A033845 -id:A033845 - cp's OEIS frontend

A086411 Greatest prime factor of 3-smooth numbers.

1, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A003586, A006530, A006899, A033845, A086410.

Reap[Do[p = FactorInteger[n][[-1, 1]]; If[p < 5, Sow[p]], {n, 1, 2*10^5}] ][[2, 1]] (* Jean-François Alcover, Dec 17 2017 *)

a(n) = A006530(A003586(n)).
A086410(n) <= a(n) <= 3.
a(A033845(n)) = A086410(A033845(n))+1; a(A006899(n)) = A086410(A006899(n)). - Reinhard Zumkeller, Sep 25 2008
Conjecture: a(n) = A049237(n+1) for n>1. - R. J. Mathar, Jun 06 2024

A147573 Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.

Original entry on oeis.org

30030, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 540540, 600600, 630630, 660660, 720720, 750750, 780780, 810810, 840840, 900900, 960960, 990990, 1051050, 1081080, 1171170, 1201200, 1261260

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575
Although 39270 has exactly 6 distinct prime divisors (39270=2*3*5*7*11*17), it is not in this sequence because the 6 distinct prime divisors may only comprise 2, 3, 5, 7, 11, and 13. - Harvey P. Dale, Oct 11 2014

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

Subsequence of A067885 and of A080197.

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 192/1001, AppendTo[a, x]], {x, 1, 100000}]; a

is(n)=if(n%30030, return(0)); my(g=30030); while(g>1, n/=g; g=gcd(n,30030)); n==1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = 30030 * A080197(n). - Charles R Greathouse IV, Sep 14 2015

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).

Original entry on oeis.org

1, 10, 18, 20, 36, 40, 54, 70, 72, 78, 80, 108, 110, 140, 144, 156, 160, 162, 174, 198, 216, 220, 222, 230, 234, 246, 280, 288, 294, 312, 320, 324, 348, 396, 414, 426, 432, 438, 440, 444, 450, 460, 468, 470, 486, 492, 534, 560, 576, 588, 594, 624, 640, 648, 666, 696, 702, 770, 792, 828, 846, 852

Offset: 1

Altug Alkan, Sep 03 2017

Squarefree terms are 1, 10, 70, 78, 110, 174, 222, 230, 246, 426, 438, ...
Common terms of this sequence and A033632 are 1, 14406, 544500, 141118050, ...
From Robert Israel, Sep 03 2017: (Start)
Includes 2^i*3^j if i >= 1 and j >= 2, i.e., 3*A033845, and A020714(n) for n >= 1.
If an even number m is in the sequence, then so is 2*m.
Are there any odd terms other than 1? (End)
a(1) = 1 is the only odd term. LHS of equation allows for 1 and 3 but only for k <= 6. RHS allows for 1 and only for k = 1. - Torlach Rush, Jul 28 2023

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

Cf. A000010, A001615, A020714, A033632, A033845.

psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
select(psi @ numtheory:-phi = numtheory:-phi @ psi, [$1..1000]); # Robert Israel, Sep 03 2017

f[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]; Select[Range[10^3], f[EulerPhi@ #] == EulerPhi[f@ #] &] (* Michael De Vlieger, Sep 03 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = eulerphi(a001615(n))==a001615(eulerphi(n)); \\ after Charles R Greathouse IV at A001615

A323329 Lesser of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

Original entry on oeis.org

1330, 2660, 3850, 5320, 6650, 7700, 10640, 11270, 13300, 14950, 15400, 18550, 19250, 21280, 22540, 26600, 29900, 30800, 33250, 37100, 38500, 42560, 45080, 53200, 59800, 61600, 66500, 73370, 74200, 74750, 77000, 78890, 85120, 90160, 92750, 96250, 106400, 119600

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

t(n) = psi(n) - n is the sum of aliquot divisors of n, d, such that n/d is squarefree. Penney & Pomerance proposed a problem to show that the "pseudo-aliquot" sequence related to this function is unbounded. This sequence lists number with pseudo-aliquot sequence of cycle 2. The sequence that is analogous to perfect numbers is A033845.

The asymptotic density of the terms relative to the positive integers is zero. See Dimitrov link. - S. I. Dimitrov, Aug 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..1000
Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025. See p. 2.
David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.

