A367377 -id:A367377 - 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.

A372181 a(n) is the smallest number whose symmetric representation of sigma consists of n copies of unimodal pattern 121 (separated by 0's if n > 1).

Original entry on oeis.org

6, 78, 1014, 12246, 171366, 1922622, 28960854, 300014754, 4174476774

Offset: 1

Hartmut F. W. Hoft, Apr 21 2024

This sequence is infinite (see A372180).

			a(2) = 78 is the smallest number whose symmetric representations of sigma has unimodal pattern 1210121.
a(4) = 12246 is the smallest number whose symmetric representations of sigma has unimodal pattern 121012101210121.

Row 2 of A367377.
Row 1 of A372180.
Cf. A237593.

(* function seqQ[ ] is defined in A372180 *)
smallest[s_, m_] := NestWhile[#+1&, s, !seqQ[#, m]&]
a372181[n_] := Rest[FoldList[smallest[#1, #2]&, 1, Range[n]]]
a372181[9] (* long computation time *)

A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

Original entry on oeis.org

1, 2, 3, 6, 5, 16, 7, 40

Offset: 1

Hartmut F. W. Hoft, Dec 05 2023

The width pattern (A341969) of the symmetric representation of sigma for a number with k >= 1 odd divisors has length 2*k - 1.
a(p) = p for any prime number p is realized by the m+1 numbers 3^(p-1), ..., 2^m * 3^(p-1) which contain m+1-p duplicates, where m = floor(log_2(3^(p-1))). Each width pattern first increases to a level 1 <= i <= p and then alternates between i and i-1 up to the diagonal of the symmetric representation of sigma resulting in p distinct patterns.
For some numbers n = 2^m * q, q odd and not prime, that are the least instantiations of a width pattern their odd parts q may not be the least instantiations of a width pattern, examples are 78, 1014, 12246 and 171366 with 4, 6, 8 and 10 odd divisors, respectively (see row 2 of the table in A367377).
Conjecture: a(9) = 28.
The least number instantiating the 28th width pattern, 12345654345654321, is n = 43356672, found in a search up to 5*10^9.
Table of width pattern counts of the symmetric representation of sigma and of all possible symmetric patterns:
# odd divisors  1   2   3   4   5    6    7    8    9   10   11    12
pattern count   1   2   3   6   5   16    7   40   28? >=47  11 >=223
A001405         1   2   3   6  10   20   35   70  126  252  462   924
The 4 symmetric patterns 10123232101, 10123432101, 12101010121 and 12123432121 cannot be instantiated as width patterns of numbers with 6 odd divisors.
30 of the 70 possible symmetric patterns of numbers n = 2^m * q, m>=0 and q odd, with 8 odd divisors cannot be instantiated as width patterns of the symmetric representation of sigma(n) since their sequence of widths contradicts the order of the odd divisors d_i of n and of the numbers 2^(m+1) * d_i and the positions of their corresponding 1's in the rows of the triangle of widths in A249223.

			In the irregular triangle below, row k lists the count and the first occurrences of successive instantiations of the distinct width patterns in the symmetric representation of sigma for numbers with k odd divisors.
# div |count|    first occurrence of distinct width patterns
      |     |    1    2    3     4     5     6      7 .. 11 .. 16 .. 40
-----------------------------------------------------------------------
1     |  1  |    1                                        .     .     .
2     |  2  |    3    6                                   .     .     .
3     |  3  |    9   18   72                              .     .     .
4     |  6  |   15   21   30    60    78   120            .     .     .
5     |  5  |   81  162  648  1296  5184                  .     .     .
6     | 16  |   45   63   75    90   147   150    180    ...  27744   .
7     |  7  |  729 1458 5832 11664 46656 93312 373248     .           .
8     | 40  |  105  135  165   189   210   231    357    ...       203808
9     | 28? |  225  441  450   882   900  1225   1800    ...
10    | >=47|  405  567  810  1134  1377  1539   1620    ...
11    | 11  |59049                 ...               1934917632
The complete sequence of first occurrences of the 11 width patterns for numbers with 11 odd divisors is: 59049, 118098, 472392, 944784, 3779136, 7558272, 30233088, 120932352, 241864704, 967458816, 1934917632.
The column labeled '1' of least occurrences of a width pattern of length 2k-1 is sequence A038547: least number with exactly k odd divisors.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596, A367377.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]
(* row n in triangle of A249223 *)
t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)
widthPattern[n_] := Map[First, Split[t262045[n]]]
nOddDivs[n_] := Length[Divisors[NestWhile[#/2&, n, EvenQ[#]&]]]
count[n_, k_] := Length[Union[Map[widthPattern, Select[Range[n], nOddDivs[#]==k&]]]]
(* count of distinct width patterns for numbers with k odd divisors in the range 1 .. n *)

A368950 Numbers with 11 odd divisors.

