A370205 -id:A370205 - cp's OEIS frontend

A370206 Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 348, 354, 366, 372, 402, 426, 438, 444, 474, 492, 498, 516, 534, 564, 582, 606, 618, 636, 642, 654, 678, 708, 732, 762, 786, 804, 820, 822, 834, 852, 860, 876, 894, 906, 940, 942, 948, 978, 996, 1002, 1038, 1060, 1068, 1074

Offset: 1

Hartmut F. W. Hoft, Feb 11 2024

nonn

Each term has 4 odd divisors and has the form 2^k * p * q, k > 0, p and q prime,  2 < p < 2^(k+1) <  2^(k+1) * p < q. The inequalities ensure that the four 1's in row a(n) of triangle in A237048 are in positions 1, p, 2^(k+1), and 2^(k+1) * p <= floor( (sqrt(8*a(n)+1) - 1)/2 ) < q and establish width pattern 1210 in SRS(a(n)) up to the diagonal. Also since p < 2^(k+1), numbers of the form 2^k * p^3 force p^2 < 2^(k+1) * p which creates a width pattern of the form 1212121.
When a(n) satisfies q = 2^(k+1) * p + 1 it is the smallest number with prime factor p  whose two parts of SRS(a(n)) meet at the diagonal since in this case 2^(k+1) * p = floor( (sqrt(8*a(n)+1) - 1)/2 ). The first 4 numbers with p = 3 are  2* 3 * 13 = 78, 2^4 * 3 * 97 = 4656, 2^5 * 3 * 193 = 18528 and 2^7 * 3 * 769 = 295296. The smallest number with prime factor p = 47 has 355 digits.
Conjecture: The subsequence of numbers m whose two parts of SRS(m) meet at the diagonal is infinite.

			a(1) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts of width pattern 121 that meet at diagonal position (54, 54).
a(38) = 4 * 5 * 41 = 820 = A262259(6)  is the smallest number in the sequence divisible by 5 and the two parts of SRS(a(38)) meet at diagonal position (570, 570).

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205.

(* function based on conditions for the odd divisors - fast computation *)
a370206Q[n_] := Module[{f=FactorInteger[n], d=Divisors[NestWhile[#/2&, n, EvenQ[#]&]]}, Length[f]==3&&f[[1, 1]]==2&&Length[d]==4&&f[[2, 1]]<2^(f[[1, 2]]+1)&&2^(f[[1, 2]]+1)*f[[2, 1]]A367377 - slow computation *)
a370206[m_, n_] :=  Select[Range[m, n], widthPattern[#]=={1, 2, 1, 0, 1, 2, 1}&]
a370206[1,1074]

A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

Original entry on oeis.org

78, 820, 6328, 62128, 5539456, 155155972096, 739936, 69342976, 431056, 31494016, 44864128, 3525354496, 3788128

Offset: 2

Hartmut F. W. Hoft, Feb 11 2024

nonn

a(n) is the smallest number of the form described above whose symmetric representation of sigma, SRS(a(n)), consists of 2 parts that have a unimodal width pattern of type 121 and that meet at the diagonal. Since floor( (sqrt(8*a(n) + 1) - 1)/2 ) = 2^(k+1) * p, the central 0 width extent of SRS(a(n)) equals 0.

Conjecture: The sequence is infinite.

			a(2) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts 121 that meet at diagonal position (54, 54).
a(4) = 6328 = 8 * 7 * 113 = A262259(11) which demonstrates that  2^k < p < 2^(k+1) need not be true.
a(15) with k = 582 and p = 47, its second prime factor 2^(k+1) * p + 1 has 178 digits so that a(15) has 355 digits.
a(16) = 24129129742336 = 2^16 * 53 * 6946817.
Table of records of number of digits a(2) through a(500):
sequence index    2  3  4  5  6   7   15    76   419    438
number of digits  2  3  4  5  7  12  355  3854  5856  20049

Chai Wah Wu, Table of n, a(n) for n = 2..75 [Notice that a(15) has 355 digits, so the graph cuts off after 14 terms. - _N. J. A. Sloane_, Feb 18 2024]

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205, A370206.

minExp[p_] := Module[{k=Floor[Log[2, p]]}, NestWhile[#+1&, k+1, !PrimeQ[2^# p+1]&]-1]/;PrimeQ[p]
a370209[p_] := Module[{k=minExp[p]}, 2^k p(2^(k+1)p+1)]/;PrimeQ[p]
Map[a370209[Prime[#]]&, Range[2, 14]] (* a(15) is too large to list *)

from itertools import count
from sympy import prime, isprime
def A370209(n):
    p = prime(n)
    return next((p<Chai Wah Wu, Feb 17 2024

a(n) = min( 2^k * p * (2^(k+1) * p + 1) : p = prime(n), 2 < p < 2^(k+1), 2^(k+1) * p + 1 is prime ), n>=2.

