cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Sometimes called the Hamming sequence, since Hamming asked for an efficient algorithm to generate the list, in ascending order, of all numbers of the form 2^i*3^j*5^k for i,j,k >= 0. The problem was popularized by Edsger Dijkstra.
Numbers k such that 8*k = EulerPhi(30*k). - Artur Jasinski, Nov 05 2008
Where record values greater than 1 occur in A165704: A165705(n) = A165704(a(n)). - Reinhard Zumkeller, Sep 26 2009
Also called "harmonic whole numbers", see Howard and Longair, 1982, Table I, page 121. - Hugo Pfoertner, Jul 16 2020
Also called ugly numbers, although it is not clear why. - Gus Wiseman, May 21 2021
Some woody bamboo species have extraordinarily long and stable flowering intervals that belong to this sequence. The model by Veller, Nowak & Davis justifies this observation from the evolutionary point of view. - Andrey Zabolotskiy, Jun 27 2021
Also those integers k for which, for every prime p > 5, p^(4*k) - 1 == 0 (mod 240*k). - Federico Provvedi, May 23 2022
As noted in the comments to A085152, Størmer's theorem implies that the only pairs of consecutive integers that appear as consecutive terms of this sequence are (1,2), (2,3), (3,4), (4,5), (5,6), (8,9), (9,10), (15,16), (24,25), and (80,81). These all represent significant musical intervals. - Hal M. Switkay, Dec 05 2022

Examples

			From _Gus Wiseman_, May 21 2021: (Start)
The sequence of terms together with their prime indices begins:
      1: {}            25: {3,3}
      2: {1}           27: {2,2,2}
      3: {2}           30: {1,2,3}
      4: {1,1}         32: {1,1,1,1,1}
      5: {3}           36: {1,1,2,2}
      6: {1,2}         40: {1,1,1,3}
      8: {1,1,1}       45: {2,2,3}
      9: {2,2}         48: {1,1,1,1,2}
     10: {1,3}         50: {1,3,3}
     12: {1,1,2}       54: {1,2,2,2}
     15: {2,3}         60: {1,1,2,3}
     16: {1,1,1,1}     64: {1,1,1,1,1,1}
     18: {1,2,2}       72: {1,1,1,2,2}
     20: {1,1,3}       75: {2,3,3}
     24: {1,1,1,2}     80: {1,1,1,1,3}
(End)

