A300250 -id:A300250 - cp's OEIS frontend

A297174 An auxiliary sequence for computing A300250. See comments and examples.

0, 1, 1, 5, 1, 19, 1, 69, 5, 19, 1, 2123, 1, 19, 19, 4165, 1, 2131, 1, 2125, 19, 19, 1, 4228171, 5, 19, 69, 2125, 1, 526631, 1, 2101317, 19, 19, 19, 268706123, 1, 19, 19, 4228237, 1, 526643, 1, 2125, 2123, 19, 1, 550026380363, 5, 2131, 19, 2125, 1, 4229203, 19, 4228237, 19, 19, 1, 8798249190555, 1, 19, 2123, 17181970501, 19, 526643, 1, 2125

Offset: 1

Antti Karttunen, Mar 07 2018

nonn

In binary representation of a(n), the distances between successive 1's (one more than the lengths of intermediate 0-runs) from the right record the prime signature ranks (A101296) of successive divisors of n, as ordered from the smallest divisor (> 1) to the largest divisor (= n).

			a(1) = 0 by convention (as 1 has no prime divisors).
a(p) = 1 for any prime p.
For any n > 1, the least significant 1-bit is at rightmost position (bit-0), signifying the smallest prime factor of n, which is always the least divisor > 1.
For n = 4 = 2*2, the next divisor of 4 after 2 is 4, for which A101296(4) = 3, thus the second least significant 1-bit comes 3-1 = 2 positions left of the rightmost 1, thus a(4) = 2^0 + 2^(3-1) = 1+4 = 5.
For n = 6 with divisors d = 2, 3 and 6 larger than one, for which A101296(d)-1 gives 1, 1 and 3, thus a(6) = 2^(1-1) + 2^(1-1+1) + 2^(1-1+1+3) = 2^0 + 2^1 + 2^4 = 19.
For n = 12 with divisors d = 2, 3, 2*2, 2*3, 2*2*3 larger than one, A101296(d)-1 gives 1, 1, 2, 3 and 5 thus a(12) = 2^0 + 2^(0+1) + 2^(0+1+2) + 2^(0+1+2+3) + 2^(0+1+2+3+5) = 2123.
For n = 18 with divisors d = 2, 3, 2*3, 3*3, 2*3*3 larger than one, A101296(d)-1 gives 1, 1, 3, 2, and 5 thus a(18) = 2^0 + 2^(0+1) + 2^(0+1+3) + 2^(0+1+3+2) + 2^(0+1+3+2+5) = 2131.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to binary expansion of n

Cf. A101296, A300250 (restricted growth sequence transform of this sequence).

Cf. also A292258, A294897.

up_to = 4096;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523.
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A297174(n) = { my(s=0,i=-1); fordiv(n, d, if(d>1, i += (A101296(d)-1); s += 2^i)); (s); };

A305800 Filter sequence for a(prime) = constant sequences.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A239968.
In the following, A stands for this sequence, A305800, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j: S(i) = S(i) => T(i) = T(j).
For example, the following implications hold:
  A -> A300247 -> A305897 -> A077462 -> A101296,
  A -> A290110 -> A300250 -> A101296.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000720, A239968.
Differs from A296073 for the first time at n=125, as a(125) = 96, while A296073(125) = 33.
Cf. also A305900, A305801, A295300, A289626 for other "upper level" filters.

Join[{1},Table[If[PrimeQ[n],2,1+n-PrimePi[n]],{n,2,150}]] (* Harvey P. Dale, Jul 12 2019 *)

A305800(n) = if(1==n,n,if(isprime(n),2,1+n-primepi(n)));

a(1) = 1; for n > 1, a(n) = 2 for prime n, and a(n) = 1+n-A000720(n) for composite n.

