A080404 -id:A080404 - cp's OEIS frontend

A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.

1, 2, 4, 6, 8, 12, 16, 18, 24, 30, 32, 36, 48, 54, 60, 64, 72, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192, 210, 216, 240, 256, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 512, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972

Offset: 1

Leroy Quet, Jul 17 2000

a(n) is also the sorted version of A057335 which is generated recursively using the formula A057335 = A057334 * A057335(repeated), where A057334 = A000040(A000120). - Alford Arnold, Nov 11 2001
Squarefree kernels of these numbers are primorial numbers. See A080404. - Labos Elemer, Mar 19 2003
If u and v are terms then so is u*v. - Reinhard Zumkeller, Nov 24 2004
Except for the initial value a(1) = 1, a(n) gives the canonical primal code of the n-th finite sequence of positive integers, where n = (prime_1)^c_1 * ... * (prime_k)^c_k is the code for the finite sequence c_1, ..., c_k. See examples of primal codes at A106177. - Jon Awbrey, Jun 22 2005
From Daniel Forgues, Jan 24 2011: (Start)
Least integer, in increasing order, of each ordered prime signature.
The least integer of each ordered prime signature are the smallest numbers with a given tuple of exponents of prime factors.
The ordered prime signature (where the order of exponents matters) of n corresponds to a given composition of Omega(n), as opposed to the prime signature of n, which corresponds to a given partition of Omega(n). (End)
Except for the initial entry 1, the entries of the sequence are the Heinz numbers of all partitions that contain all parts 1,2,...,k, where k is the largest part. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436. The number 150 (= 2*3*5*5) is in the sequence because it is the Heinz number of the partition [1,2,3,3]. - Emeric Deutsch, May 22 2015
Numbers n such that A053669(n) > A006530(n). - Anthony Browne, Jun 06 2016
From David W. Wilson, Dec 28 2018: (Start)
Numbers n such that for primes p > q, p | n => q | n.
Numbers n such that prime p | n => A034386(p) | n. (End)

			60 is included because 60 = 2^2 * 3 * 5 and 2, 3 and 5 are consecutive primes beginning at 2.
Sequence A057335 begins
1..2..4..6..8..12..18..30..16..24..36..60..54..90..150..210... which is equal to
1..2..2..3..2...3...3...5...2...3...3...5...3...5....5....7... times
1..1..2..2..4...4...6...6...8...8..12..12..18..18...30...30...

Michael De Vlieger, Table of n, a(n) for n = 1..10000, first 1001 terms from Franklin T. Adams-Watters.
Jon Awbrey, Riffs and Rotes.
Michael De Vlieger, Extended table of n, a(n) for n = 1..100000.
Robert Vajda, Computational Exploration of the Degree Sequence of the Malyshev Polynomials, Proceedings of the 11th International Conference on Applied Informatics (Eger, Hungary, 2020).
Index entries for sequences related to prime signature.

Cf. A057335 (permuted), A056808, A025487, A007947, A002110, A080404, A034386, A106177, A124829, A124830, A124831, A124833, A080259 (complement), A215366.

Cf. A005867, A324939, A345974.

[1] cat [k:k in[2..1000 by 2]|forall{i:i in [1..#PrimeDivisors(k)-1]|NextPrime(pd[i]) in pd where pd is PrimeDivisors(k)}]; // Marius A. Burtea, Feb 01 2020

isA055932 := proc(n)
    local s,p ;
    s := numtheory[factorset](n) ;
    for p in s do
        if p > 2 and not prevprime(p)  in s then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 2 to 100 do
    if isA055932(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 02 2012

Select[Range[1000], #==1||FactorInteger[ # ][[ -1, 1]]==Prime[Length[FactorInteger[ # ]]]&]
cpQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},f=={1}||f==Prime[ Range[Length[f]]]]; Select[Range[1000],cpQ] (* Harvey P. Dale, Jul 14 2012 *)

is(n)=my(f=factor(n)[,1]~);f==primes(#f) \\ Charles R Greathouse IV, Aug 22 2011

list(lim,p=2)=my(v=[1],q=nextprime(p+1),t=1);while((t*=p)<=lim,v=concat(v,t*list(lim\t,q))); vecsort(v) \\ Charles R Greathouse IV, Oct 02 2012

from itertools import count, islice
from sympy import primepi, primefactors
def A055932_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        p = list(map(primepi,primefactors(k)))
        if k==1 or (min(p)==1 and max(p)==len(p)):
            yield k
A055932_list = list(islice(A055932_gen(),40)) # Chai Wah Wu, Aug 07 2025

Sum_{n>=1} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

Edited by Daniel Forgues, Jan 24 2011

A100827 Highly cototient numbers: records for a(n) in A063741.

