sequence.

			1235 is in the sequence because 1235 = 5*13*19 and 19 - (5+13) = 1.

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

Cf. A033845.

with(numtheory):for n from 2 to 2000 do:x:=factorset(n):m:=nops(x):s:=0: s:=sum( '
x[i] ', 'i'=1..m):q:=s-x[m]:if x[m]-q =1 then printf(`%d, `,n):else fi:od:

gpdQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},Max[f]-Total[ Most[ f]] == 1]; Select[Range[2,2000],gpdQ] (* Harvey P. Dale, Aug 28 2013 *)

A336772 Sums s of positive exponents such that no prime of the form 2^j*3^k + 1 with j + k = s exists.

Original entry on oeis.org

12, 24, 33, 46, 48, 60, 72, 74, 80, 96, 102, 111, 118, 120, 130, 132, 141, 142, 144, 147, 159, 162, 165, 166, 168, 186, 200, 216, 234, 240, 242, 252, 258, 288, 306, 309, 312, 318, 358, 370, 374, 375, 384, 399, 405, 408, 414, 420, 432, 435, 462, 464, 468, 478

Offset: 1

Hugo Pfoertner, based on a suggestion from Rainer Rosenthal, Aug 24 2020

nonn

			a(1) = 12, because none of the 11 numbers {2^1*3^11+1, 2^2*3^10+1, ..., 2^11*3^1+1} = {354295, 236197, 157465, 104977, 69985, 46657, 31105, 20737, 13825, 9217, 6145} is prime,
a(2) = 24: none of the 23 numbers {2^1*3^23+1, 2^2*3^22+1, ..., 2^23*3^1+1} = {188286357655, 125524238437, 83682825625, 55788550417, ..., 56623105, 37748737, 25165825} is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..500

Cf. A033845, A058383, A336773.

for(s=2,500, my(t=1); for(j=1,s-1, my(k=s-j); if(isprime(2^j*3^k+1),t=0;break)); if(t,print1(s,", ")))

A339465 Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

19, 31, 37, 43, 61, 67, 73, 79, 103, 109, 127, 139, 157, 163, 181, 199, 223, 229, 241, 271, 277, 283, 307, 313, 337, 349, 367, 373, 379, 397, 409, 433, 439, 457, 487, 499, 523, 541, 577, 607, 613, 619, 643, 673, 709, 733, 739, 757, 787, 811, 823, 829, 853, 877, 907, 919

Offset: 1

Bernard Schott, Dec 09 2020

nonn

Paul Erdős asked if there are infinitely many primes p such that (p-1)/A006530(p-1) = 2^k or = 2^q*3^r (see Richard K. Guy reference).
It is not known if this sequence is infinite.
Proposition: if prime p is a term, then p is of the form 6*m+1 (A002476).

			31 is prime, 30/5 = 6 = 2*3 hence 31 is a term.
37 is prime, 36/3 = 12 = 2^2*3 hence 37 is a term.
127 is prime, 126/7 = 18 = 2*3^2 hence 127 is a term.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321.

Cf. A006093, A006530, A033845, A052126, A339464.
Cf. A074781 (ratio=2^k), A339466 (ratio <> 2^k and <> 2^q*3^r).
Subsequence of A002476.

s:=func; [p:p in PrimesInInterval(3,1000)|PrimeDivisors(a) eq [2,3] where a is (p-1) div s(p-1)]; // Marius A. Burtea, Dec 09 2020

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 3}:
select(is_a, [$3..919]); # Peter Luschny, Dec 13 2020

q[n_] := PrimeQ[n] && Module[{f = FactorInteger[n - 1]}, (Length[f] == 2 && f[[2, 1]] == 3 && f[[2, 2]] > 1) || (Length[f] == 3 && f[[2, 1]] == 3 && f[[3, 2]] == 1)]; Select[Range[1000], q] (* Amiram Eldar, Dec 09 2020 *)

More terms from Marius A. Burtea, Dec 09 2020

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

Original entry on oeis.org

7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633, 578415690713089, 1141260857376769, 57711166318706689

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Larger of twin primes p such that p-1 = (2^u)*(3^w), u,w >= 1.

			a(4) = 73, {71,73} are twin primes and (71 + 73)/2 = 72 = 2*2*2*3*3.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000)
Harsh Aggarwal, Table of n, a(n) for n = 62..91 (terms from 10^1000 to 10^20000)

Cf. A001454, A002822, A027856, A033845, A058383, A059960.

