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.

User: Carlos Alves

Carlos Alves's wiki page.

Carlos Alves has authored 75 sequences. Here are the ten most recent ones:

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]

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]

A323049 Numbers that are neither 5-smooth nor a sum of two 5-smooth numbers.

Original entry on oeis.org

71, 119, 142, 191, 211, 213, 223, 238, 239, 284, 299, 311, 319, 355, 357, 359, 367, 373, 382, 397, 419, 422, 426, 431, 446, 461, 463, 467, 473, 476, 478, 479, 497, 523, 529, 547, 551, 553, 559, 568, 569, 571, 573, 583, 589, 595, 598, 599, 607, 613, 617, 619, 622, 623, 638, 639, 659, 669, 671, 703, 709, 710, 713, 714
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Comments

Complementary set of (A323048).
Also numbers k such that at least three five-smooth numbers are needed to sum to k. - David A. Corneth, Jan 04 2019
Contains all k == 71 or 119 (mod 120). - Robert Israel, Apr 02 2019

Links

Crossrefs

Cf. A003586, A051037, A323048.
Similar to A323046 (for 3-smooth numbers).

Programs

  • Maple
    N:= 1000: # to get all terms <= N
V:= {seq(seq(seq(2^a*3^b*5^c, a = 0 .. floor(log[2](N/3^b/5^c))),b = 0 .. floor(log[3](N/5^c))),c=0..floor(log[5](N)))}:
S:= {$1..N} minus V minus {seq(seq(V[i]+V[j],i=1..j),j=1..nops(V))}:
A:= sort(convert(S,list)): # Robert Israel, Apr 02 2019
  • Mathematica
    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)]}]; b = Block[{nn = 800, s}, s = f[nn]; {0, 1}~Join~
   Select[Union@Flatten@Outer[Plus, s, s], # <= nn &]];
Complement[Range[800], 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

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]

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

A164291 a(n) = p is the first twin prime (p, p+2) for which p+1 has n prime factors (n>=2, multiplicity counted).

Original entry on oeis.org

3, 11, 59, 71, 239, 191, 2111, 1151, 14591, 26111, 15359, 139967, 138239, 675839, 2101247, 737279, 4866047, 786431, 22118399, 36175871, 194641919, 63700991, 138412031, 169869311, 1321205759, 11123294207, 16357785599, 4076863487, 25165823999, 10871635967
Offset: 2

Views

Author

Carlos Alves, Aug 12 2009

Keywords

Comments

a(3)-a(6) are the first elements of A060213, A102168, A164289, A164290 respectively with n=3,4,5,6 (prime factors in the middle number).
This gives the first p with (p,p+2) twin primes and Omega(p+1)=n with n>=2 (n=1 is impossible).

Examples

			a(7)=191 because in (191, 192, 193) we have Omega(192)=Omega(2*2*2*2*2*2*3)=7 and 191, 193 are twin primes.
The sequence oscillates and here we see that a(7)<a(6)=239.

Links

Crossrefs

Cf. A060213, A102168, A164289, A164290.

Programs

  • Mathematica
    Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 10000000]]; Aseq=(Flatten@Position[Partition[Wmil, 3, 1], {1, #, 1}])[[1]] & /@ Range[3,19]

Extensions

Definition and comments corrected, a(2) and a(20)-a(29) from Donovan Johnson, Aug 20 2009

A164293 Decimal expansion of the twin prime constant associated with the binary constant in A164292.

Original entry on oeis.org

1, 6, 4, 6, 8, 2, 3, 9, 0, 6, 3, 4, 5, 3, 8, 9, 3, 5, 3, 9, 6, 2, 3, 8, 1, 0, 9, 7, 7, 4, 7, 4, 0, 4, 4, 6, 9, 2, 6, 4, 3, 3, 6, 6, 1, 0, 2, 4, 3, 6, 3, 1, 7, 8, 4, 4, 4, 0, 7, 1, 7, 8, 1, 9, 9, 6, 6, 2, 2, 7, 9, 5, 3, 4, 9, 6, 2, 1, 5, 1, 7, 9, 3, 0, 5, 5, 0, 2, 9, 4, 7, 0, 7, 2, 7, 5, 6, 8, 6, 6, 1, 7, 6, 2, 6
Offset: 1

Views

Author

Carlos Alves, Aug 12 2009

Keywords

Comments

Twin prime constant decimals (A164292)_2 = 0.1646823906345389353962381...
Similar: prime constant decimals A051006 = 0.4146825098511116602481096...
If twin prime conjecture is false then this twin prime constant is rational! As a counterpart if it is not rational, the conjecture is true.
If the conjecture is proved by other means, it will remain to see if it is irrational or transcendental.

Crossrefs

Cf. A164292, A051006.

A164290 Sequence of twin prime p where the middle term p+1 has 6 prime factors (here p+2 is the associated twin prime, not listed).

Original entry on oeis.org

239, 599, 809, 1319, 1487, 2087, 2339, 2969, 3299, 4157, 4271, 4787, 5021, 5099, 5231, 5639, 5849, 6359, 6659, 7307, 7349, 9431, 9767, 10007, 10139, 10331, 10709, 10889, 11069, 11171, 11351, 11549, 11717, 11831, 11969, 12539, 13007, 13337
Offset: 1

Views

Author

Carlos Alves, Aug 12 2009

Keywords

Comments

This sequence is similar to A060213, A102168, A164289 respectively with 3, 4 and 5 prime factors in the middle number.
These sequences are of the form (p,p+1,p+2) with (p,p+2) twin primes and Omega(p+1)=m with m>=3 (m=1 or m=2 is impossible). Here m=6.

Examples

			(239, 240, 241): Omega(240)=Omega(2*2*2*2*3*5)=6 and 239, 241 are twin primes.

Links

Crossrefs

Cf. A060213, A102168, A164289, A164291.

Programs

  • Mathematica
    Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 30000]]; Asequence = Flatten@Position[Partition[Wmil, 3, 1], {1, 6, 1}]
Transpose[Select[Partition[Prime[Range[1600]],2,1],#[[2]]-#[[1]]==2 && PrimeOmega[ #[[1]]+1]==6&]][[1]] (* Harvey P. Dale, May 15 2012 *)

A163947 Number of functions on a finite set that are not obtainable by any composition power (excluding identity as power).

Original entry on oeis.org

0, 0, 6, 84, 1400, 25590, 516432
Offset: 1

Views

Author

Carlos Alves, Aug 06 2009

Keywords

Comments

a(n) is the number of functions on a finite set {1,...,n} that are not composition powers of any other function or powers(>1) of itself.
Hard to compute for n>7, as the number of functions to test is n^n.

Examples

			For n=2, the set is {1,2} and we have 4 functions: the constants 1 and 2, the identity, and the transposition. Any composition power of a constant function or of identity is the function itself. Odd composition powers of the transposition give the transposition. Thus all 4 functions are represented.
For n=3, the set is {1,2,3} and f:{1,2,3}->{1,1,2} cannot be represented by composition powers of any other function, or powers of itself (as fof gives the constant function=1). There are 6 functions in this situation (similar).

Crossrefs

Cf. A102687, A163859, A163860, A163861, A163948.

Formula

a(n) = n^n - A163948(n).

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