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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A337946 a(1)=1; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n*(n+1) different entries (the maximum possible).

Original entry on oeis.org

1, 3, 7, 12, 22, 30, 47, 61, 85, 113, 126, 177, 193, 246, 279, 321, 341, 428, 499, 571, 616, 686, 754, 854, 975, 1052, 1150, 1317, 1376, 1457, 1513, 1664, 1761, 1961, 2307, 2434, 2591, 2795, 2843, 3057, 3226, 3405, 3508, 3776, 3930, 4023, 4196, 4575, 4731
Offset: 1

Views

Author

Peter Kagey, Oct 02 2020

Keywords

Examples

			The addition table of a(k) for k=1..5:
   + | 1 3  7 12 22
  ---+-------------
   1 | 2 4  8 13 23
   3 |   6 10 15 25
   7 |     14 19 29
  12 |        24 34
  22 |           44
The multiplication table of a(k) for k=1..5:
   * | 1 3  7  12  22
  ---+---------------
   1 | 1 3  7  12  22
   3 |   9 21  36  66
   7 |     49  84 154
  12 |        144 264
  22 |            484
These two tables contain the 5*(5+1) = 30 values {1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 19, 21, 22, 23, 24, 25, 29, 34, 36, 44, 49, 66, 84, 144, 154, 264, 484}.

Links

Crossrefs

Cf. A005282 (addition table), A066720 (multiplication table), A337655, A337656, A337947.

Programs

  • Mathematica
    j={k=1};Do[While[l=Join[j,{++k}];g=Union[Sort/@Tuples[l,{2}]];p=Times@@#&/@g;s=Total/@g;!SameQ@@Flatten[{Length@Union@Flatten@{p,s},Length@l(Length@l+1)}]];j=Join[j,{k}];k=Last@j,48];j (* Giorgos Kalogeropoulos, Nov 16 2021 *)

A079852 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number such that all a(i)*a(j)*a(k) are different.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 210, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 307, 311, 313, 317, 331
Offset: 1

Views

Author

Amarnath Murthy, Feb 19 2003

Keywords

Comments

Note that a(57) = 210 = 2*3*5*7, while 330 = 2*3*5*11 is not in the sequence. This demonstrates that this sequence is not determined by prime signature alone. - Charles R Greathouse IV, Oct 17 2015

Links

Crossrefs

Cf. A000045, A079850, A066720, A026477.

Programs

  • Mathematica
    f[l_List] := Block[{k = 1,p2 = Times @@@ Subsets[l, {2}], p3 = Times @@@ Subsets[l, {3}]},While[Intersection[p3, p2*k] != {}, k++ ]; Append[l, k]]; Nest[f, {1, 2, 3}, 62] (* Ray Chandler, Feb 12 2007 *)

Extensions

Extended by Ray Chandler, Feb 12 2007
Typo in name fixed by Zak Seidov, Jul 06 2013

A079850 a(1) = 1 and then the smallest primes such that all a(k)-a(j) are distinct composite numbers.

Original entry on oeis.org

1, 5, 11, 19, 31, 47, 71, 103, 151, 227, 277, 389, 463, 541, 599, 733, 797, 887, 1087, 1217, 1361, 1579, 1693, 1861, 2129, 2267, 2887, 3137, 3301, 3389, 3967, 4133, 4567, 4801, 5021, 5581, 5879, 6983, 7027, 7333, 8123, 8677, 8971, 9949, 10289, 10937
Offset: 1

Views

Author

Amarnath Murthy, Feb 18 2003

Keywords

Links

Crossrefs

Cf. A066720, A079852.

Programs

  • Mathematica
    CompositeQ[n_] := ! (Abs[n] == 1 || PrimeQ[n]);f[l_List] := Block[{pi = 1, d = Subtract @@@ Subsets[l, {2}], p},While[p = Prime[pi]; Intersection[d, l - p] != {} || Nand @@ (CompositeQ /@ (l - p)), pi++ ];Append[l, p]];Nest[f, {1}, 46] (* Ray Chandler, Feb 12 2007 *)

Extensions

Extended by Ray Chandler, Feb 12 2007

A347498 Least k such that there exists an n-element subset S of {1,2,...,k} with the property that all products i * j are distinct for i <= j.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 20, 23, 25, 28, 29, 31, 33, 37, 40, 41, 42, 43, 47, 51, 53, 55, 57, 59, 61, 67, 69, 71, 73, 75, 79, 83
Offset: 1

Views

Author

Peter Kagey, Sep 03 2021

Keywords

Comments

a(n) <= A066720(n) and a(n+1) >= a(n) + 1

Examples

			   n | example set
-----+-------------------------------------------------------
   1 | {1}
   2 | {1, 2}
   3 | {1, 2, 3}
   4 | {1, 2, 3, 5}
   5 | {1, 3, 4, 5,  6}
   6 | {1, 3, 4, 5,  6,  7}
   7 | {1, 2, 5, 6,  7,  8,  9}
   8 | {1, 2, 5, 6,  7,  8,  9, 11}
   9 | {1, 2, 5, 6,  7,  8,  9, 11, 13}
  10 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15}
  11 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15, 17}
  12 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15, 17, 19}
  13 | {1, 5, 6, 7,  9, 11, 13, 14, 15, 16, 17, 19, 20}
  14 | {1, 2, 5, 7, 11, 12, 13, 16, 17, 18, 19, 20, 21, 23}
For n = 4, the set {1,2,3,4} does not have distinct products because 2*2 = 1*4. However, the set {1,2,3,5} does have distinct products because 1*1, 1*2, 1*3, 1*5, 2*2, 2*3, 2*5, 3*3, 3*5, and 5*5 are all distinct.

