A056472 -id:A056472 - cp's OEIS frontend

A162247 Irregular triangle in which row n lists all factorizations of n, sorted by the number of factors in each factorization.

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

Offset: 1

T. D. Noe, Jun 28 2009

Row n begins with n because it is a factorization of length 1. In each factorization, the factors are in nondecreasing order. This sequence is A056472 with the factorizations in a different order. Sequence A001055(n) gives the number of factorizations of n; A066637(n) gives the number of numbers in row n. In the Mathematica program, the function f returns a list of the factorizations of n.

These factorizations are useful in determining the forms of numbers that have a given number of divisors. For example, to find the forms of numbers that have 12 divisors, we look at the four factorizations of 12 (12, 2*6, 3*4, 2*2*3), subtract 1 from each factor, and find the forms to be p^11, p q^5, p^2 q^3, and p q r^2, where p, q, and r are prime numbers.

			1;
2;
3;
4,2*2;
5;
6,2*3;
7;
8,2*4,2*2*2;
9,3*3;
10,2*5;
11;
12,2*6,3*4,2*2*3;

See A001055.

T. D. Noe, Rows n=1..1000 of triangle, flattened
R. J. Mathar, Factorizations of n=1..1100

Cf. A001055, A056472, A066637, A316439.

import Data.List (sortBy)
import Data.Ord (comparing)
a162247 n k = a162247_tabl !! (n-1) !! (k-1)
a162247_row n = a162247_tabl !! (n-1)
a162247_tabl = map (concat . sortBy (comparing length)) $ tail fss where
   fss = [] : map fact [1..] where
         fact x = [x] : [d : fs | d <- [2..x], let (x',r) = divMod x d,
                                  r == 0, fs <- fss !! x', d <= head fs]
-- Reinhard Zumkeller, Jan 08 2013

g[lst_,p_] := Module[{t,i,j}, Union[Flatten[Table[t=lst[[i]]; t[[j]]=p*t[[j]]; Sort[t], {i,Length[lst]}, {j,Length[lst[[i]]]}], 1], Table[Sort[Append[lst[[i]],p]], {i,Length[lst]}]]]; f[n_] := Module[{i,j,p,e,lst={{}}}, {p,e}=Transpose[FactorInteger[n]]; Do[lst=g[lst,p[[i]]], {i,Length[p]}, {j,e[[i]]}]; lst]; Flatten[Table[f[n], {n,25}]]

A337080 Complement of A337037.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248

Offset: 1

Matej Veselovac, Aug 14 2020

Numbers with a pair of unordered factorizations whose sums of factors are the same.
All terms of the sequence are composite.
The smallest odd term of the sequence is a(174) = 675. This is a term of the sequence because 675 = 27*5*5 = 9*3*25 and 27+5+5 = 9+3+25 = 37.
Terms of the sequence are used in variations of a logic puzzle known as "Ages of Three Children Puzzle" or "Census-taker problem". For the original puzzle, see A334911.
If a number m is in the sequence, then all multiples of m are in the sequence. For example, multiples of 4 are in the sequence because there always exist at least two factorizations 4*k = 2*2*k whose factors sum to the same value 4+k = 2+2+k.
Numbers m such that A069016(m) < A001055(m). - Michel Marcus, Aug 15 2020

			All unordered factorization of 90 are 90 = 45*2 = 30*3 = 18*5 = 15*6 = 15*3*2 = 10*9 = 9*5*2 = 10*3*3 = 6*5*3 = 5*3*3*2. Corresponding sums of factors are not all distinct: 90, 57, 33, 23, 21, 20, 19, 16, 16, 14, 13 because the sum 16 = 10+3+3 = 9+5+2 appears twice. Therefore 90 is in the sequence.
All unordered factorization of 30 are 30 = 15*2 = 10*3 = 6*5 = 5*3*2. Corresponding sums of factors are all distinct: 30 = 30, 17 = 15+2, 13 = 10+3, 11 = 6+5, 10 = 2+3+5. Therefore 30 is not in the sequence.

Eric Weisstein's World of Mathematics, Unordered Factorization.

Cf. A334911 (census-taker numbers).
Cf. A337037 (complement), A337081.
Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).
Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).

factz(n, minn) = {my(v=[]); fordiv(n, d, if ((d>=minn) && (d<=sqrtint(n)), w = factz(n/d, d); for (i=1, #w, w[i] = concat([d], w[i]);); v = concat(v, w););); concat(v, [[n]]);}
factorz(n) = factz(n, 2);
isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs != #Set(vs); \\ Michel Marcus, Aug 14 2020

Edited by N. J. A. Sloane, Sep 14 2020

A337081 Primitive complement of A337037: terms of A337080 that are not multiples of previous terms.