A077462 Prime factor configuration patterns.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 6, 4, 4, 2, 9, 3, 4, 5, 6, 2, 10, 2, 11, 4, 4, 4, 12, 2, 4, 4, 9, 2, 10, 2, 6, 6, 4, 2, 13, 3, 8, 4, 6, 2, 14, 4, 9, 4, 4, 2, 15, 2, 4, 6, 16, 4, 10, 2, 6, 4, 10, 2, 17, 2, 4, 8, 6, 4, 10, 2, 13, 7, 4

Offset: 0

Michael Somos, Nov 07 2002

Call two numbers equivalent if they have the same prime factorization exponents (in the same order). This sequence enumerates the equivalence classes.
A055932(a(n)) = A071364(n). - David Wasserman, Dec 21 2004
From Antti Karttunen, Jun 13 2018: (Start)
After a(0) = 0, this is the restricted growth sequence transform of A071364. The latter sequence is an "ordered variant" of A046523, and because A101296 is the rgs-transform of A046523, it follows that for all i, j: a(i) = a(j) => A101296(i) = A101296(j).
(End)

			12 = 2^2*3^1 has exponents {2,1}, and is the first number with that pattern, so its value is one more than the largest previous value; a(12) = 6. Contrast that with 18 = 2^1*3^2 having exponents {1,2}, which is different from {2,1}, so a(18) is not equal to a(12). - _Franklin T. Adams-Watters_, Aug 01 2012

Antti Karttunen, Table of n, a(n) for n = 0..100000 (terms 0..10000 from T. D. Noe)
S. Whealton, Prime Factor Configuration Pattern Numbers.

Cf. A037916, A071364, A101296, A290110, A300250.

One more than A079616.

fList = {{0}}; Join[{0, 1}, Table[e = Transpose[FactorInteger[n]][[2]]; pos = Position[fList, e]; If[pos == {}, AppendTo[fList, e]; Length[fList], pos[[1, 1]]], {n, 2, 100}]] (* T. D. Noe, Aug 01 2012 *)

a(n)=local(vn); if(n<1,return(0)); vn=factor(n)[,2]; for(i=1,n,if(vn==factor(i)[,2],return(#Set(vector(i,j,factor(j)[,2])))))

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A071364(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ From A071364
v077462 = rgs_transform(vector(up_to,n,A071364(n)));
A077462(n) = if(!n,n,v077462[n]); \\ Antti Karttunen, Jun 13 2018

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

A355445 Numbers of the form p^2 * q where p and q are primes with p^2 < q.

Original entry on oeis.org

20, 28, 44, 52, 68, 76, 92, 99, 116, 117, 124, 148, 153, 164, 171, 172, 188, 207, 212, 236, 244, 261, 268, 279, 284, 292, 316, 332, 333, 356, 369, 387, 388, 404, 412, 423, 428, 436, 452, 477, 508, 524, 531, 548, 549, 556, 596, 603, 604, 628, 639, 652, 657, 668, 692, 711, 716, 724, 725, 747, 764, 772, 775, 788, 796

Offset: 1

Antti Karttunen, Jul 02 2022

nonn

Numbers whose number of divisors of n (A000005) is equal to 3 + the number of prime factors of n (with multiplicity, A001222), and the third smallest divisor is a square of a prime (A001248).

			20 = 2^2 * 5 is included because 2 < 5, and of the divisors of 20, [1, 2, 4, 5, 10, 20], the third one (4) is a square of prime as 2^2 < 5.

Setwise difference A096156 \ A355446. Subsequence of A119315.
Positions of 9's in A290110 and in A300250.
Cf. A000005, A001222, A001248, A355443 (characteristic function).

