A055932
Numbers all of whose prime divisors are consecutive primes starting at 2.
Original entry on oeis.org
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
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.
-
[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
A005867
a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).
Original entry on oeis.org
1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000
Offset: 0
a(3): the mod 30 prime remainder set sieve representation yields the remainder set: {1, 7, 11, 13, 17, 19, 23, 29}, 8 elements.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 0..99
- Larry Deering, The Black Key Sieve, Box 275, Bellport NY 11713-0275, 1998.
- Alphonse de Polignac, Six propositions arithmologiques déduites du crible d'Ératosthène, Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale, Série 1, Tome 8 (1849), pp. 423-429. See p. 425.
- Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
- Ken Hicks and Kevin Ward, Series and Product Relations Made from Primes, arXiv:2108.03268 [math.NT], 2021.
- Dennis Martin, Proofs Regarding Primorial Patterns [via Internet Archive Wayback-machine]
- Dennis Martin, Proofs Regarding Primorial Patterns [Cached copy, with permission of the author]
- Francis E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991.
- Travis Near, Improving MATLAB's isprime performance without arbitrary-precision arithmetic, arXiv:2108.04791 [cs.MS], 2021.
- John K. Sellers, Distribution of twin primes in repeating sequences of prime factors, arXiv:2108.00288 [math.GM], 2021. See Table 1 p. 11.
- Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
-
a005867 n = a005867_list !! n
a005867_list = scanl (*) 1 a006093_list
-- Reinhard Zumkeller, May 01 2013
-
A005867 := proc(n)
mul(ithprime(j)-1,j=1..n) ;
end proc: # Zerinvary Lajos, Aug 24 2008, R. J. Mathar, May 03 2017
-
Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
RecurrenceTable[{a[0]==1,a[n]==(Prime[n]-1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Dec 09 2013 *)
EulerPhi@ FoldList[Times, 1, Prime@ Range@ 18] (* Michael De Vlieger, Mar 18 2016 *)
-
for(n=0, 22, print1(prod(k=1,n, prime(k)-1), ", "))
Offset changed to 0, Name changed, and Comments and Examples sections edited by
T. D. Noe, Apr 04 2010
A057335
a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.
Original entry on oeis.org
1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410
Offset: 0
From _Gus Wiseman_, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
18: {1,2,2}
30: {1,2,3}
16: {1,1,1,1}
24: {1,1,1,2}
36: {1,1,2,2}
60: {1,1,2,3}
54: {1,2,2,2}
90: {1,2,2,3}
150: {1,2,3,3}
210: {1,2,3,4}
32: {1,1,1,1,1}
48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)
Unsorted prime signature is
A124010.
Numbers whose prime signature is aperiodic are
A329139.
All of the following pertain to compositions in standard order (
A066099):
- Constant compositions are
A272919.
- Aperiodic compositions are
A328594.
Cf.
A005867,
A029931,
A048793,
A052409,
A056239,
A066099,
A112798,
A124767,
A228351,
A233249,
A333220,
A345974.
-
Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)
-
mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); v;} \\ Michel Marcus, Feb 09 2014
-
A057335(n) = if(0==n,1,prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
A056808
Members of A055932 which are not least prime signatures (cf. A025487).
Original entry on oeis.org
18, 54, 90, 108, 150, 162, 270, 300, 324, 450, 486, 540, 600, 630, 648, 750, 810, 972, 1050, 1200, 1350, 1458, 1470, 1500, 1620, 1890, 1944, 2100, 2250, 2400, 2430, 2700, 2916, 2940, 3000, 3150, 3240, 3750, 3780, 3888, 4050, 4200, 4374, 4410, 4500, 4800
Offset: 1
18 = 2*3*3 and all prime divisors are consecutive primes but the least prime signature is 12 = 2*2*3; so a(1) = 18.
-
With[{nn = 4800}, Select[Range[2, nn], And[#1 != Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[#3, Greater]], Last[#2] == Prime@ Length[#2]] & @@ Apply[Join, {{#1}, Transpose@ #2}] & @@ {#, FactorInteger[#]} &] ] (* Michael De Vlieger, Feb 06 2020 *)
More terms from Larry Reeves (larryr(AT)acm.org), Nov 28 2000
Showing 1-4 of 4 results.
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