A160811 -id:A160811 - cp's OEIS frontend

A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).

4, 6, 8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Omar E. Pol, Jun 20 2009

Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=2 and cannot be eliminated by any sieve s >= 3. - R. J. Mathar, Jun 24 2009
After a(3)=8 all terms are 2*prime; for n > 3, a(n) = 2*prime(n-1) = 2*A000040(n-1). - Zak Seidov, Jul 18 2009
From Omar E. Pol, Jul 18 2009: (Start)
A classification of the natural numbers A000027.
=============================================================
Numbers k whose largest divisor <= sqrt(k) equals j
=============================================================
j       Sequence     Comment
=============================================================
1 ..... A008578      1 together with the prime numbers
2 ..... A161344      This sequence
3 ..... A161345
4 ..... A161424
5 ..... A161835
6 ..... A162526
7 ..... A162527
8 ..... A162528
9 ..... A162529
10 .... A162530
11 .... A162531
12 .... A162532
... And so on. (End)
The numbers k whose largest divisor <= sqrt(k) is j are exactly those numbers j*m where m is either a prime >= k or one of the numbers in row j of A163925. - Franklin T. Adams-Watters, Aug 06 2009
See also A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Also A100484 UNION 8. - Omar E. Pol, Nov 29 2012 (after Seidov and Hasler)
Is this the union of {4} and A073582? - R. J. Mathar, May 30 2025

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Illustration of initial terms
Omar E. Pol. Illustration of initial terms of A008578, A161344, A161345, A161424

Cf. A000005, A018253, A160811, A160812, A161205, A161346, A033676, A008578, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532, A163925.

Second column of array in A163280. Also, second row of array in A163990.

isA := proc(n,s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161344 := proc(n) for s from 3 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,2) ; end: for n from 1 to 3000 do if isA161344(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009

a[n_] := If[n <= 3, 2n+2, 2*Prime[n-1]]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Nov 26 2012, after Zak Seidov *)

a(n)=if(n>3,prime(n-1),n+1)*2 \\ M. F. Hasler, Nov 27 2012

Equals 2*A000040 union {8}. - M. F. Hasler, Nov 27 2012

a(n) = 2*A046022(n+1) = 2*A175787(n). - Omar E. Pol, Nov 27 2012

More terms from R. J. Mathar, Jun 24 2009

Definition added by R. J. Mathar, Jun 28 2009

A161345 Numbers k whose largest divisor <= sqrt(k) is 3.

Original entry on oeis.org

9, 12, 15, 18, 21, 27, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723

Offset: 1

Omar E. Pol, Jun 20 2009

Define a sieve operation with parameter s that eliminates integers of the form s^2+s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=3 and cannot be eliminated by any sieve s >= 4. - R. J. Mathar, Jun 24 2009
See A161344 for more information. - Omar E. Pol, Jul 05 2009
See also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Union of {12, 18, 27} and all the numbers of the form 3*p, where p is an odd prime. - Amiram Eldar, Apr 17 2024

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Omar E. Pol, Illustration: Divisors and pi(x).
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424. [From _Omar E. Pol_, Oct 24 2009]

Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161346, A033676, A008578, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.

Third column of the array in A163280. Also, third row of array in A163990. - Omar E. Pol, Oct 24 2009

isA := proc(n,s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161345 := proc(n) for s from 4 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,3) ; end: for n from 1 to 3000 do if isA161345(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009

md3Q[n_]:=Max[Select[Divisors[n],#<=Sqrt[n]&]]==3; Select[Range[800],md3Q] (* Harvey P. Dale, Aug 12 2013 *)

Numbers k such that A033676(k)=3. - Omar E. Pol, Jul 05 2009

Terms beyond a(10) from R. J. Mathar, Jun 24 2009

Definition added by R. J. Mathar, Jun 28 2009

A161424 Numbers k whose largest divisor <= sqrt(k) equals 4.

Original entry on oeis.org

16, 20, 24, 28, 32, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964

Offset: 1

Omar E. Pol, Jun 20 2009

Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=4 and cannot be eliminated by any sieve s >= 5. - R. J. Mathar, Jun 24 2009
See A161344 for more information. - Omar E. Pol, Jul 05 2009
See also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424 [From _Omar E. Pol_, Oct 24 2009]

Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161425, A161428, A033676, A008578, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.