Select[Range[800], (f = FactorInteger[#])[[;; , 2]] == {2, 1} && f[[1, 1]]^2 < f[[2, 1]] &] (* Amiram Eldar, Jul 07 2022 *)

A355443(n) = ((numdiv(n) == (3+bigomega(n))) && issquare(divisors(n)[3]));
isA355445(n) = A355443(n);

A300716 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1A101296(d)-1).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 2, 4, 1, 60, 1, 4, 4, 42, 1, 60, 1, 60, 4, 4, 1, 4620, 2, 4, 6, 60, 1, 1000, 1, 546, 4, 4, 4, 21780, 1, 4, 4, 4620, 1, 1000, 1, 60, 60, 4, 1, 1021020, 2, 60, 4, 60, 1, 4620, 4, 4620, 4, 4, 1, 6897000, 1, 4, 60, 12558, 4, 1000, 1, 60, 4, 1000, 1, 75162780, 1, 4, 60, 60, 4, 1000, 1, 1021020, 42, 4, 1

Offset: 1

Antti Karttunen, Mar 13 2018

nonn

a(n) = Product formed from the primes indexed with the prime signatures of proper divisors of n.
The restricted growth sequence transform of this sequence is A101296 because from the set of prime signatures of the proper divisors of n it is always possible to determine the prime signature of n itself, and vice versa, from the prime signature of n, we can form the set of prime signatures of all its proper divisors.
For all i, j: a(i) = a(j) <=> A101296(i) = A101296(j).

			For n = 12, whose proper divisors > 1 are 2, 3, 4, 6, their prime signature ranks from A101296 are: 2, 2, 3, 4. We subtract one from each, to form product prime(1)*prime(1)*prime(2)*prime(3) = 2*2*3*5 = 60, which is thus value of a(12).

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from exponents in factorization of n

Cf. A046523, A101296, A297174, A300250.

Cf. also A292258, A294897, A300827.

Block[{nn = 83, s}, s = Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@#2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}]; Table[If[n == 1, 0, Times @@ Map[Prime[FirstPosition[Keys@ s, #][[1]] - 1] &, Most@ Rest@ Divisors@ n]], {n, nn}]] (* Michael De Vlieger, Mar 13 2018 *)

up_to = 8192;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A300716(n) = { my(m=1); if(1==n, 0, fordiv(n,d,if((d>1)&(dA101296(d)-1))); (m)); };
for(n=1,up_to,write("b300716.txt", n, " ", A300716(n)));

A355446 Numbers of the form p^2 * q where p and q are primes with p < q < p^2.

Original entry on oeis.org

12, 45, 63, 175, 275, 325, 425, 475, 539, 575, 637, 833, 931, 1127, 1421, 1519, 1573, 1813, 2009, 2057, 2107, 2299, 2303, 2783, 2873, 3211, 3509, 3751, 3887, 4477, 4901, 4961, 5203, 5239, 5491, 5687, 6253, 6413, 6647, 6929, 7139, 7267, 7381, 7943, 8107, 8303, 8381, 8591, 8833, 8957, 8959, 9559, 9971, 10043, 10309, 10469

Offset: 1

Antti Karttunen, Jul 02 2022

nonn

Numbers whose number of divisors of n (A000005) is equal to 3 + the number of prime factors of n (with multiplicity, A001222), and the fourth smallest divisor is a square of a prime (A001248).

			12 = 2^2 * 3 is included because 2 < 3, and of the divisors of 12, [1, 2, 3, 4, 6, 12], the fourth one (4) is a square of prime as 2^2 > 3.

Setwise difference A096156 \ A355445.
Positions of 6's in A290110 and in A300250.
Subsequence of A066680, and of A355455.
A251720 is a subsequence.
Cf. A000005, A001222, A001248, A355444 (characteristic function).

Select[Range[10^4], (f = FactorInteger[#])[[;; , 2]] == {2, 1} && f[[1, 1]]^2 > f[[2, 1]] &] (* Amiram Eldar, Jul 07 2022 *)

A355444(n) = ((numdiv(n) == (3+bigomega(n))) && issquare(divisors(n)[4]));
isA355446(n) = A355444(n);

cp's OEIS Frontend

A297174 An auxiliary sequence for computing A300250. See comments and examples.

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A305800 Filter sequence for a(prime) = constant sequences.

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A077462 Prime factor configuration patterns.

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

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A355445 Numbers of the form p^2 * q where p and q are primes with p^2 < q.

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A300716 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1A101296(d)-1).

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A355446 Numbers of the form p^2 * q where p and q are primes with p < q < p^2.

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