Cf. A001615, A002025, A033845 (Dedekind psi perfect numbers), A323327, A323328, A323330.

psi[n_] := n*Times@@(1+1/Transpose[FactorInteger[n]][[1]]); t[n_]:= psi[n] - n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s,m]], {m, 1, 120000}]; s

A147574 Numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17}.

Original entry on oeis.org

510510, 1021020, 1531530, 2042040, 2552550, 3063060, 3573570, 4084080, 4594590, 5105100, 5615610, 6126120, 6636630, 7147140, 7657650, 8168160, 8678670, 9189180, 10210200, 10720710, 11231220, 12252240, 12762750, 13273260, 13783770

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A060735, A080681, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x 510510] == 92160 x, AppendTo[a, 510510 x]], {x, 1, 100}]; a
sdpdQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Length[f]==7&&Max[f]==17]; Select[Range[510510,138*10^5,510510],sdpdQ] (* Harvey P. Dale, Aug 03 2019 *)

a(n) = 510510 * A080681(n). - Amiram Eldar, Mar 10 2020

Sum_{n>=1} 1/a(n) = 1/92160. - Amiram Eldar, Nov 12 2020

A288162 Numbers whose prime factors are 2 and 13.

Original entry on oeis.org

26, 52, 104, 208, 338, 416, 676, 832, 1352, 1664, 2704, 3328, 4394, 5408, 6656, 8788, 10816, 13312, 17576, 21632, 26624, 35152, 43264, 53248, 57122, 70304, 86528, 106496, 114244, 140608, 173056, 212992, 228488, 281216, 346112, 425984, 456976, 562432, 692224, 742586, 851968, 913952

Offset: 1

Bernard Schott, Jun 06 2017

nonn

Numbers k such that phi(k)/k = 6/13.

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

Cf. A033845, A033846, A033847, A033848, A033849, A033850, A033851, A086780, A107326, A143207.

[n:n in [1..100000] | Set(PrimeDivisors(n)) eq {2,13}];  // Marius A. Burtea, May 10 2019

Select[Range[920000],FactorInteger[#][[All,1]]=={2,13}&] (* Harvey P. Dale, Jun 18 2021 *)

is(n) = factor(n)[, 1]~==[2, 13] \\ Felix Fröhlich, Jun 06 2017

list(lim)=my(v=List(),t); for(n=1,logint(lim\2,13), t=13^n; while((t<<=1)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 11 2017

a(n) = 26 * A107326(n). - David A. Corneth, Jun 06 2017

Sum_{n>=1} 1/a(n) = 1/12. - Amiram Eldar, Dec 22 2020

A323330 Larger of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

Original entry on oeis.org

1550, 3100, 4790, 6200, 7750, 9580, 12400, 12922, 15500, 15290, 19160, 20330, 23950, 24800, 25844, 31000, 30580, 38320, 38750, 40660, 47900, 49600, 51688, 62000, 61160, 76640, 77500, 82150, 81320, 76450, 95800, 90454, 99200, 103376, 101650, 119750, 124000, 122320

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The terms are ordered according to the order of their lesser counterparts (A323329).

Amiram Eldar, Table of n, a(n) for n = 1..1000
S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025. See p. 2.

Cf. A001615, A002046, A033845 (Dedekind psi perfect numbers), A323327, A323328, A323329.

psi[n_] := n*Times@@(1+1/Transpose[FactorInteger[n]][[1]]); t[n_]:= psi[n] - n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s,n]], {m, 1, 120000}]; s

A336773 a(n) is the least prime of the form 2^j*3^k + 1, j > 0, k > 0, j + k = n. a(n) = 0 if no such prime exists.

Original entry on oeis.org

7, 13, 37, 73, 97, 193, 577, 769, 3457, 10369, 0, 12289, 629857, 839809, 147457, 995329, 1990657, 786433, 5308417, 120932353, 14155777, 28311553, 0, 113246209, 29386561537, 3439853569, 6879707137, 1811939329, 18345885697, 3221225473, 1253826625537, 0, 85691213438977

Offset: 2

Hugo Pfoertner, Aug 28 2020

nonn

Robert Israel, Table of n, a(n) for n = 2..2193

Cf. A033845, A058383, A336772 (positions of 0).

f:= proc(n) local k,p;
   for k from 1 to n-1 do
     p:= 2^(n-k)*3^k+1;
     if isprime(p) then return p fi
   od;
   0
end proc:
map(f, [$2..40]); # Robert Israel, Aug 30 2020

for(n=2,34, my(pm=oo); for(j=1,n-1, my(k=n-j,p=2^j*3^k+1);if(isprime(p),pm=min(p,pm))); print1(if(pm==oo,0,pm),", "))

A382248 Smallest number k that is neither squarefree nor a prime power such that k is coprime to n.