Links

Crossrefs

Subsequences: A003592, A003593, A051916 , A257997.
For p-smooth numbers with other values of p, see A003586, A002473, A051038, A080197, A080681, A080682, A080683.
Cf. A006530, A112757, A112758, A112759, A112763, A112764, A291719.
The partitions with these Heinz numbers are counted by A001399.
The conjugate opposite is A033942, counted by A004250.
The opposite is A059485, counted by A004250.
The non-3-smooth case is A080193, counted by A069905.
The conjugate is A037144, counted by A001399.
The complement is A279622, counted by A035300.
Requiring the sum of prime indices to be even gives A344297.
Cf. A000244, A002182, A002183, A035301, A056239, A112798, A261144, A344293.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a051037 n = a051037_list !! (n-1)
a051037_list = f $ singleton 1 where
   f s = y : f (insert (5 * y) $ insert (3 * y) $ insert (2 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015
  • Magma
    [n: n in [1..500] | PrimeDivisors(n) subset [2,3,5]]; // Bruno Berselli, Sep 24 2012
  • Maple
    A051037 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            numtheory[factorset](a) minus {2, 3,5 } ;
            if % = {} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A051037(n),n=1..100) ; # R. J. Mathar, Nov 05 2017
  • Mathematica
    mx = 405; Sort@ Flatten@ Table[ 2^a*3^b*5^c, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}] (* Or *)
Select[ Range@ 405, Last@ Map[First, FactorInteger@ #] < 7 &] (* Robert G. Wilson v *)
With[{nn=10},Select[Union[Times@@@Flatten[Table[Tuples[{2,3,5},n],{n,0,nn}],1]],#<=2^nn&]] (* Harvey P. Dale, Feb 28 2022 *)
  • PARI
    test(n)= {m=n; forprime(p=2,5, while(m%p==0,m=m/p)); return(m==1)}
for(n=1,500,if(test(n),print1(n",")))
  • PARI
    a(n)=local(m); if(n<1,0,n=a(n-1); until(if(m=n, forprime(p=2,5, while(m%p==0,m/=p)); m==1),n++); n)
  • PARI
    list(lim)=my(v=List(),s,t); for(i=0,logint(lim\=1,5), t=5^i; for(j=0,logint(lim\t,3), s=t*3^j; while(s<=lim, listput(v,s); s<<=1))); Set(v) \\ Charles R Greathouse IV, Sep 21 2011; updated Sep 19 2016
  • PARI
    smooth(P:vec,lim)={ my(v=List([1]),nxt=vector(#P,i,1),indx,t);
while(1, t=vecmin(vector(#P,i,v[nxt[i]]*P[i]),&indx);
if(t>lim,break); if(t>v[#v],listput(v,t)); nxt[indx]++);
Vec(v)
};
smooth([2,3,5], 1e4) \\ Charles R Greathouse IV, Dec 03 2013
  • PARI
    is_A051037(n)=n<7||vecmax(factor(n,6)[, 1])<7 \\ M. F. Hasler, Jan 16 2015
  • Python
    def isok(n):
  while n & 1 == 0: n >>= 1
  while n % 3 == 0: n //= 3
  while n % 5 == 0: n //= 5
  return n == 1 #  Darío Clavijo, Dec 30 2022
  • Python
    from sympy import integer_log
def A051037(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,5)[0]+1):
            for j in range(integer_log(y:=x//5**i,3)[0]+1):
                c -= (y//3**j).bit_length()
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024
  • Python
    # faster for initial segment of sequence
import heapq
from itertools import islice
def A051037gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                    heapq.heappush(h, v*p)
print(list(islice(A051037gen(), 65))) # Michael S. Branicky, Sep 17 2024

Formula

Let s(n) = Card(k | a(k)Benoit Cloitre, Dec 30 2001
The characteristic function of this sequence is given by:
Sum_{n>=1} x^a(n) = Sum_{n>=1} -Möbius(30*n)*x^n/(1-x^n). - Paul D. Hanna, Sep 18 2011
a(n) = A143207(n) / 30. - Reinhard Zumkeller, Sep 13 2011
A204455(15*a(n)) = 15, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
A006530(a(n)) <= 5. - Reinhard Zumkeller, May 16 2015
Sum_{n>=1} 1/a(n) = Product_{primes p <= 5} p/(p-1) = (2*3*5)/(1*2*4) = 15/4. - Amiram Eldar, Sep 22 2020

A004250 Number of partitions of n into 3 or more parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, 127, 168, 222, 288, 375, 480, 616, 781, 990, 1243, 1562, 1945, 2422, 2996, 3703, 4550, 5588, 6826, 8332, 10126, 12292, 14865, 17958, 21618, 25995, 31165, 37317, 44562
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of (n+1)-vertex spider graphs: trees with n+1 vertices and exactly 1 vertex of degree at least 3 (i.e. branching vertex). There is a trivial bijection with the objects described in the definition. - Emeric Deutsch, Feb 22 2014
Also the number of graphical partitions of 2n into n parts. - Gus Wiseman, Jan 08 2021

Examples

			a(6)=7 because there are three partitions of n=6 with i=3 parts: [4, 1, 1], [3, 2, 1], [2, 2, 2] and two partitions with i=4 parts: [3, 1, 1, 1], [2, 2, 1, 1] and one partition with i=5 parts: [2, 1, 1, 1, 1] and one partition with i=6 parts: [1, 1, 1, 1, 1, 1].
From _Gus Wiseman_, Jan 18 2021: (Start)
The a(3) = 1 through a(7) = 11 graphical partitions of 2n into n parts:
  (222)  (2222)  (22222)  (222222)  (2222222)
         (3221)  (32221)  (322221)  (3222221)
                 (33211)  (332211)  (3322211)
                 (42211)  (333111)  (3332111)
                          (422211)  (4222211)
                          (432111)  (4322111)
                          (522111)  (4331111)
                                    (4421111)
                                    (5222111)
                                    (5321111)
                                    (6221111)
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

