A191743 -id:A191743 - cp's OEIS frontend

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

isA025487(n)=my(k=valuation(n,2),t);n>>=k;forprime(p=3,default(primelimit),t=valuation(n,p);if(t>k,return(0),k=t);if(k,n/=p^k,return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(Nmax)=vecsort(concat(vector(logint(Nmax,2),n,select(t->t<=Nmax,if(n>1,[factorback(primes(#p),Vecrev(p)) || p<-partitions(n)],[1,2]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A066680 Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 23, 27, 29, 30, 31, 37, 41, 43, 45, 47, 50, 53, 59, 61, 63, 67, 70, 71, 73, 75, 79, 80, 83, 89, 97, 98, 101, 103, 105, 107, 109, 112, 113, 125, 127, 128, 131, 137, 139, 147, 149, 151, 154, 157, 163

Offset: 1

Reinhard Zumkeller, Dec 31 2001

A099104(a(n)) = 1.
a(A207432(n)) = A000040(n). - Reinhard Zumkeller, Feb 17 2012
Obviously all primes and cubes of primes are in the sequence, while squares of primes are not. In fact, A000225 tells us which exponents prime powers in the sequence will exhibit.
But where it gets really interesting is in what happens to the Achilles numbers: the smallest badly sieved numbers that are also Achilles numbers are 864 and 972. - Alonso del Arte, Feb 21 2012
From Peter Munn, Aug 09 2019: (Start)
The factorization pattern of a number's divisors (as defined in A191743) determines whether a number is a term.
There are no semiprimes in the sequence, and a 3-almost prime is present if and only if its largest prime factor is less than its square root. The first term that is a 4-almost prime is 220.
The effect of this sieve can be compared against the A270877 trapezoidal sieve. Each unmarked number k marks k-1 numbers in both sieves; but the largest number marked by k in this sieve is k^2, about twice the largest number marked by k in A270877 (the triangular number T_k = k(k+1)/2). The relative densities early in the two sequences are illustrated by a(10) = 18 < A270877(10) = 19, a(100) = 312 > A270877(100) = 268, a(1000) = 4297 > A270877(1000) = 2894. (End)

			For 2, the first unmarked number, there is only one multiple <= 4=2^2:
giving 2 3 [4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
for 3, the next unmarked number, we mark 6=2*3 and 9=3*3
giving 2 3 [4] 5 [6] 7 8 [9] 10 11 12 13 14 15 16 17 18 19 20 ...
for 5, the next unmarked number, we mark 10=2*5, 15=3*5, 20=4*5 and 25=5*5
giving 2 3 [4] 5 [6] 7 8 [9] [10] 11 12 13 14 [15] 16 17 18 19 [20] ... and so on.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Sieve
Wikipedia, Sieve theory
Index entries for sequences generated by sieves

A066681, A066682, A066683, A099042, A099043, A207432 have analysis of this sequence.
Cf. A056875, A075362, A099104 (characteristic function), A191743.
Sequences generated by a closely related sieving process: A000040 (also a subsequence), A026424, A270877.

a066680 n = a066680_list !! (n-1)
a066680_list = s [2..] where
   s (b:bs) = b : s [x | x <- bs, x > b ^ 2 || mod x b > 0]
-- Reinhard Zumkeller, Feb 17 2012

A099104[1] = 0; A099104[n_] := A099104[n] = Product[If[n > d^2, 1, 1 - A099104[d]], {d, Select[ Range[n-1], Mod[n, #] == 0 &]}]; Select[ Range[200], A099104[#] == 1 &] (* Jean-François Alcover, Feb 15 2012 *)
max = 200; badPrimes = Range[2, max]; len = max; iter = 1; While[iter <= len, curr = badPrimes[[iter]]; badPrimes = Complement[badPrimes, Range[2, curr]curr]; len = Length[badPrimes]; iter++]; badPrimes (* Alonso del Arte, Feb 21 2012 *)

A290110 a(n) = the discovery rank of the factorization pattern of the sequence of divisors of n.