Take[Select[Sort[Flatten[Table[2^a 3^b,{a,250},{b,250}]]],AllTrue[#+{1,-1},PrimeQ]&]+1,23] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2019 *)

isok(p) = isprime(p) && isprime(p-2) && (vecmax(factor(p-1)[,1]) == 3); \\ Michel Marcus, Sep 05 2017

a(n) = A027856(n+1) + 1. - Amiram Eldar, Mar 17 2025

Name corrected by Sean A. Irvine, Oct 31 2022

A343300 a(n) is p1^1 + p2^2 + ... + pk^k where {p1,p2,...,pk} are the distinct prime factors in ascending order in the prime factorization of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 2, 3, 27, 11, 11, 13, 51, 28, 2, 17, 11, 19, 27, 52, 123, 23, 11, 5, 171, 3, 51, 29, 136, 31, 2, 124, 291, 54, 11, 37, 363, 172, 27, 41, 354, 43, 123, 28, 531, 47, 11, 7, 27, 292, 171, 53, 11, 126, 51, 364, 843, 59, 136, 61, 963, 52, 2, 174, 1342, 67, 291, 532, 370, 71, 11, 73

Offset: 1

Giorgos Kalogeropoulos, Apr 11 2021

nonn

From Bernard Schott, May 07 2021: (Start)
a(n) depends only on prime factors of n (see formulas).
Primes are fixed points of this sequence.
Terms are in increasing order in A344023. (End)

			a(60) = 136 because the distinct prime factors of 60 are {2, 3, 5} and 2^1 + 3^2 + 5^3 = 136.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A027748, A344023 (terms ordered).

A372972 Numbers k such that A372720(k) is negative.

Original entry on oeis.org

162, 250, 324, 384, 486, 648, 686, 768, 972, 1152, 1250, 1296, 1372, 1458, 1536, 1728, 1875, 1944, 2058, 2250, 2304, 2430, 2500, 2560, 2592, 2662, 2738, 2916, 3000, 3072, 3362, 3402, 3456, 3698, 3750, 3840, 3888, 3993, 4050, 4116, 4374, 4394, 4418, 4500, 4608

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
For s > 1, an infinite number of k in R are such that g(k) is negative. For example, with s = 6, all terms k > 864 in A033845 are in this sequence.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 162 = 2*3^4, since tau(162) - f(162)
     = (1+1)*(4+1) - card(A369609(162))
     = 10 - 12 = -2.
a(2) = 250 = 2*5^3, since tau(250) - f(250)
     = (1+1)*(3+1) - card(A369609(250))
     = 8 - 9 = -1.
a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
     = (2+1)*(4+1) - card(A369609(324))
     = 15 - 16 = -1, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 5000}], ?(# < 0 &)][[All, 1]]

A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

Original entry on oeis.org

36, 144, 216, 324, 576, 900, 1296, 1728, 2304, 2916, 3600, 5184, 5832, 7776, 8100, 9216, 11664, 13824, 14400, 20736, 22500, 26244, 27000, 32400, 36864, 44100, 46656, 57600, 72900, 82944, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 186624, 202500, 216000

Offset: 1

Michael De Vlieger, Jul 25 2025

nonn

Perfect powers k^m, m > 1, for k in A055932.
Union of {k^m : rad(k) | P(i), m >= 2}, rad = A007947, P = A002110. Therefore perfect powers in A033845, A143207, A147571, A147572, etc. are proper subsets.
Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading. Terms that also appear in A368682 are marked by "#":
                         Exponents
 n      a(n)             2.3.5.7.11
-----------------------------------
 1       36 =    6^2  #  2.2
 2      144 =   12^2  #  4.2
 3      216 =    6^3  #  3.3
 4      324 =   18^2     2.4
 5      576 =   24^2  #  6.2
 6      900 =   30^2  #  2.2.2
 7     1296 =    6^4  #  4.4
 8     1728 =   12^3  #  6.3
 9     2304 =   48^2  #  8.2
10     2916 =   54^2     2.6
11     3600 =   60^2  #  4.2.2
12     5184 =   72^2  #  6.4
26    44100 =  210^2  #  2.2.2.2
90  5336100 = 2310^2  #  2.2.2.2.2

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fast Mathematica algorithm for A055932.