A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

Original entry on oeis.org

6, 12, 78, 20, 102, 1014, 24, 114, 1734, 12246, 28, 138, 2166, 12714, 171366, 40, 174, 3174, 13026, 501126, 1922622, 48, 186, 5046, 13182, 781926, 2057406, 28960854, 56, 222, 5766, 13494, 1679046, 2067546, 144825414, 300014754, 80, 246, 8214, 13962, 4243686, 2072382, 282275286, 300137214, 4174476774

Offset: 1

Hartmut F. W. Hoft, Apr 21 2024

Every number in this sequence is even since the symmetric representation of sigma for an odd number q starts 101. Each number in column m of T(n,m) has 2*m odd divisors.
Since u(m) = 2 * 3 * 13^(m-1), m>=1, has 2m odd divisors and 1 < 3 < 4 < 4*3 < 13 < 3*13 < 4*13 < 3*4*13 < 13^2 < ..., the symmetric representation of sigma for u(m) consists of m copies of unimodal pattern 121. Therefore, every column in the table T(n,m), m>=1, contains infinitely many entries. Number u(m) is the smallest entry in the m-th column when m is prime.
In general: If m>1 then T(n,m) = 2^k * q, k>=1, q odd, has at least 4 odd divisors which satisfy
d_(2i+2)  < 2^(k+1) * d_(2i+1) < 2^(k+1) * d_(2i+2) < d_(2i+3), i>=0,
with the odd divisors d_j of n in increasing order.

			a(1) = T(1,1) = 6, its symmetric representation of sigma, SRS(6), has unimodal pattern 121 and a single unit of width 2 at the diagonal.
a(3) = T(1,2) = 78, SRS(78) has unimodal pattern 1210121;
a(10) = T(1,4) = 12246, SRS(12246) has unimodal pattern 121012101210121;
both symmetric representations of sigma have width 0 at the diagonal where two parts meets.
Each number in the m-th column has 2m odd divisors. T(1,9) = 4174476774.
  -------------------------------------------------------------------------
   n\m  1    2     3     4       5         6          7          8
  -------------------------------------------------------------------------
   1|   6   78   1014  12246   171366   1922622    28960854  300014754 ...
   2|  12  102   1734  12714   501126   2057406   144825414  300137214 ...
   3|  20  114   2166  13026   781926   2067546   282275286  300235182 ...
   4|  24  138   3174  13182   1679046  2072382   888215334  300357642 ...
   5|  28  174   5046  13494   4243686  2081742  3568939926  300431118 ...
   6|  40  186   5766  13962   5541126  2091882     ...      300602562 ...
   7|  48  222   8214  14118   8487372  2097966              300651546 ...
   8|  56  246  10086  14898  11082252  2110134              300896466 ...
   9|  80  258  10092  15054  11244966  2112162              301165878 ...
  10|  88  282  11094  15366  16954566  2116218              301386306 ...
  ...

Row 1 gives A372181.
Column 1 gives A370205.
Column 2 gives A370206.
Cf. A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A341971, A342592, A342594, A342595, A342596, A367377, A370209.

divQ[k_, {d1_, d2_, d3_}] := d2<2^(k+1)d1&&2^(k+1)d2

T(n,1) = 2^k * p with odd prime p satisfying p < 2^(k+1), see A370205.

T(n,2) = 2^k * p * q, k > 0, p and q prime, 2 < p < 2^(k+1) < 2^(k+1) * p < q, see A370206.

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 63, 66, 70, 75, 77, 78, 80, 88, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 130, 132, 135, 138, 143, 150, 153, 154, 156, 160, 162, 165, 170, 174, 175, 176, 182, 186, 187, 189, 190, 192, 195, 196, 200

Offset: 1

Hartmut F. W. Hoft, Aug 21 2024

nonn

Number m = 2^k * q, k >= 0 and q odd, is in this sequence precisely when for any divisor s <= A003056(m) of q there is at most one divisor t of q satisfying s < t <= min(2^(k+1) * s, A003056(m)), and at least one such pair s < t of successive odd divisors exists. Equivalently, row m of the triangle in A249223 contains at least one 2, but no number larger than 2.

			a(4) = 18 has width pattern 1 2 1 2 1 in its symmetric representation of sigma consisting of a single part, and row 18 in the triangle of A249223 is 1 1 2 1 1.
a(9) = 35 has width pattern 1 0 1 2 1 0 1 in its symmetric representation of sigma consisting of 3 parts, and row 35 in the triangle of A249223 is 1 0 0 0 1 1 2.
Irregular triangle of rows a(n) in triangle of A341970, i.e. of positions of 1's in triangle of A237048, and for the corresponding widths to the diagonal in triangle of A341969:
a(n)| row in A341970      left half of row in A341969
6   | 1   3               1   2
12  | 1   3               1   2
15  | 1   2   3   5       1   0   1   2
18  | 1   3   4           1   2   1
20  | 1   5               1   2
24  | 1   3               1   2
28  | 1   7               1   2
30  | 1   3   4   5       1   2   1   2
35  | 1   2   5   7       1   0   1   2
36  | 1   3   8           1   2   1
...

Column 2 of A253258.
Subsequence of A005279.
Some subsequences are A352030, A370205, A370206, A370209.
Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2
a375611[m_, n_] := Select[Range[m, n], mW2Q]
a375611[1, 200]

cp's OEIS Frontend

A370206 Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

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A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

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A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

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A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

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