Original entry on oeis.org

12, 45, 20, 45, 12, 175, 12, 45, 20, 63, 12, 175, 12, 45, 28, 45, 12, 175, 12, 63, 20, 45, 12, 175, 12, 45, 20, 45, 12, 539, 12, 45, 20, 45, 12, 175, 12, 45, 20, 63, 12, 275, 12, 45, 28, 45, 12, 175, 12, 63, 20, 45, 12, 175, 12, 45, 20, 45, 12, 539, 12, 45, 20

Offset: 1

Michael De Vlieger, Mar 31 2025

Let p be the smallest prime that is coprime to n and let q be the second smallest prime that is coprime to n. Then a(n) = p^2 * q.

Records in this sequence are set by n in A002110.

			a(1) = 12 = 2^2*3, since p = 2, q = 3.
a(2) = 45 = 3^2*5, since p = 3, q = 5.
a(3) = 20 = 2^2*5, since p = 2, q = 5.
a(4) = 45 = 3^2*5, since p = 3, q = 5, a(2^i) = 45 for i > 0.
a(6) = 175 = 5^2*7, since p = 5, q = 7.
a(9) = 20 = 2^2*5, since p = 2, q = 5, a(3^i) = 20 for i > 0.
a(10) = 63 = 3^2*7, since p = 3, q = 7.
a(12) = 175 = 5^2*7, since p = 5, q = 7, a(k) = 175 for n in A033845 (i.e., n such that rad(n) = 6).
a(20) = 63 = 3^2*7, since p = 3, q = 7, a(k) = 63 for n in A033846 (i.e., n such that rad(n) = 10).
a(30) = 539 = 7^2*11, since p = 7, q = 11, etc.

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

Cf. A002110, A007947, A053669, A126706, A380539.

Table[c = 0; q = 2; Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]

a(n) = my(k=2); while (isprimepower(k) || issquarefree(k) || (gcd(k, n) != 1) , k++); k; \\ Michel Marcus, Apr 01 2025

a(n) = A053669(n)^2 * A380539(n).

For k and m such that rad(k) = rad(m), a(k) = a(m), where rad = A007947.

A000423 a(n) is smallest number > a(n-1) of form a(i)*a(j), i < j < n.

Original entry on oeis.org

2, 3, 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

Offset: 1

R. Muller

Sequence contains 2, 3 and all numbers of form 2^a*3^b where a >= 1 and b >= 1. - David W. Wilson, Aug 15 1996

Main entry for this sequence is A033845, which is this sequence starting at 6. - Charles R Greathouse IV, Feb 27 2012

Amarnath Murthy, The sum of the reciprocals of the Smarandache multiplicative sequence, (to be published in Smarandache Notions Journal).
F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ
M. Myers, Smarandache Multiplicative Numbers, in Memorables 1998, Bristol Banner Books, Bristol, p. 37, 1998.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Subsequence of A003586 (3-smooth numbers).

A007335 and A033845 are subsequences.

a[1] = 2; a[2] = 3; a[n_] := a[n] = For[k = a[n - 1] + 1, True, k++, If[ AnyTrue[Table[a[i] a[j], {i, 1, n-2}, {j, i+1, n-1}] // Flatten, # == k& ], Return[k]]]; Table[an = a[n]; Print[an]; an, {n, 1, 50}] (* Jean-François Alcover, Feb 08 2016 *)

Sum_{n>=1} 1/a(n) = 4/3. - Amiram Eldar, Jul 31 2022

More terms from David W. Wilson, Aug 15 1996

cp's OEIS Frontend

A086411 Greatest prime factor of 3-smooth numbers.

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A147573 Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.

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A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).

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Maple

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A323329 Lesser of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

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A147574 Numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17}.

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A288162 Numbers whose prime factors are 2 and 13.

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Magma

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A323330 Larger of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

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A336773 a(n) is the least prime of the form 2^j*3^k + 1, j > 0, k > 0, j + k = n. a(n) = 0 if no such prime exists.

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