Original entry on oeis.org

4, 90, 546, 675, 850, 918, 945, 1026, 1050, 1134, 1242, 1365, 1386, 1575, 1650, 1750, 1782, 1950, 2205, 2295, 2310, 2450, 2475, 2646, 2793, 2850, 3250, 3366, 3465, 3626, 3654, 3762, 3850, 3969, 3990, 4218, 4290, 4374, 4455, 4510, 4550, 4650, 4875, 4998, 5022, 5166, 5382, 5390, 5610

Offset: 1

Matej Veselovac, Aug 14 2020

The only semiprime in the sequence is a(1) = 4, and there are no terms with exactly 3 prime factors.

Numbers of form p^k where p >= 5 is a prime number are terms of the sequence if and only if k = 4p+6. The only terms of the form 2^k or 3^k have k = 2, 12 respectively.

			Numbers of the form m = 2*p*q*((p-1)*q-(p-2)) where p, q and (p-1)*q-(p-2) are odd prime numbers are even terms of the sequence. First, notice that m is a term of A337080 because the factorizations m = (2*((p-1)*q-(p-2)))*(p)*(q) = (2)*(((p-1)*q-(p-2)))*(p*q) have equal sums of factors. Second, m is not a multiple of any of the previous terms of the sequence because m has exactly 4 prime factors and the only term with less than 4 prime factors is 4, but 4 does not divide m.

David A. Corneth, Table of n, a(n) for n = 1..10000
Math StackExchange, Smallest power of a prime whose factorizations don't have distinct sums of factors, 2020.

Cf. A337037, A337080, A337112 (smallest term with n factors).
Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).
Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).

factz(n, minn) = {my(v=[]); fordiv(n, d, if ((d>=minn) && (d<=sqrtint(n)), w = factz(n/d, d); for (i=1, #w, w[i] = concat([d], w[i]);); v = concat(v, w););); concat(v, [[n]]);}
factorz(n) = factz(n, 2);
isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs != #Set(vs);
isprimitive(n, va) = {for (k=1, #va, if ((n % va[k]) == 0, return (0));); return (1);}
lista(nn) = {my(va = []); for (n=1, nn, if (isok(n) && isprimitive(n, va), va = concat(va, n));); va;} \\ Michel Marcus, Aug 15 2020

A337037 Numbers whose every unordered factorization has a distinct sum of factors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101

Offset: 1

Matej Veselovac, Aug 12 2020

The number 1 is in the sequence by convention.
All primes p are trivially in the sequence.
All semiprimes greater than 4 are in the sequence because they have only two unordered factorizations pq = p*q whose sums are distinct. They are distinct because the only solution to p*q = p+q is p=q=2.
If a number m is not in the sequence, then all multiples of m are not in the sequence. For example, multiples of 4 are not in the sequence because there always exist at least two factorizations 4*k = 2*2*k whose factors sum to the same value 4+k = 2+2+k.
The complement is in A337080.
Numbers m such that A069016(m) = A001055(m). - Michel Marcus, Aug 15 2020

			All unordered factorization of 30 are 30 = 2*15 = 3*10 = 5*6 = 2*3*5. Corresponding sums of factors are distinct: 30, 17 = 15+2, 13 = 10+3, 11 = 6+5, 10 = 2+3+5. Therefore 30 is in the sequence.
All unordered factorization of 90 are 90 = 45*2 = 30*3 = 18*5 = 15*6 = 15*3*2 = 10*9 = 9*5*2 = 10*3*3 = 6*5*3 = 5*3*3*2. Corresponding sums of factors are not all distinct: 90, 57, 33, 23, 21, 20, 19, 16, 16, 14, 13 because the sum 16 = 10+3+3 = 9+5+2 appears twice. Therefore 90 is not in the sequence.

Eric Weisstein's World of Mathematics, Unordered Factorization.