Original entry on oeis.org

59049, 118098, 236196, 472392, 944784, 1889568, 3779136, 7558272, 9765625, 15116544, 19531250, 30233088, 39062500, 60466176, 78125000, 120932352, 156250000, 241864704, 282475249, 312500000, 483729408, 564950498, 625000000, 967458816, 1129900996, 1250000000, 1934917632

Offset: 1

Hartmut F. W. Hoft, Jan 10 2024

nonn

Every number in this sequence has the form 2^k * p^10, k >= 0, where p is an odd prime. Exactly 11 different width patterns (A341969) of the symmetric representation of sigma are instantiated by the numbers in this sequence. The width pattern becomes unimodal for k >= floor(log_2(p^10)), see A367370 and A367377.

			a(1) = 59049 = 3^10, a(9) = 5^10 = 9765625 is the smallest number with prime factor 5, a(19) = 282475249 is the smallest number with prime factor 7 and a(27) = 2^floor(log_2(3^10)) * 3^10 = 32768 * 59049 = 1934917632 is the smallest whose width pattern of its symmetric representation of sigma is unimodal.

Cf. A267983 (lists the sequences of numbers with 1 .. 10 odd divisors), A367370, A367377.

Cf. A235791, A237048, A237270, A237593, A241008, A241010, A249223, A341969.

N:= 10^10: # for terms <= N
R:= NULL: p:= 2:
do
  p:= nextprime(p);
  if p^10 > N then break fi;
  R:= R, seq(2^i*p^10, i = 0 .. floor(log[2](N/p^10)))
od:
sort([R]); # Robert Israel, Jan 16 2024

numL[p_, b_] := Map[2^# p^10&, Range[0, Floor[Log[2, b/p^10]]]]
primeL[b_] := Most[NestWhileList[NextPrime[#]&, 3, #^10<=b&]]
a368950[b_] := Union[Flatten[Map[numL[#, b]&, primeL[b]]]]
a368950[2 10^9]

A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

Original entry on oeis.org

3, 78, 10728, 28920, 53752896, 4157280, 278628512256, 90323520, 1658908800, 21499810560, 7487812494923563008, 13005699840, 155267279705546496147456, 111451576596480, 8599694054400, 468208581120, 4172630516011611848266349543424, 5202323481600, 21630916595004029113587563614961664, 67421367982080

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Let the nonincreasing multiset cL = { c_1, ... , c_s } be a factorization of n, let dL = { d_1, ... , d_s } be any set of s distinct odd primes, let q = dL^(cL - 1) = d_1^(c_1 - 1) * ... * d_s^(c_s - 1), and let k satisfy 2^k < q < 2^(k+1). Then SRS(2^k * q) is unimodal of maximum height n, 2^k * q has 2n odd divisors and its width pattern has 2n-1 entries. The smallest possible choice for 2^k * q is with the increasing sequence of odd primes d_i = p_(i+1), 1 <= i <= s. The overall smallest 2^k * q is the minimum among all factorizations of n. The smallest number m for which SRS(m) has two unimodal parts of maximum width n requires the additional prime factor r > 2^(k+1) * q which yields m = 2^k * q * r.

This sequence is column 2 in the array of A367377 and a(2) = A370206(1).

			a(2) = 78  is in the sequence since SRS(78) consists of two parts with width pattern 1 2 1 0 1 2 1 and 78 is the smallest number with those properties.
a(3) = 10728 = 2^3 * 3^2 * 149 is in the sequence since SRS(10728) consists of two parts with width pattern 1 2 3 2 1 0 1 2 3 2 1 and 10728 is the smallest number with those properties.
a(6) = 4157280 = 2^5 * 3^2 * 5 * 2887 is in the sequence. The two factorizations of 6 are {6} and {3, 2} so that with 3^5 = 243 and 3^2 * 5^1 = 45 the inequality 2^5 < 45 < 2^6 determines the single unimodular SRS(32 * 45) of maximum width 6, A250071(6) = 1440. Since 2887 is the smallest prime exceeding 2^6 * 3^2 * 5, 4157280 is the smallest number with SRS(4157280) consisting of two unimodular parts of maximum width 6.

Cf. A003056, A082662, A162247, A235791, A237270, A237591, A237593, A239929, A250071, A341969, A370206.

(* function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f2U[n_] := Module[{s=sF[n], k, p}, k=Floor[Log[2, s]]; p=NextPrime[2^(k+1) s]; 2^k s p]
a378470[n_] := Map[f2U, Range[n]]
a378470[20]

a(p) = 2^k * 3^(p-1) * r, for odd primes p, with 2^k < 3^(p-1) < 2^(k+1) and r > 2^(k+1) * 3^(p-1) least prime, i.e., k = floor( (p-1)*(log_2 (3)) ) and r = prime( primepi(2^(k+1) * 3^(p-1)) + 1 ).

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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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A372181 a(n) is the smallest number whose symmetric representation of sigma consists of n copies of unimodal pattern 121 (separated by 0's if n > 1).

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A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

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A368950 Numbers with 11 odd divisors.

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A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

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