A160811 -id:A160811 - cp's OEIS frontend

A348520 Pentaphobe or 5-phobe numbers: integers that are not pentaphile numbers.

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, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 48, 49, 50, 51, 52, 53, 54, 56, 60, 65, 66, 68, 72, 74, 80, 84, 97, 98, 102, 104, 108, 120, 132, 144, 168, 194, 240

Offset: 1

Bernard Schott, Nov 02 2021

Pentaphile numbers are described in A348518.
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 and the smallest one is always 1; here, there exist 68 pentaphobe numbers and the largest one is 240.

			There are no 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 32 = b_1 + b_2 + b_3 + b_4 + b_5, hence 32 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), A348519 (k=4), this sequence (k=5).

isok(k) = forpart(p=k, if (#Set(p) == 5, if (!(p[2] % p[1]) && !(p[3] % p[2]) && !(p[4] % p[3]) && !(p[5] % p[4]), return(0))), , [5, 5]); return(1); \\ Michel Marcus, Nov 14 2021

A161425 a(n) = A161424(n)/2.

Original entry on oeis.org

8, 10, 12, 14, 16, 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

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161428.

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

A161339 Partial sums of A161205.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 24, 29, 35, 41, 47, 53, 59, 65, 72, 80, 88, 96, 104, 112, 120, 128, 136, 145, 155, 165, 175, 185, 195, 205, 215, 225, 235, 245, 256, 268, 280, 292, 304, 316, 328, 340, 352, 364, 376, 388, 400, 413, 427, 441, 455, 469, 483, 497, 511

Offset: 1

Omar E. Pol, Jun 19 2009

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000005, A000720, A018253, A160811, A160812, A161205.

A161339 := proc(n) option remember: local s: if(n=1)then return 1: fi: s:=sqrt(n): if(frac(s)=0)then return procname(n-1)+2*s-1: else return procname(n-1)+2*floor(s): fi: end: seq(A161339(n), n=1..60); # Nathaniel Johnston, May 06 2011

A161428 a(n) = A161424(n)/4.

Original entry on oeis.org

4, 5, 6, 7, 8, 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, 277

Offset: 1

Omar E. Pol, Jun 20 2009

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161425.

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

A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 3, 5, 6, 1, 2, 4, 7, 8, 9, 10, 1, 5, 11, 12, 1, 2, 3, 6, 13, 14, 15, 16, 1, 7, 17, 18, 1, 2, 4, 8, 19, 20, 21, 22, 1, 3, 9, 23, 24, 25, 26, 27, 28, 1, 2, 5, 10, 29, 30, 1, 11, 31, 32, 33, 34, 35, 36, 1, 2, 3, 4, 6, 12, 37, 38, 39, 40

Offset: 0

Omar E. Pol, Jun 30 2009

			Triangle begins:
1;
1,(2);
1,.2,(3),4;
1,....3,...(5),6;
1,.2,....4,......(7),8,.9,10;
1,..........5,..............(11),12;
1,.2,.3,.......6,..................(13),14,15,16;
1,................7,............................(17),18;
1,.2,....4,..........8,................................(19),20,21,22;

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

Cf. A000005, A000040, A000720, A027750, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161827, A161828, A161835.

A349188 Largest n-phobe number.

Original entry on oeis.org

2, 24, 48, 240, 1440, 2400, 7440, 25920, 72000, 234000

Offset: 2

Bernard Schott, Nov 09 2021

A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n such that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n. A number that is not n-phile is called n-phobe.
The idea for this sequence and the words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).
The number of n-phobe numbers is always finite (A349189), the smallest one is always 1, and this sequence lists the largest n-phobe numbers.
a(6) >= 720. - Michel Marcus, Nov 14 2021
From David A. Corneth, Nov 14 2021: (Start)
a(6) >= 1440. If a(6) > 1440 then a(6) > 50000.
a(7) >= 2400. If a(7) > 2400 then a(7) > 50000.
a(8) >= 7440. If a(8) > 7440 then a(8) > 100000.
a(9) >= 25920. If a(9) > 25920 then a(9) > 100000. (End)
Indeed, all these bounds are the corresponding values of a(6), a(7), a(8), a(9). Proof in link. For n >= 5, the five known terms are divisible by 240. - Bernard Schott, Nov 19 2021

			For n = 2, integers 1 and 2 are 2-phobe, then for m >= 3, every m = 1 + (m-1) with 1 < m-1 and 1 divides m-1, so, each m >= 3 is 2-phile number and a(2) = 2.