Crossrefs

Analogous for sums: A003022 and A227590.
Cf. A066720, A338006, A347499.

Programs

  • Mathematica
    Table[k=1;While[!Or@@(Length[s=Union[Sort/@Tuples[#,{2}]]]==Length@Union[Times@@@s]&/@Subsets[Range@k,{n}]),k++];k,{n,12}] (* Giorgos Kalogeropoulos, Sep 08 2021 *)
  • Python
    from itertools import combinations, combinations_with_replacement
def a(n):
    k = n
    while True:
        for Srest in combinations(range(1, k), n-1):
            S = Srest + (k, )
            allprods = set()
            for i, j in combinations_with_replacement(S, 2):
                if i*j in allprods: break
                else: allprods.add(i*j)
            else: return k
        k += 1
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Sep 08 2021

Formula

a(n) = min {k >= 1; A338006(k) = n}. - Pontus von Brömssen, Sep 09 2021

Extensions

a(15)-a(20) from Michael S. Branicky, Sep 08 2021
a(21)-a(38) (based on the terms in A338006) from Pontus von Brömssen, Sep 09 2021

A076941 a(n) = 2^A066657(n) * 3^A066658(n).

Original entry on oeis.org

6, 18, 54, 108, 486, 972, 1944, 4374, 8748, 17496, 69984, 13122, 162, 52488, 209952, 839808, 354294, 708588, 1417176, 5668704, 22674816, 45349632, 3188646, 6377292, 12754584, 51018336, 204073344, 408146688, 3265173504, 258280326, 516560652, 1033121304
Offset: 0

Views

Author

Amarnath Murthy, Oct 19 2002

Keywords

Links

Crossrefs

Cf. A066657, A066658, A076940.
Cf. A066720, subsequence of A003586.

Programs

  • Haskell
    a076941 n = 2 ^ (a066657 n) * 3 ^ (a066658 n)
-- Reinhard Zumkeller, Nov 19 2013

Extensions

Edited by Max Alekseyev, Aug 11 2013
Offset changed by Reinhard Zumkeller, Nov 19 2013

A079854 a(1) = 1, a(k) divides a(k+r) for all k and r and the ratios a(k+r)/a(k) are all different.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 840, 6720, 60480, 604800, 6652800, 86486400, 1210809600, 18162144000, 308756448000, 4940103168000, 88921857024000, 1689515283456000, 35479820952576000, 780556060956672000, 17952789402003456000
Offset: 1

Views

Author

Amarnath Murthy, Feb 19 2003

Keywords

Examples

			a(7) != 120*1, 120*2, ..., 120*6 as the ratios 1,2,3,...,6 appeared as 1/1, 2/1, 6/2, 24/6, 120/24, 6/1. So a(7) = 7*120 = 840.

Crossrefs

Cf. A000045, A066720, A079852, A036241.

Programs

  • Mathematica
    f[l_List] := Block[{n = Length[l], w, k = 1, r = l[[ -1]]/l},w = Flatten[Table[Take[l, i - n]/l[[i]], {i, n}]];While[Intersection[w, k*r] != {}, k++ ];Append[l, k*l[[ -1]]]];Nest[f, {1}, 21] (* Ray Chandler, Feb 08 2007 *)

Extensions

Corrected and extended by Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 30 2006
Edited and further extended by Ray Chandler, Feb 08 2007

A381063 Lexicographically earliest sequence of positive integers such that each nonempty subset has a distinct geometric mean.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 176, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251
Offset: 1

Views

Author

Neal Gersh Tolunsky, Feb 12 2025

Keywords

Comments

Every prime occurs in the sequence.

Crossrefs

Cf. A066720, A260873.

Programs

  • PARI
    is(v, w, k) = my(f=factor(k), x, y, z); for(i=1, #f~, if(setsearch(v, f[i, 1]), listput(w, f[i, 1]))); w=Set(w); forsubset(#w, r, x=#r; y=k*prod(i=1, x, w[r[i]]); forsubset(#w, s, z=#s; if(y^z==prod(i=1, z, w[s[i]])^(x+1)&&z>0, return(0)))); 1;
lista(nn) = my(f, k=1, v=[1], w=List([1])); for(n=2, nn, while(!isprime(k++), if(is(v, w, k), f=factor(k); for(i=1, #f~, if(setsearch(v, f[i, 1]), listput(w, f[i, 1]))); listput(w, k); break)); v=concat(v, k)); v; \\ Jinyuan Wang, Feb 13 2025

Extensions

More terms from Jinyuan Wang, Feb 13 2025
Previous Showing 11-17 of 17 results.

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