Original entry on oeis.org

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

Offset: 1

Luc Rousseau, Jul 19 2017

nonn

The definition for the factorization pattern of the sequence of divisors of a number n is the same as in sequence A191743. Let's use the abbreviation FPSD. One can generate a list of distinct FPSD by trying all integers, 1, 2, 3, ..., and ignoring duplicates. a(n) is the index of the FPSD of n in this list.
From Antti Karttunen, Mar 07 & 08 2018: (Start)
This is NOT restricted growth sequence transform of A297174, but instead A300250 is, from which this differs for the first time at n=858, where a(858) = 115, while A300250(858) = 75.
This gives a finer partitioning of natural numbers than A300250, and indeed we have:
For all i, j:
  a(i) = a(j) => A300250(i) = A300250(j) => A101296(i) = A101296(j).
(End)

			The divisors of 17 are {1, 17}. They follow the pattern {1, p} which is pattern number 2 in discovery order. a(17)=2.
The divisors of 28 are {1, 2, 4, 7, 14, 28}. They follow the pattern {1, p, p^2, q, p*q, p^2*q}, which is pattern number 9 in discovery order. a(28)=9.
From _Michael De Vlieger_ and _Antti Karttunen_, Mar 07 & 08 2018: (Start)
Divisors of 462 = 2*3*7*11 (p=2, q=3, r=7, s=11) are 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462, thus the factorization patterns in the order of increasing divisors are: 1, p, q, pq, r, s, pr, qr, ps, qs, pqr, pqs, rs, prs, qrs and pqrs.
Divisors of 546 = 2*3*7*13 (p=2, q=3, r=7, s=13) are 1, 2, 3, 6, 7, 13, 14, 21, 26, 39, 42, 78, 91, 182, 273, 546, thus the factorization patterns are 1, p, q, pq, r, s, pr, qr, ps, qs, pqr, pqs, rs, prs, qrs and pqrs, that is, identical with those of 462, thus a(546) = a(462).
Divisors of 858 = 2*3*11*13 (p=2, q=3, r=11, s=13) are 1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858, thus the factorization patterns are 1, p, q, pq, r, s, pr, ps, qr, qs, pqr, pqs, rs, prs, qrs and pqrs. At the 8th divisor (26), we see that pattern ps is different from pattern qr of the 8th divisor of 546 (21), thus a(858) is not equal to a(546).
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..65537

Cf. A191743, A300250.

FactorizationPattern[n_] := Module[
  {pn, fd, f1, f2, d},
  pn = First /@ FactorInteger[n];
  fd = FactorInteger[ReplacePart[Divisors[n], 1 -> {}]];
  f1 = (ReplacePart[#,
      1 -> FromCharacterCode[
        111 + First[Position[pn, First[#]]]]]) &;
  f2 = (f1 /@ #) &;
  fd = f2 /@ fd;
  f1 = (Power[First[#], Last[#]]) &;
  For[i = 1, i <= Length[fd], i++,
   d = fd[[i]];
   For[j = 1, j <= Length[d], j++,d[[j]] = f1[d[[j]]];];
   d = Product[x, {x, d}];
   fd[[i]] = d;
  ];
  fd
]
ListFactorizationPatternIndices[n_] := Module[
  {mem, k, i, p, a},
  mem = Association[];
  a = {}; k = 0;
  For[i = 1, i \[LessSlantEqual] n, i++,
   p = FactorizationPattern[i];
   If[KeyExistsQ[mem, p],,
    k++;
    mem = Append[mem, p -> k]
   ];
   a = Append[a, mem[p]]
  ];
  a
]
ListFactorizationPatternIndices[80]
(* or *)
f[n_] := If[n==1, 1, Block[{p = First /@ FactorInteger@n, z,x}, z= Table[p[[i]] -> x[i], {i, Length@p}]; Times @@ (((#[[1]] /. z)^#[[2]]) & /@ FactorInteger@ #) & /@ Divisors[n]]]; A = <||>; Table[k = f[n]; If[ KeyExistsQ[A, k], A[k], t = 1 + Length@A; A[k] = t], {n, 80}] (* Giovanni Resta, Jul 20 2017 *)

A191743(n) = MIN(k such that a(k)=n).
a(p) = 2, for p prime;
a(p^2) = 3, for p prime;
a(p*q) = 4, for p, q distinct primes.

More terms from Michael De Vlieger and Antti Karttunen, Mar 07 2018

A366250 Numbers k that are not powerful and do not have a strictly superior squarefree divisor.

Original entry on oeis.org

48, 54, 96, 160, 162, 192, 224, 250, 320, 375, 384, 405, 448, 486, 567, 640, 686, 704, 768, 832, 896, 960, 1029, 1080, 1200, 1215, 1250, 1280, 1350, 1408, 1440, 1458, 1500, 1536, 1620, 1664, 1701, 1715, 1792, 1875, 1920, 2016, 2058, 2160, 2176, 2250, 2268, 2352