Cf. A337080 (complement), A337081 (primitive complement).
Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).
Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).

factz(n, minn) = {my(v=[]); fordiv(n, d, if ((d>=minn) && (d<=sqrtint(n)), w = factz(n/d, d); for (i=1, #w, w[i] = concat([d], w[i]);); v = concat(v, w););); concat(v, [[n]]);}
factorz(n) = factz(n, 2);
isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs == #Set(vs); \\ Michel Marcus, Aug 13 2020

A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

Original entry on oeis.org

0, 4, 0, 90, 675, 1134, 6318, 4374, 32805, 255879, 1003833, 531441, 327544803, 20751953125, 225830078125, 91552734375, 1068115234375, 23651123046875, 316619873046875, 1697540283203125, 13256072998046875, 85353851318359375, 541210174560546875, 4518032073974609375, 58233737945556640625

Offset: 1

Matej Veselovac, Aug 16 2020

nonn

a(n) is the smallest product of n primes that has unordered factorizations whose sums of factors are the same (is a term of A337080) and all of whose proper divisors have the complementary property: that every unordered factorization has a distinct sum of factors (i.e., all proper divisors are terms of A337037).

Cf. A337113 (factors of terms).
Cf. A337037, A337080, A337081.
Cf. A056472 (all factorizations of n).
Cf. r-almost primes: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).

a(14) onward from David A. Corneth, Aug 26 2020

A076709 Factors in factorizations of composite numbers into at least 2 factors > 1.

Original entry on oeis.org

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

Offset: 1

Donald S. McDonald, Oct 26 2002

nonn

Each factorization is given in nondecreasing order. To determine which of two factorizations a_1 * a_2 * ... * a_r and b_1 * ... * b_s (of the same number) comes first, find the smallest index k such that a_k != b_k. If k=r then the a-factorization comes first. If k=s the b-factorization comes first. Otherwise, if a_k < b_k then the a-factorization comes first; if b_k < a_k the b-factorization comes first.

			The first 20 terms come from the factorizations of 4, 6, 8, 9, 10 and 12: 4 = 2*2, 6 = 2*3, 8 = 2*4 = 2*2*2, 9 = 3*3, 10 = 2*5, 12 = 2*6 = 2*2*3 = 3*4.

Donald S. McDonald, Posting to sci.math newsgroup, Feb 07 1999

Cf. A028422, A056472.

mf[1, ds_] := {{}}; mf[n_, {}] := {}; mf[n_, ds_] := mf[n, ds]=If[Mod[n, ds[[1]]]==0, RotateRight[Join[Prepend[ #, ds[[1]]]&/@mf[n/ds[[1]], ds], RotateLeft[mf[n, Drop[ds, 1]]]]], mf[n, Drop[ds, 1]]]; mf[n_] := mf[n, Drop[Divisors[n], 1]]; Flatten[Drop[mf[ # ], 1]&/@Range[50]]

Edited by Dean Hickerson, Dec 06 2002

A337113 Triangle read by rows in which row n lists all prime factors of A337112(n) in increasing order.

Original entry on oeis.org

0, 2, 2, 0, 0, 0, 2, 3, 3, 5, 3, 3, 3, 5, 5, 2, 3, 3, 3, 3, 7, 2, 3, 3, 3, 3, 3, 13, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 13, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 17, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 43, 43

Offset: 1

Matej Veselovac, Aug 16 2020

If A337112(n) = 0, then n 0's are listed instead.

			A337112(1)  = 0.
A337112(2)  = 2*2.
A337112(3)  = 0*0*0.
A337112(4)  = 2*3*3*5.
A337112(5)  = 3*3*3*5*5.
A337112(6)  = 2*3*3*3*3*7.
A337112(7)  = 2*3*3*3*3*3*13.
A337112(8)  = 2*3*3*3*3*3*3*3.
A337112(9)  = 3*3*3*3*3*3*3*3*5.
A337112(10) = 3*3*3*3*3*3*3*3*3*13.
A337112(11) = 3*3*3*3*3*3*3*3*3*3*17.
A337112(12) = 3*3*3*3*3*3*3*3*3*3*3*3.
A337112(13) = 3*3*3*3*3*3*3*3*3*3*3*43*43.

Cf. A337112 (products of rows).
Cf. A337037, A337080, A337081.
Cf. A056472 (all factorizations of n).

cp's OEIS Frontend

A162247 Irregular triangle in which row n lists all factorizations of n, sorted by the number of factors in each factorization.

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A337080 Complement of A337037.

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A337081 Primitive complement of A337037: terms of A337080 that are not multiples of previous terms.

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A337037 Numbers whose every unordered factorization has a distinct sum of factors.

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A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

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A076709 Factors in factorizations of composite numbers into at least 2 factors > 1.

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A337113 Triangle read by rows in which row n lists all prime factors of A337112(n) in increasing order.

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