Cf. A001597, A002110, A006530, A007947, A033845, A053669, A055932, A126706, A131605, A143207, A147571, A147572, A246547, A304250, A365308 (subset), A368682 (subset), A369374 (superset).

(* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Intersection of A131605 and A055932 = A304250 \ A246547.

A083263 Numbers k such that the difference of the largest and smallest prime factors of k divides k.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 48, 54, 60, 70, 72, 90, 96, 108, 120, 140, 144, 150, 162, 180, 192, 198, 210, 216, 240, 270, 280, 286, 288, 300, 324, 350, 360, 384, 396, 420, 432, 450, 480, 486, 490, 510, 540, 560, 572, 576, 594, 600, 630, 646, 648, 700, 720, 750, 768

Offset: 1

Labos Elemer, May 12 2003

nonn

			Every number k of the form 2^i * 3^j * m is a term because 3 - 2 = 1 is always a divisor of k.
Every number k of the form 2 * p * (p+2) * m is a term if p and p+2 form a twin prime pair.
Other terms include some in which the difference d = gpf(k) - lpf(k) > 2 is prime (e.g., 30 = 2*3*5 = 3*10; d = 5 - 2 = 3) and some in which it is composite (e.g., 8710 = 2*5*13*67 = 65*134; d = 67 - 2 = 65).
All terms are even. - _Jon E. Schoenfield_, Jul 10 2018

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

Cf. A033845, A071141, A006530, A020639.

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; mi[x_] := Min[ba[x]] Do[s=ma[ba[n]]-mi[ba[n]]; If[Mod[n, s]==0, Print[{n, ba[n], s}]], {n, 1, 10000}]

Solutions to x mod (A006530(x) - A020639(x)) = 0.

Edited by Jon E. Schoenfield, Jul 10 2018

A228104 Numbers of form 2^(2i-1)*3^j, with i,j > 0.

Original entry on oeis.org

6, 18, 24, 54, 72, 96, 162, 216, 288, 384, 486, 648, 864, 1152, 1458, 1536, 1944, 2592, 3456, 4374, 4608, 5832, 6144, 7776, 10368, 13122, 13824, 17496, 18432, 23328, 24576, 31104, 39366, 41472, 52488, 55296, 69984, 73728, 93312, 98304, 118098, 124416, 157464, 165888

Offset: 1

Ralf Stephan, Aug 10 2013

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

Subsequence of A033845 and A011775.

N:= 10^6: # for terms <= N
sort([seq(seq(2^i * 3^j, j = 1 .. ilog[3](N/2^i)),i=1..ilog2(N/3),2)]); # Robert Israel, Oct 14 2024

With[{max = 2*10^5}, Flatten[Table[2^(2*i-1)*3^j, {i, 1, (Log2[max]+1)/2}, {j, 1, Log[3, max/2^(2*i-1)]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

vecsort(vector(10000,n,2^(2*((n-1)%100)+1)*3^((n\100)+1))) /* (first 100 values) */

Sum_{n>=1} 1/a(n) = 1/3. - Amiram Eldar, Mar 29 2025

cp's OEIS Frontend

A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

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A215142 Numbers n such that the difference between the greatest prime divisor of n and the sum of the other distinct prime divisors equals 1.

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A336772 Sums s of positive exponents such that no prime of the form 2^j*3^k + 1 with j + k = s exists.

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A339465 Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

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A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

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A343300 a(n) is p1^1 + p2^2 + ... + pk^k where {p1,p2,...,pk} are the distinct prime factors in ascending order in the prime factorization of n.

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A372972 Numbers k such that A372720(k) is negative.

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