Offset: 1

Peter Munn and Michael De Vlieger, Feb 08 2024

nonn

A number k does not have a strictly superior squarefree divisor if and only if k is at least as large as the square of rad(k), the largest squarefree divisor of k. All powerful numbers (A001694) have this property. This sequence lists the other such numbers.
Let rad(k) = A007947(k), the largest squarefree divisor, i.e., the squarefree kernel of k. A341645 lists the numbers without a strictly superior squarefree divisor.
A341645 = { k : rad(k) <= k/rad(k) } = { k : A007947(k) <= A003557(k) }, and it is evident that rad(k) <= k/rad(k) is true for powerful k, that is, k in A001694.
Since A001694 contains A001597, the above is also true for perfect powers k; A001597 is a proper subset of A341645.
This sequence contains "weak" k (in A052485) such that rad(k) < k/rad(k).
The presence of a number, k, in this sequence depends only upon A290110(k), i.e., upon the factorization pattern of its sequence of divisors as defined in A191743.
Let S = A006939 and let P = A002110. Almost all superprimorials are in this sequence: S \ {1, 2, 12, 360} is a proper subset. S(i) = S(i-1)*P(i), where S(i-1) = A003557(S(i)) and P(i) = rad(S(i)), and for i > 4, S(i-1) > P(i). Since prime(i) | S(i) but prime(i)^2 does not divide S(i), S(i) is not powerful. Corollary: almost all superprimorials are in A341645, since this sequence is a proper subset of A341645.

			Let b(n) = A364702(n).
a(1) = b(1) = 48 since rad(48) < 48/rad(48), 6 < 8.
b(2) = 50 is not in the sequence since rad(50) > 50/rad(50), 10 > 5.
a(2) = b(3) = 54 since 6 < 9, etc.

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

Cf. A001694, A002110, A003557, A006939, A007947, A052485, A191743, A290110, A341645, A364702.

Select[Range[2, 2400], And[! AllTrue[#2[[All, -1]], # > 1 &], #1 >= Apply[Times, #2[[All, 1]]^2]] & @@ {#, FactorInteger[#]} &]

isok(m) = if (!ispowerful(m), my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0); \\ Michel Marcus, Feb 11 2024

Set difference of A341645 and A001694.
Intersection of A341645 and A364702 where the latter is a proper subset of A052485.
Sequence contains infinite intersections of A052485 and { k = m*s : s is squarefree, rad(m) | s, 1 < s < m }.
{a(n)} = union of { k = s*m : s > 1 is squarefree, rad(m) | s, m >= s, k is not powerful }.
{a(n)} = { k in A364702 : k >= rad(k)^2 }.

A319070 a(n) is the area of the surface made of the rectangles with vertices (d, n/d), (D, n/d), (D, n/D), (d, n/D) for all (d, D), pair of consecutive divisors of n.

Original entry on oeis.org

0, 1, 4, 4, 16, 7, 36, 12, 24, 19, 100, 17, 144, 39, 44, 32, 256, 33, 324, 41, 72, 103, 484, 40, 160, 147, 108, 65, 784, 57, 900, 80, 152, 259, 228, 66, 1296, 327, 204, 93, 1600, 99, 1764, 137, 160, 487, 2116, 92, 504, 165, 332, 185, 2704, 135, 388

Offset: 1

Luc Rousseau, Sep 09 2018

			The divisors of n=12 are {1, 2, 3, 4, 6, 12}. The widths of the rectangles from the definition are obtained by difference: {1, 1, 1, 2, 6}. By symmetry, their heights are the same, but in reverse order: {6, 2, 1, 1, 1}. The sought total area is the sum of products width*height of each rectangle, in other words it is the dot product 1*6 + 1*2 + 1*1 + 2*1 + 6*1. Result: 17. So, a(12)=17.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Luc Rousseau, Diagram illustrating a(10)=19.

Cf. A191743, A290110 (introducing factorization patterns of sequences of divisors).

Cf. A165900 (the Fibonacci polynomial).

a[n_] := Module[{x = Differences[Divisors[n]]}, Plus @@ (x*Reverse[x])];
Table[a[n], {n, 1, 55}]

arect(n, d, D) = (D-d)*(n/d - n/D);
a(n) = my(vd = divisors(n)); sum(k=1, #vd-1, arect(n, vd[k], vd[k+1])); \\ Michel Marcus, Oct 28 2018

