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 1-5 of 5 results.

A323046 Numbers that are neither 3-smooth nor a sum of two 3-smooth numbers.

Original entry on oeis.org

23, 46, 47, 53, 61, 69, 71, 77, 79, 92, 94, 95, 101, 103, 106, 107, 115, 119, 121, 122, 125, 127, 133, 138, 139, 141, 142, 143, 149, 151, 154, 157, 158, 159, 161, 167, 169, 173, 175, 179, 181, 183, 184, 185, 187, 188, 190, 191, 197, 199, 202, 203, 205, 206, 207, 211, 212, 213, 214, 215, 221, 223, 227, 229, 230, 231, 233
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Examples

			23 is not in A003586; also 22 (23-1), 21 (23-2), 20 (23-3), 19 (23-2*2), 17 (23-2*3), 15 (23-2*2*2), 14 (23-3*3), 11 (23-2*2*3), 7 (23-2*2*2*2), 5 (23-2*3*3) are not in A003586.

Links

Crossrefs

Cf. A003586, A237442, A323047, A323049, A323050. Subsequence of A081329.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
S:= {seq(seq(2^i*3^j,i=0..ilog2(N/3^j)),j=0..floor(log[3](N)))}:
sort(convert({$1..N} minus S minus map(t -> op(map(`+`, S,t)), S), list)); # Robert Israel, May 19 2019

A323050 Numbers that cannot be written as a sum of two or fewer 7-smooth numbers (A002473).

Original entry on oeis.org

311, 479, 551, 619, 622, 671, 719, 839, 851, 933, 937, 958, 1102, 1103, 1117, 1151, 1193, 1238, 1244, 1291, 1319, 1342, 1391, 1433, 1437, 1438, 1487, 1499, 1511, 1531, 1553, 1555, 1559, 1619, 1651, 1653, 1667, 1678, 1679, 1697, 1857, 1866, 1871, 1874, 1913, 1916, 1919, 1933, 1937, 1991, 2011, 2013, 2077, 2113, 2117, 2157
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Comments

Numbers that are not of the form (2^i * 3^j * 5^k * 7^l)*a + (2^m * 3^n * 5^p * 7^q)*b, with i,j,k,m,n,p >= 0, and a,b = 0 or 1. The first number excluded is 311.
These numbers are also included in A323046 and A323049.

Crossrefs

Similar to A323046 (for 3-smooth) and A323049 (for 5-smooth). Cf. A002473.

Programs

  • Mathematica
    f[n_] := Union@Flatten@Table[2^a*3^b*5^c*7^d, {a, 0, Log2[n]}, {b, 0, Log[3, n/2^a]}, {c, 0, Log[5, n/(2^a*3^b)]}, {d, 0, Log[7, n/(2^a*3^b*5^c)]}];
b = Block[{nn = 3000, s}, s = f[nn]; {0, 1}~Join~Select[Union@Flatten@Outer[Plus, s, s], # <= nn &]];
Complement[Range[3000], b]

A323047 Numbers that are not the sum of three (or fewer) 3-smooth numbers.

Original entry on oeis.org

431, 485, 509, 565, 637, 671, 719, 725, 727, 862, 887, 935, 941, 943, 959, 967, 970, 1130, 1151, 1175, 1199, 1205, 1274, 1293, 1319, 1342, 1367, 1373, 1391, 1415, 1421, 1423, 1438, 1439, 1445, 1447, 1450, 1453, 1454, 1455, 1481, 1527, 1535, 1559
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Comments

Numbers below 431 may be written as a sum of three (or fewer) elements in A003586. These are the first exceptions.
Below 18431 every number can be written as a sum of 4 or fewer 3-smooth numbers, and below 3448733 every number can be written as a sum of 5 or fewer 3-smooth numbers (cf. sequence A018899).

Links

Crossrefs

Cf. A003586, A018899, A237442, A323046, A323048, A323049, A323050.

Programs

  • Maple
    N:= 1000: # for all terms <= N
S:= {seq(seq(2^i*3^j,i=0..ilog2(N/3^j)),j=0..floor(log[3](N)))}:
S2:= select(`<=`,map(t -> op(map(`+`, S,t)), S),N):
S3:= select(`<=`,map(t -> op(map(`+`, S,t)), S2), N):
A:= {$1..N} minus S minus S2 minus S3:
sort(convert(A,list)); # Robert Israel, May 19 2019
  • Mathematica
    f[n_] := Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2[n]}, {b, 0, Log[3, n/2^a]}];
b=Block[{nn = 2000, s}, s = f[nn]; {0, 1, 2}~Join~Select[Union@ Flatten@ Outer[Plus, s, s, s], # <= nn &]]; Complement[Range[2000], b]

A323048 Sums of no more than two 5-smooth numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Comments

Sequence includes 5-smooth numbers (A051037).
Numbers that are of the form (2^i * 3^j * 5^k)*a + (2^m * 3^n * 5^p)*b, with i,j,k,m,n,p >= 0, and a,b = 0 or 1. The first number excluded is 71. The numbers excluded are in A323049.

Examples

			70 = 2*2*3*5 + 2*5, 72 = 2*2*2*3*3 = 2*2*3*5 + 2*2*3, but 71 is not in the sequence.

Crossrefs

Cf. A051037, A323049 (complementary sequence).

Programs

  • Mathematica
    S5 = Join[{0}, Select[Range[500], FactorInteger[#][[-1, 1]] <= 5 &]];
Union@Flatten@Outer[Plus, S5, S5]
(* more efficient code by Michael De Vlieger *)
f[n_] := Union@Flatten@Table[2^a*3^b*5^c, {a, 0, Log2[n]}, {b, 0, Log[3, n/2^a]}, {c, 0, Log[5, n/(2^a*3^b)]}]; Block[{nn = 500, s}, s = f[nn]; {0, 1}~Join~
  Select[Union@Flatten@Outer[Plus, s, s], # <= nn &]]

Extensions

Name edited by Jianing Song, Jun 11 2019

A323051 Numbers that cannot be written as a sum of two or fewer 11-smooth numbers (A051038).

Original entry on oeis.org

479, 958, 1151, 1319, 1437, 1559, 1679, 1916, 2302, 2351, 2395, 2638, 2874, 2999, 3013, 3071, 3118, 3353, 3358, 3453, 3671, 3737, 3769, 3832, 3911, 3957, 4199, 4309, 4311, 4604, 4677, 4702, 4703, 4751, 4790, 4919, 5037, 5057, 5269, 5276, 5389, 5443, 5519, 5597, 5683
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Comments

Similar to A323046 (3-smooth), A323049 (5-smooth) or A323050 (7-smooth).
This sequence is a subsequence of A323046, A323049, and A323050.
Notice that A045535(4) = a(1) = 479.

Crossrefs

See A323046 (3-smooth), A323049 (5-smooth) or A323050 (7-smooth). Cf. A051038, A045535 (or A062241).

Programs

  • Mathematica
    f[n_] := Union@Flatten@Table[2^a*3^b*5^c*7^d, {a, 0, Log2[n]}, {b, 0, Log[3, n/2^a]}, {c, 0, Log[5, n/(2^a*3^b)]}, {d, 0, Log[7, n/(2^a*3^b*5^c)]}];
b = Block[{nn = 3000, s}, s = f[nn]; {0, 1}~Join~
    Select[Union@Flatten@Outer[Plus, s, s], # <= nn &]];
Complement[Range[3000], b]
Showing 1-5 of 5 results.

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