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.

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A062292 A B_2 sequence: a(n) is the smallest cube such that the pairwise sums of {a(1)...a(n)} are all distinct.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 35937, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336
Offset: 1

Views

Author

Labos Elemer, Jul 02 2001

Keywords

Comments

A Mian-Chowla sequence consisting only of cubes.

Examples

			During recursive construction of this set, for n=1-50, the cubes of 12,18,24,32,34,36,48 are left out to keep all sums of distinct cubes distinct from each other.

Crossrefs

Cf. A011185, A010672, A025582, A005282, A062294.

Programs

  • Python
    from itertools import count, islice
def A062292_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in (n**3 for n in count(1)):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2.update(bset2)
A062292_list = list(islice(A062292_gen(),30)) # Chai Wah Wu, Sep 05 2023

A133604 Elements of A005282 that are also the sum of a pair of not necessarily distinct elements of A005282.

Original entry on oeis.org

2, 4, 8, 21, 66, 97, 204, 565, 662, 775, 970, 1821, 2780, 6374, 8730, 8942, 10898, 24596, 55307, 67189, 79047, 84345, 164868, 231694, 233570, 234619, 271511, 298414, 433973, 474668, 475800, 567408, 829129, 839728, 889285, 1394240
Offset: 1

Views

Author

Klaus Brockhaus, Sep 18 2007

Keywords

Comments

A005282 is the sequence of smallest numbers such that the pairwise sums of not necessarily distinct elements are all distinct.
Conjecture: 2, 4 and 8 are the only terms n such that n = 2*A005282(k) for some k.

Examples

			A005282(3) = 4 + 4 = 8 = A005282(4), hence 8 is in the sequence.
A005282(10) = 81, A005282(12) = 123. 81 + 123 = 204 = A005282(15), hence 204 is in the sequence.

Links

Crossrefs

Cf. A005282, A011185, A133605.

Programs

  • Python
    from itertools import count, islice
def A133604_gen(): # generator of terms
    aset2, alist = set(), []
    for k in count(1):
        bset2 = {r:=k<<1}
        if r not in aset2:
            for d in alist:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                if k in aset2:
                    yield k
                alist.append(k)
                aset2.update(bset2)
A133604_list = list(islice(A133604_gen(),30)) # Chai Wah Wu, Sep 11 2023

A349777 Lexicographically first sequence of positive integers such that all disjoint equivalent sets of K terms have distinct sums for 1 <= K <= 4.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 25, 45, 85, 162, 310, 595, 1107, 2052, 3515, 5925, 9798, 16169, 23295, 34303, 53259, 72215, 112624, 153552, 198523, 283570, 370114, 497383, 700022, 840817, 1145415, 1398434, 1717972, 2279969, 2819186, 3436864, 4299205, 5239007, 6335442, 7650495, 9219214
Offset: 1

Views

Author

Santanu Banerjee, Nov 29 2021

Keywords

Links

Crossrefs

Cf. A011185 (k=1..2), A036241 (k=1..3).
A005318 and A276661 are similar sequences.

Programs

  • Mathematica
    a={};k=1;Do[While[(t=1;While[t<=4&&DuplicateFreeQ[Total/@Subsets[Join[a,{k}],{t}]],t++];t)<=4,k++];AppendTo[a,k];Print@k,30] (* Giorgos Kalogeropoulos, Dec 02 2021 *)
  • Python
    # See links.

Extensions

a(13)-a(26) from Alois P. Heinz, Dec 01 2021
a(27)-a(41) from Gleb Ivanov, Dec 02 2021

A080432 Even numbers such that all a(i) + a(j) are distinct.

Original entry on oeis.org

2, 4, 6, 10, 16, 26, 42, 60, 78, 106, 148, 190, 256, 304, 364, 424, 516, 632, 748, 826, 952, 1062, 1092, 1216, 1434, 1596, 1724, 1930, 2120, 2322, 2614, 2772, 2870, 3112, 3444, 3668, 3868, 4116, 4522, 4994, 5398, 5748, 6122, 6394, 6664, 7258, 7424, 7736
Offset: 1

Views

Author

Amarnath Murthy, Feb 20 2003

Keywords

Crossrefs

Cf. A080430, A080431.
2 * A011185(n).