a(1) = 0.
a(p) = (p-1)^2 for p a prime number.
a(p^k) = (p-1)^2*k*p^(k-1) for p^k a prime power.
a(p*q) = 2*(p-1)^2*q + (q-p)^2 for p and q primes (p < q).
a(n) = (n/2 - 1)^2 + 3 if n=2*p with p a prime greater than 2.
a(n) = (n/p + F(p-1))^2 + p^2 - F(p-1)^2 if n = p*q, p < q primes; where F denotes the Fibonacci polynomial, F(x) = x^2 - x - 1 (see A165900).
For more complex factorization patterns of n, the formula depends on the factorization pattern of the sequence of divisors of n (see A191743 or A290110), e.g.:
a(p^2*q) = 4*p*q*(p-1)^2 + (q-p^2)^2 if 1 < p < p^2 < q < p*q < p^2*q,
but
a(p^2*q) = 2*p*q*(p-1)^2 + 2*p*(q-p)^2 + (p^2-q)^2 if 1 < p < q < p^2 < p*q < p^2*q.
a(n) = Sum_{i=1..tau(n)-1} (d_[tau(n)-i+1] - d_[tau(n)-i])*(d_[i+1] - d_[i]), where {d_i}, i=1..tau(n) is the increasing sequence of divisors of n. - Ridouane Oudra, Oct 17 2021

A355474 Square array T(m,n) = Card({ (i, j) : 1 <= i <= m, 1 <= j <= min(n, i), GCD(i, j) = 1 }), read by antidiagonals upwards.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 5, 4, 2, 1, 6, 7, 6, 4, 2, 1, 7, 8, 9, 6, 4, 2, 1, 8, 10, 10, 10, 6, 4, 2, 1, 9, 11, 13, 11, 10, 6, 4, 2, 1, 10, 13, 15, 15, 12, 10, 6, 4, 2, 1, 11, 14, 17, 17, 17, 12, 10, 6, 4, 2, 1, 12, 16, 19, 20, 20, 18, 12, 10, 6, 4, 2, 1

Offset: 1

Luc Rousseau, Jul 03 2022

Also the number of regions in the 0 < x < y sector of the plane that are delimited by the lines with equations i*x + j*y = 0, where i and j are integers, not both 0, and |i| <= m, |j| <= n. This remark is motivated by Factorization Patterns (FPs) and Factorization Patterns of Sequences of Divisors (FPSD) concerns, as defined in A191743 and A290110. This is the case k=2 of a more general problem where k is omega(z)=A001221(z), the number of distinct primes dividing z, for which we would define T(n1,n2,...,nk) instead of T(m,n). The idea is the following: two numbers (e.g., 12 and 20) can have the same FP (p^2*q) without having the same FPSD ([1 < p < q < p^2 < p*q < p^2*q] != [1 < p < p^2 < q < p*q < p^2*q]). T(m,n) tells how many distinct FPSDs share the same FP of the p^m*q^n form. See the illustration for (m,n) = (2,1), section Links.

			Let m=2 and n=1. There are exactly two lattice points (i, j) that satisfy 1 <= i <= 2 and 1 <= j <= min(1, i) and GCD(i, j) = 1, namely (1, 1) and (2, 1). So T(2,1) = 2.
Array begins:
  m\n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
  ---+----------------------------------------------------
   1 |  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
   2 |  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
   3 |  3  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4
   4 |  4  5  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6
   5 |  5  7  9 10 10 10 10 10 10 10 10 10 10 10 10 10 10
   6 |  6  8 10 11 12 12 12 12 12 12 12 12 12 12 12 12 12
   7 |  7 10 13 15 17 18 18 18 18 18 18 18 18 18 18 18 18
   8 |  8 11 15 17 20 21 22 22 22 22 22 22 22 22 22 22 22
   9 |  9 13 17 20 24 25 27 28 28 28 28 28 28 28 28 28 28
  10 | 10 14 19 22 26 27 30 31 32 32 32 32 32 32 32 32 32
  11 | 11 16 22 26 31 33 37 39 41 42 42 42 42 42 42 42 42
  12 | 12 17 23 27 33 35 40 42 44 45 46 46 46 46 46 46 46
  13 | 13 19 26 31 38 41 47 50 53 55 57 58 58 58 58 58 58
  14 | 14 20 28 33 41 44 50 53 57 59 62 63 64 64 64 64 64
  15 | 15 22 30 36 44 47 54 58 62 64 68 69 71 72 72 72 72
  16 | 16 23 32 38 47 50 58 62 67 69 74 75 78 79 80 80 80
  17 | 17 25 35 42 52 56 65 70 76 79 85 87 91 93 95 96 96

Luc Rousseau, Illustration of T(2,1)=2 and its relationship with the p^2*q factorization pattern.

Cf. A191743, A290110, A002088, A001221.

T(m, n) = sum(i=1, m, sum(j=1, min(n, i), gcd(i, j)==1))
for(d=2,10,for(n=1,d-1,my(m=d-n);print1(T(m,n),", ")))

T(n,n) = A002088(n).

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