Cf. Fourth column of array in A163280. Also, fourth row of array in A163990. - Omar E. Pol, Oct 24 2009

isA := proc(n,s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161424 := proc(n) for s from 5 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,4) ; end: for n from 1 to 3000 do if isA161424(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009

Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)

Numbers n such that A033676(n)=4. - Omar E. Pol, Jul 05 2009

Terms beyond a(8) from R. J. Mathar, Jun 24 2009

Definition added by R. J. Mathar, Jun 28 2009

A161205 Triangle read by rows in which row n lists 2n-1 followed by 2n numbers 2n.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 1

Omar E. Pol, Jun 19 2009

Row sums: A125202(n+1). - R. J. Mathar, Feb 16 2010

			Triangle begins:
  1,  2,  2;
  3,  4,  4,  4,  4;
  5,  6,  6,  6,  6,  6,  6;
  7,  8,  8,  8,  8,  8,  8,  8,  8;
  9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10;

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A000720, A018253, A125202, A160811, A160812, A161339.

A161205 := proc(n,k) if k=1 then 2*n-1; else 2*n; end if; end proc: seq(seq(A161205(n,k),k=1..2*n+1),n=1..12) ; # R. J. Mathar, Feb 16 2010

If n is a perfect square, then a(n) = 2*sqrt(n)-1; otherwise a(n) = 2*floor(sqrt(n)). - Nathaniel Johnston, May 06 2011

a(n) = A000196(n-1) + A000196(n) = floor(sqrt(n-1)) + floor(sqrt(n)). - Ridouane Oudra, Jun 07 2019

More terms from R. J. Mathar, Feb 16 2010

A160812 a(n) = A161205(n)-A000005(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 4, 0, 4, 2, 2, 2, 6, 2, 6, 2, 4, 4, 6, 0, 6, 6, 6, 4, 8, 2, 8, 4, 6, 6, 6, 2, 10, 8, 8, 4, 10, 4, 10, 6, 6, 8, 10, 2, 10, 8, 10, 8, 12, 6, 10, 6, 10, 10, 12, 2, 12, 10, 8, 8, 12, 8, 14, 10, 12, 8, 14, 4, 14, 12, 10, 10, 12, 8, 14, 6, 12, 14, 16, 6, 14, 14, 14, 10, 16, 6

Offset: 1

Omar E. Pol, Jun 19 2009

It appears that a(n)= 0 if and only if n divides 24 (See also the comments in A018253).

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A000720, A018253, A160811, A161205.

a(n) = 2*(A000196(n) - A038548(n)) = 2*A236627(n). - Omar E. Pol, Feb 05 2014

Edited by Omar E. Pol, Aug 02 2009

A019532 Numbers k such that Fibonacci(k) divides k!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 12, 24

Offset: 1

N. J. A. Sloane

These terms m are called “triphobe” or “3-phobe” numbers, by the French website Diophante (see link), because there are no 3 positive integers b_1 < b_2 < b_3 such that b_1 divides b_2, b_2 divides b_3, and m = b_1 + b_2 + b_3. A number that is not “triphobe” is called “triphile” or “3-phile” (A160811). The set of k-phobe numbers is always finite, there exist 9 triphobe numbers and the largest one is 24. - Bernard Schott, Oct 23 2021

Posting to math-fun(AT)cs.arizona.edu by R. W. Gosper Nov 06 1996.

Diophante, A496 - Pentaphiles et pentaphobes (in French).

k-phobe numbers: this sequence (k=3), A348519 (k=4), A348520 (k=5).
k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
Cf. A000045, A000142, A160811.

Select[Range[30],Divisible[#!,Fibonacci[#]]&] (* Harvey P. Dale, Jun 14 2020 *)

Offset changed to 1 by David A. Corneth, Oct 27 2021

A348517 Positive integers m with the property that there are 4 positive integers b_1 < b_2 < b_3 < b_4 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, and m = b_1 + b_2 + b_3 + b_4.

Original entry on oeis.org

15, 19, 21, 22, 23, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Bernard Schott, Oct 21 2021

nonn

The idea for this sequence comes from the French website Diophante (see link) where these numbers are called “tetraphile” or “4-phile”. A number that is not tetraphile is called "tetraphobe" or "4-phobe".
It is possible to generalize for "k-phile" or "k-phobe" numbers (see Crossrefs).
Some results:
The smallest tetraphile number is 15 = 1 + 2 + 4 + 8 and the largest tetraphobe is 48, so this sequence is infinite since every integer >= 49 is a term.
If m is tetraphile, q* m, q > 1, is another tetraphile number.
Numbers equal to 1 + 2*triphile (A160811) are tetraphile numbers, but there are other terms not of this form, as even terms.
There exist 23 tetraphobe numbers.

			As 22 = 1 + 3 + 6 + 12, 22 is a term.
As 33 = 1 + 2 + 6 + 24, 33 is another term.