Original entry on oeis.org

2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, 2099, 2309, 2729, 3149, 3359, 3569, 3989, 4199, 4289, 4409, 4619, 5249, 5459, 5879, 6089, 6509, 6719, 6929

Offset: 1

Alonso del Arte, Jan 06 2005

nonn

Each number k on this list has more solutions to the equation x - phi(x) = k (where phi is Euler's totient function, A000010) than any preceding k except 1.
This sequence is a subset of A063741. As noted in that sequence, there are infinitely many solutions to x - phi(x) = 1. Unlike A097942, the highly totient numbers, this sequence has many odd numbers besides 1.
With the expection of 2, 4, 8, all of the known terms are congruent to -1 mod a primorial (A002110). The specific primorial satisfying this congruence would result in a sequence similar to A080404 a(n)=A007947[A055932(n)]. - Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Dec 28 2006
Because most of the solutions to x - phi(x) = k are semiprimes p*q with p+q=k+1, it appears that this sequence eventually has terms that are one less than the Goldbach-related sequence A082917. In fact, terms a(108) to a(176) are A082917(n)-1 for n=106..174. [T. D. Noe, Mar 16 2010] This holds through a(229).  [Jud McCranie, May 18 2017]

			a(3) = 8 since x - phi(x) = 8 has three solutions, {12, 14, 16}, one more than a(2) = 4 which has two solutions, {6, 8}.

Jud McCranie, Table of n, a(n) for n = 1..229 (terms 1..176 by T. D. Noe)
Wikipedia, Highly cototient number

Cf. A063741, A097942, A007374, A101373.

searchMax = 4000; coPhiAnsYldList = Table[0, {searchMax}]; Do[coPhiAns = m - EulerPhi[m]; If[coPhiAns <= searchMax, coPhiAnsYldList[[coPhiAns]]++ ], {m, 1, searchMax^2}]; highlyCototientList = {2}; currHigh = 2; Do[If[coPhiAnsYldList[[n]] > coPhiAnsYldList[[currHigh]], highlyCototientList = {highlyCototientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyCototientList]

More terms from Robert G. Wilson v, Jan 08 2005

A084918 Numbers n >= 1000, such that if prime P divides n, then so does each smaller prime.

Original entry on oeis.org

1024, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1728, 1800, 1890, 1920, 1944, 2048, 2100, 2160, 2250, 2304, 2310, 2400, 2430, 2520, 2592, 2700, 2880, 2916, 2940, 3000, 3072, 3150, 3240, 3360, 3456, 3600, 3750

Offset: 0

Alford Arnold, Jul 15 2003

A055932 lists terms below 1000.

Cf. A055932, A056808, A057335, A066099, A071364, A080259, A080404.

espQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Prime[Range[ PrimePi[ Max[f]]]] == f]; Select[Range[1000,4000],espQ] (* Harvey P. Dale, Mar 09 2019 *)

Edited by Don Reble, Nov 03 2003

A373515 Numbers k, divisible by 2 but not by 4, such that rad(k) is primorial.

Original entry on oeis.org

2, 6, 18, 30, 54, 90, 150, 162, 210, 270, 450, 486, 630, 750, 810, 1050, 1350, 1458, 1470, 1890, 2250, 2310, 2430, 3150, 3750, 4050, 4374, 4410, 5250, 5670, 6750, 6930, 7290, 7350, 9450, 10290, 11250, 11550, 12150, 13122, 13230, 15750, 16170, 17010, 18750, 20250

Offset: 1

David James Sycamore, Jun 07 2024

nonn

Intersection of A055932 and A016825. In other words, numbers k congruent to 2 (mod 4) such that the squarefree kernel of k is a term in A002110.  A term m in A055932 is in this sequence iff m/2 is an odd number.
If x, y are terms in this sequence then x*y is not. All primorial numbers >= 2 are terms.
For i >= 1, primorial A002110(i) is a term in this sequence, since primorials are squarefree. - Michael De Vlieger, Jun 08 2024

			6 is a term because 2|6 but 4!|6 and rad(6) = 6 = A002110(2) is a primorial number.
A primorial number m > 1 is a term since m is squarefree and == 2 (mod 4).

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

Cf. A002110, A007947, A016825, A055932, A080404.

Select[Range[2, 25000, 4], Or[# == {2}, Union@ Differences@ PrimePi[#] == {1}] &@
FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Jun 08 2024 *)

lista(kmax) = {my(f); forstep(k = 2, kmax, 4, f = factor(k); if(primepi(f[#f~, 1]) == #f~, print1(k, ", ")));} \\ Amiram Eldar, Jun 08 2024

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A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.

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A100827 Highly cototient numbers: records for a(n) in A063741.

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A084918 Numbers n >= 1000, such that if prime P divides n, then so does each smaller prime.

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A373515 Numbers k, divisible by 2 but not by 4, such that rad(k) is primorial.

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