Diophante, A496 - Pentaphiles et pentaphobes (in French).
Bernard Schott, Proof that a(6) = 1440 and other proofs.

Cf. A349189.
k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
k-phobe numbers: A019532 (k=3), A348519 (k=4), A348520 (k=5).

a(6)-a(11) from David A. Corneth, Nov 19 2021

A349189 Number of n-phobe numbers.

Original entry on oeis.org

2, 9, 23, 68, 177, 459, 1162, 2947, 7306, 18202

Offset: 2

Bernard Schott, Nov 14 2021

A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n with the property that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n. A number that is not n-phile is called n-phobe.
The words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).
The number of n-phobe numbers is always finite, the smallest one is always 1 and the largest n-phobe number is in A349188.
a(6) >= 176. - Michel Marcus, Nov 15 2021
a(6) >= 177, a(7) >= 459, a(8) >= 1162, a(9) >= 2947. - David A. Corneth, Nov 15 2021
Indeed, all these bounds are the corresponding values of a(6), a(7), a(8) and a(9). Proof comes from Proof link in A349188. - Bernard Schott, Nov 19 2021

			For n = 2, integers 1 and 2 are 2-phobe, then for m >= 3, every m = 1 + (m-1) with 1 < m-1 and 1 divides m-1, so, each m >= 3 is 2-phile number and a(2) = 2.

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), A348519 (k=4), A348520 (k=5).
Cf. A349188.

a(6)..a(11) from David A. Corneth, Nov 19 2021

A161416 Partial sums of A056737.

Original entry on oeis.org

0, 1, 3, 3, 7, 8, 14, 16, 16, 19, 29, 30, 42, 47, 49, 49, 65, 68, 86, 87, 91, 100, 122, 124, 124, 135, 141, 144, 172, 173, 203, 207, 215, 230, 232, 232, 268, 285, 295, 298, 338, 339, 381, 388, 392, 413, 459, 461, 461, 466, 480, 489, 541, 544, 550, 551, 567, 594

Offset: 1

Omar E. Pol, Jun 21 2009

nonn

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

Cf. A056737, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161828, A161835, A219729, A219730.

a(n) = A219730(n) - A219729(n). - Tamas Sandor Nagy, Jan 20 2024

Extended beyond a(16) by R. J. Mathar, Aug 01 2009

A162192 Triangle read by rows in which row n lists the divisors of n, prime(n), the consecutive composites that are greater than prime(n), and prime (n+1), but row 0 is formed by 1 and 2.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 3, 5, 6, 7, 1, 2, 4, 7, 8, 9, 10, 11, 1, 5, 11, 12, 13, 1, 2, 3, 6, 13, 14, 15, 16, 17, 1, 7, 17, 18, 19, 1, 2, 4, 8, 19, 20, 21, 22, 23, 1, 3, 9, 23, 24, 25, 26, 27, 28, 29, 1, 2, 5, 10, 29, 30, 31

Offset: 0

Omar E. Pol, Jun 30 2009

See also A162190, a sequence with a similar structure.

			Triangle begins:
1,(2);
1,(2),(3);
1,.2.,(3),4,(5);
1,.....3,...(5),6,(7);
1,.2,.....4,......(7),8,.9,10,(11);
1,...........5,...............(11),12,(13);
1,.2,..3,.......6,....................(13),14,15,16,(17);
1,.................7,...............................(17),18,(19);
1,.2,.....4,..........8,....................................(19),20,21,22,(23);

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

Cf. A000005, A000040, A000720, A027750, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161827, A161828, A161835, A162190.

cp's OEIS Frontend

A348520 Pentaphobe or 5-phobe numbers: integers that are not pentaphile numbers.

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PARI

A161425 a(n) = A161424(n)/2.

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A161339 Partial sums of A161205.

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Maple

A161428 a(n) = A161424(n)/4.

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A162190 Triangle read by rows in which row n lists the divisors of n, the n-th prime and the consecutive composites that are greater than the n-th prime, with a(0)=1.

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A349188 Largest n-phobe number.

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A349189 Number of n-phobe numbers.

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A161416 Partial sums of A056737.

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A162192 Triangle read by rows in which row n lists the divisors of n, prime(n), the consecutive composites that are greater than prime(n), and prime (n+1), but row 0 is formed by 1 and 2.

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