Programs

  • Maple
    sumset := {6}: keepA := array(1..10000): for m from 1 to 10000 do keepA[m] := 0 od: keepA[2] := 2: keepA[4] := 4: for n from 6 to 10000 by 2 do mytest := 0: for j from 2 to n-2 by 2 do if keepA[j]>0 then if member(keepA[j]+n, sumset) then mytest := 1; break; fi: fi: od: if mytest=0 then keepA[n] := n; for j from 2 to n-2 by 2 do sumset := sumset union {keepA[j]+n} od: fi: od: for i from 2 to 10000 by 2 do if keepA[i]>0 then printf(`%d,`, keepA[i]) fi: od: # James Sellers, Feb 26 2003

Extensions

More terms from James Sellers, Feb 26 2003

A317778 Starting with 1,2,3,4,5,6: a(n) is the next smallest number greater than a(n-1) such that a[i] + a[j] + a[k] != a[x] + a[y] + a[z] for 1 <= i,j,k,x,y,z <= n all distinct.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 13, 22, 39, 72, 131, 229, 386, 641, 896, 1164, 1419, 1855, 2831, 3545, 5036, 5750, 8034, 10022, 12227, 14377, 17455, 19951, 24701, 27197, 36455, 42303, 49751, 57232, 65684, 83879, 94391, 110073, 124015, 137442, 156835, 175130, 209215, 229396, 242692
Offset: 1

Views

Author

Ben Paul Thurston, Aug 06 2018

Keywords

Comments

a(n) <= a(n-1) + a(n-2) + a(n-3) - 2. - Charlie Neder, Feb 09 2019

Examples

			After 1,2,3,4,5,6: 7 cannot be the next term because 1+3+7 = 2+4+5.

Links

Crossrefs

Cf. A011185.

Programs

  • Python
    def u(series):
    for i in range(0, len(series)):
        for j in range(i+1, len(series)):
            for k in range(j+1, len(series)):
                for l in range(0, len(series)):
                    for m in range(l+1, len(series)):
                        for n in range(m+1, len(series)):
                            if len(set([i,j,k,l,m,n]))==6:
                                if series[i]+series[j]+series[k]==series[l]+series[m]+series[n]:
                                    return False
    return True
def a(series, n):
    a = []
    for i in range(0, len(series)):
        a.append(series[i])
    a.append(n)
    return a
series = [1, 2, 3,4,5,6]
for i in range(7, 1000):
    print(i)
    nseries = a(series, i)
    if u(nseries):
        series.append(i)
        print(series)
print(series)

Extensions

a(24)-a(45) from Charlie Neder, Feb 09 2019

A062559 A B2 sequence consisting of powers of primes but not primes: a(n) = least value from A025475 such that sequence increases and pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 49, 64, 121, 243, 256, 343, 512, 729, 961, 1024, 1331, 1369, 1849, 2048, 2187, 2197, 2401, 3125, 3481, 4096, 4913, 5329, 6561, 6859, 6889, 8192, 10201, 12769, 14641, 15625, 16384, 16807, 19683, 22201, 22801, 27889, 28561, 29791, 32768
Offset: 0

Views

Author

Labos Elemer, Jul 03 2001

Keywords

Comments

32 is the first prime power > 1 not in this sequence.

Crossrefs

Cf. A000961, A025475, A010672, A011185, A025582, A005282.

Programs

  • Python
    from itertools import count, islice
from sympy import factorint
def A062559_gen(): # generator of terms
    aset2, alist = set(), []
    for k in count(0):
        if len(f:=factorint(k).values()) == 1 and max(f) > 1:
            bset2 = set()
            for a in alist:
                if (b:=a+k) in aset2:
                    break
                bset2.add(b)
            else:
                yield k
                alist.append(k)
                aset2.update(bset2)
A062559_list = list(islice(A062559_gen(),30)) # Chai Wah Wu, Sep 11 2023

Extensions

More terms from Sean A. Irvine, Apr 03 2023
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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