Links

Crossrefs

Cf. A004251, A029889, A035300, A095268, A058984.
Rightmost column of A259873.
Central column of A339659.
A000041 counts partitions of 2n into n parts, ranked by A340387.
A000569 counts graphical partitions, ranked by A320922.
A008284 counts partitions by sum and length.
A027187 counts partitions of even length.
A309356 ranks simple covering graphs.
The following count vertex-degree partitions and give their Heinz numbers:
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339617 counts non-graphical partitions of 2n (A339618).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A000219, A339560, A339561, A339661.
Partial sums of A117995.

Programs

  • Maple
    with(combinat);
for i from 1 to 15 do pik(i,3) od;
pik:= proc(n::integer, k::integer)
# Thomas Wieder, Jan 30 2007
local i, Liste, Result;
if k > n or n < 0 or k < 1 then
return fail
end if;
Result := 0;
for i from k to n do
Liste:= PartitionList(n,i);
#print(Liste);
Result := Result + nops(Liste);
end do;
return Result;
end proc;
PartitionList := proc (n, k)
# Authors: Herbert S. Wilf and Joanna Nordlicht. Source: Lecture Notes
# "East Side West Side,..." University of Pennsylvania, USA, 2002.
# Available at: http://www.cis.upenn.edu/~wilf/lecnotes.html
# Calculates the partition of n into k parts.
# E.g. PartitionList(5,2) --> [[4, 1], [3, 2]].
local East, West;
if n < 1 or k < 1 or n < k then
RETURN([])
elif n = 1 then
RETURN([[1]])
else if n < 2 or k < 2 or n < k then
West := []
else
West := map(proc (x) options operator, arrow;
[op(x), 1] end proc,PartitionList(n-1,k-1)) end if;
if k <= n-k then
East := map(proc (y) options operator, arrow;
map(proc (x) options operator, arrow; x+1 end proc,y) end proc,PartitionList(n-k,k))
else East := [] end if;
RETURN([op(West), op(East)])
end if;
end proc;
#  Thomas Wieder, Feb 01 2007
ZL :=[S, {S = Set(Cycle(Z),3 <= card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..41); # Zerinvary Lajos, Mar 25 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card >=3)},unlabelled]: seq(combstruct[count](B, size=n), n=1..41); # Zerinvary Lajos, Mar 21 2009
  • Mathematica
    Length /@ Table[Select[Partitions[n], Length[#] > 2 &], {n, 20}] (* Eric W. Weisstein, May 16 2007 *)
Table[Count[Length /@ Partitions[n], ?(# > 2 &)], {n, 20}] (* _Eric W. Weisstein, May 16 2017 *)
Table[PartitionsP[n] - Floor[n/2] - 1, {n, 20}] (* Eric W. Weisstein, May 16 2017 *)
Length /@ Table[IntegerPartitions[n, {3, n}], {n, 20}] (* Eric W. Weisstein, May 16 2017 *)
  • PARI
    a(n) = numbpart(n) - (n+2)\2; /* Joerg Arndt, Apr 03 2013 */

Formula

G.f.: Sum_{n>=0} (q^n / Product_{k=1..n+3} (1 - q^k)). - N. J. A. Sloane
a(n) = A000041(n) - floor((n+2)/2) = A000041(n)-A004526(n+2) = A058984(n)-1. - Vladeta Jovovic, Jun 18 2003
Let P(n,i) denote the number of partitions of n into i parts. Then a(n) = Sum_{i=3..n} P(n,i). - Thomas Wieder, Feb 01 2007
a(n) = A259873(n,n). - Gus Wiseman, Jan 08 2021

Extensions

Definition corrected by Thomas Wieder, Feb 01 2007 and by Eric W. Weisstein, May 16 2007

A058399 Triangle of partial row sums of partition triangle A008284.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1, 11, 10, 7, 4, 2, 1, 15, 14, 11, 7, 4, 2, 1, 22, 21, 17, 12, 7, 4, 2, 1, 30, 29, 25, 18, 12, 7, 4, 2, 1, 42, 41, 36, 28, 19, 12, 7, 4, 2, 1, 56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1, 77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1, 101, 100, 94, 80, 62
Offset: 1

Views

Author

Wolfdieter Lang, Dec 11 2000

Keywords

Comments

T(n,m) is also the number of m-th largest elements in all partitions of n. - Omar E. Pol, Feb 14 2012
It appears that reversed rows converge to A000070. - Omar E. Pol, Mar 10 2012
The row sums give A006128. - Omar E. Pol, Mar 26 2012
T(n,m) is also the number of regions traversed by the m-th column of the section model of partitions with n sections (Cf. A135010, A206437). - Omar E. Pol, Apr 20 2012

Examples

			From _Omar E. Pol_, Mar 10 2012: (Start)
Triangle begins:
   1;
   2,  1;
   3,  2,  1;
   5,  4,  2,  1;
   7,  6,  4,  2,  1;
  11, 10,  7,  4,  2,  1;
  15, 14, 11,  7,  4,  2,  1;
  22, 21, 17, 12,  7,  4,  2,  1;
  30, 29, 25, 18, 12,  7,  4,  2,  1;
  42, 41, 36, 28, 19, 12,  7,  4,  2,  1;
  56, 55, 50, 40, 29, 19, 12,  7,  4,  2,  1;
  77, 76, 70, 58, 43, 30, 19, 12,  7,  4,  2,  1;
(End)

Links

Crossrefs

Columns 1-5: A000041(n), A000065(n+1), A004250(n+2), A035300(n-1), A035301(n-1), n >= 1.
Cf. A008284.

Programs

  • Maple
    b:= proc(n, k) option remember;
      `if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
    end:
T:= (n, m)-> b(n,n) -b(n,m-1):
seq (seq (T(n, m), m=1..n), n=1..15);  # Alois P. Heinz, Apr 20 2012
  • Mathematica
    t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

T(n, m) = Sum_{k=m..n} A008284(n, k).
G.f. for m-th column: Sum_{n>=1} x^(n)/Product_{k=1..n+m-1} (1 - x^k).
T(n, m) = Sum_{k=1..n} A207379(k, m). - Omar E. Pol, Apr 22 2012

A347542 Number of partitions of n into 6 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 44, 65, 92, 130, 178, 244, 326, 435, 571, 747, 964, 1242, 1581, 2009, 2530, 3178, 3962, 4930, 6094, 7518, 9225, 11296, 13768, 16751, 20295, 24546, 29583, 35591, 42685, 51112, 61028, 72757, 86523, 102740, 121720, 144007, 170018, 200461, 235910, 277270
Offset: 6

Views

Author

Ilya Gutkovskiy, Sep 06 2021

Keywords

Crossrefs

Cf. A000065, A001402, A004250, A035300, A035301, A347543, A347544, A347545, A347547.

Programs

  • Mathematica
    nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 6, nmax}], {x, 0, nmax}], x] // Drop[#, 6] &

Formula

G.f.: Sum_{k>=6} x^k / Product_{j=1..k} (1 - x^j).

A347543 Number of partitions of n into 7 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 134, 186, 255, 345, 461, 611, 801, 1043, 1346, 1727, 2199, 2787, 3508, 4398, 5482, 6809, 8414, 10365, 12711, 15545, 18935, 23006, 27854, 33646, 40513, 48680, 58326, 69748, 83192, 99048, 117650, 139513, 165083, 195034, 229968, 270760
Offset: 7

Views

Author

Ilya Gutkovskiy, Sep 06 2021

Keywords

Crossrefs

Cf. A000065, A004250, A008636, A035300, A035301, A347542, A347544, A347545, A347547.

Programs

  • Mathematica
    nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 7, nmax}], {x, 0, nmax}], x] // Drop[#, 7] &

Formula

G.f.: Sum_{k>=7} x^k / Product_{j=1..k} (1 - x^j).

A347544 Number of partitions of n into 8 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 96, 137, 190, 263, 356, 480, 637, 842, 1098, 1427, 1835, 2351, 2986, 3780, 4749, 5949, 7405, 9190, 11344, 13966, 17111, 20913, 25454, 30908, 37393, 45141, 54315, 65222, 78090, 93317, 111220, 132323, 157050, 186088, 220015, 259716
Offset: 8

Views

Author

Ilya Gutkovskiy, Sep 06 2021

Keywords

Crossrefs

Cf. A000065, A004250, A008637, A035300, A035301, A347542, A347543, A347545, A347547.

Programs

  • Mathematica
    nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 8, nmax}], {x, 0, nmax}], x] // Drop[#, 8] &

Formula

G.f.: Sum_{k>=8} x^k / Product_{j=1..k} (1 - x^j).

A347545 Number of partitions of n into 9 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 138, 193, 267, 364, 491, 656, 868, 1139, 1483, 1917, 2461, 3142, 3985, 5030, 6315, 7893, 9817, 12165, 15007, 18451, 22597, 27589, 33565, 40724, 49249, 59410, 71460, 85753, 102632, 122574, 146032, 173638, 206003, 243951, 288296, 340124
Offset: 9

Views

Author

Ilya Gutkovskiy, Sep 06 2021

Keywords

Crossrefs

Cf. A000065, A004250, A008638, A035300, A035301, A347542, A347543, A347544, A347547.

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 9, nmax}], {x, 0, nmax}], x] // Drop[#, 9] &

Formula

G.f.: Sum_{k>=9} x^k / Product_{j=1..k} (1 - x^j).

A347547 Number of partitions of n into 10 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 270, 368, 499, 667, 887, 1165, 1524, 1973, 2544, 3253, 4143, 5239, 6602, 8268, 10320, 12813, 15859, 19537, 24000, 29359, 35820, 43541, 52795, 63803, 76929, 92476, 110926, 132694, 158414, 188649, 224231, 265916, 314793
Offset: 10

Views

Author

Ilya Gutkovskiy, Sep 06 2021

Keywords

Crossrefs

Cf. A000065, A004250, A008639, A035300, A035301, A347542, A347543, A347544, A347545.

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 10, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: Sum_{k>=10} x^k / Product_{j=1..k} (1 - x^j).

A127745 Counts Bell numbers (except for Catalans) associated with the partition number [n].

Original entry on oeis.org

0, 0, 0, 1, 8, 50, 294, 1717, 10194, 62284, 394346, 2597266, 17827166, 127575414, 951411752, 7386583917, 59623674472, 499648882838, 4340548090590, 39033489125836, 362871600781796, 3482858492844510, 34471940635650958, 351444263328831458
Offset: 1

Views

Author

Alford Arnold, Feb 25 2007

Keywords

Comments

A074664 counts the Bell Numbers associated with the partition number [n]. A000108 counts the corresponding Catalan numbers and here we count the remaining Bell numbers associated with the partition number [n].

Examples

			There are 15 Bell objects when n = 4, 14 are also Catalans so a(4) = 1.
There are 52 Bell objects when n = 5, 42 are also Catalans; we know that 5 = 4+1 = 1+4 which accounts for two of the non-Catalan Bells so, a(5) = 52 - 42 - 2 = 8.

Crossrefs

Cf. A000041, A000108, A000110, A001399, A016098, A035300, A127743, A074664.

Formula

a(n) = A074664(n) - A000108(n-1)
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