Diophante, A496 - Pentaphiles et pentaphobes (in French).

k-phile numbers: A160811 \ {5} (k=3), this sequence (k=4), A348518 (k=5).

k-phobe numbers: A019532 (k=3).

Select[Range@92,Select[Select[IntegerPartitions[#,{4}],Length@Union@#==4&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]!={}&] (* Giorgos Kalogeropoulos, Oct 22 2021 *)

A161346 a(n) = A161345(n)/3.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Omar E. Pol, Jun 20 2009

Union of {4, 6, 9} and all the odd primes. - Amiram Eldar, Apr 17 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A018253, A160811, A160812, A161205, A161344, A161345.

Select[Range[271], Function[{n, s}, Max[TakeWhile[Divisors[n], # <= s &]] == 3] @@ {#, Sqrt@ #} &[3 #] &] (* Michael De Vlieger, Feb 14 2020 *)

Terms beyond a(10) from R. J. Mathar, Jun 24 2009

A348518 Positive integers m with the property that there are 5 positive integers b_1 < b_2 < b_3 < b_4 < b_5 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, b_4 divides b_5, and m = b_1 + b_2 + b_3 + b_4 + b_5.

Original entry on oeis.org

31, 39, 43, 45, 46, 47, 55, 57, 58, 59, 61, 62, 63, 64, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 99, 100, 101, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Bernard Schott, Oct 21 2021

nonn

The idea for this sequence comes from the French website Diophante (see link) where these numbers are called “pentaphile” or “5-phile”. A number that is not pentaphile is called “pentaphobe” or “5-phobe”.
It is possible to generalize for “k-phile” or “k-phobe” numbers (see Crossrefs).
Some results:
The smallest pentaphile number is 31 = 1 + 2 + 4 + 8 + 16 and the largest pentaphobe number is 240, so, this sequence is infinite since all integers >= 241 are terms.
Every term m = r * (1+s*t) with r > 0, s > 1 and t is a tetraphile number (A348517).
Odd numbers equal to 1 + 2*t where t is tetraphile (A348517) are pentaphile numbers, so odd numbers >= 99 are pentaphile.
If m is pentaphile, q* m, q > 1, is another pentaphile number.
There exist 68 pentaphobe numbers.

			As 43 = 1 + 2 + 4 + 12 + 24, 43 is a term.
As 89 = 1 + 4 + 12 + 24 + 48, 89 is another term.

Diophante, A496 - Pentaphiles et pentaphobes (in French).

k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), this sequence (k=5).

k-phobe numbers: A019532 (k=3).

Select[Range@100,Select[Select[IntegerPartitions[#,{5}],Length@Union@#==5&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]!={}&] (* Giorgos Kalogeropoulos, Oct 22 2021 *)

A348519 Tetraphobe or 4-phobe numbers: integers that are not tetraphile numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 24, 25, 26, 32, 48

Offset: 1

Bernard Schott, Oct 23 2021

Tetraphile numbers are described in A348517.
The idea for this sequence comes from the French website Diophante (see link).
It is possible to generalize for "k-phile" or "k-phobe" numbers (see Crossrefs).
The set of k-phobe numbers is always finite, the smallest one is always 1; here, there exist 23 tetraphobe numbers and the largest one is 48.

			There are no 4 positive integers b_1 < b_2 < b_3 < b_4 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, and 17 = b_1 + b_2 + b_3 + b_4, hence 17 is a term.

Diophante, A496 - Pentaphiles et pentaphobes (in French).

k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).

k-phobe numbers: A019532 (k=3), this sequence (k=4), A348520 (k=5).

Select[Range@48,Select[Select[IntegerPartitions[#,{4}],Length@Union@#==4&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]=={}&] (* Giorgos Kalogeropoulos, Oct 24 2021 *)

isok(k) = forpart(p=k, if (#Set(p) == 4, if (!(p[2] % p[1]) && !(p[3] % p[2]) && !(p[4] % p[3]), return(0))), , [4, 4]); return(1); \\ Michel Marcus, Nov 14 2021

cp's OEIS Frontend

A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A161345 Numbers k whose largest divisor <= sqrt(k) is 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A161424 Numbers k whose largest divisor <= sqrt(k) equals 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A161205 Triangle read by rows in which row n lists 2n-1 followed by 2n numbers 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A160812 a(n) = A161205(n)-A000005(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A019532 Numbers k such that Fibonacci(k) divides k!.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A348517 Positive integers m with the property that there are 4 positive integers b_1 < b_2 < b_3 < b_4 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, and m = b_1 + b_2 + b_3 + b_4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica