A337080 -id:A337080 - cp's OEIS frontend

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

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

A334911 Census-taker numbers: Numbers k such that exactly two unordered triples of positive numbers have product k and equal sums.

Original entry on oeis.org

36, 40, 72, 96, 126, 176, 200, 225, 234, 252, 280, 297, 320, 408, 520, 550, 576, 588, 600, 648, 690, 714, 735, 736, 768, 780, 784, 816, 850, 855, 896, 945, 972, 1026, 1040, 1064, 1092, 1160, 1188, 1216, 1242, 1248, 1275, 1280, 1296, 1300, 1350, 1404, 1530

Offset: 1

Thomas Gawlick, May 16 2020

nonn

First differs from A060292 at n = 4.

If neither of the two unordered triples contains a 1, then the corresponding term is also in A337080. - Matej Veselovac, Sep 14 2020

			36 = 6*6*1 = 9*2*2. 6+6+1 = 9+2+2. So 36 is in the sequence.
This example also explains the name of the sequence:
"A census taker knocks on a door. A mother answers.
The census taker says, "I need to know the number of children you have, and their ages." The woman responds in puzzle-ese, "I have three daughters, the product of their ages is 36, and the sum of their ages is equal to the house number next door."
The census taker, who never wastes questions, computes for a while and then asks, "Does your oldest daughter love dogs?" The mother answers affirmatively. The census taker says, "Thank you. I now know the ages."
What are the ages of the children?" (Garces and Loyola 2010, 28)
3264 is not in the sequence: (34, 32, 3) and (48, 17, 4) give sum 69, but (48, 34, 2), (64, 17, 3) and (68, 12, 4) give sum 84.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
L. F. Meyers and R. See, The Census-Taker Problem, Math. Mag. 63 (1990) 86-88
I. J. L. Garces and M. L. Loyola, Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem, Intersection, 11 (2010), pp. 28-38; arXiv:1204.2071 [math.HO], 2012.

Subset of A060292 (at least two unordered triples of positive numbers have product n and equal sums).

Cf. A337080 (at least two unordered factorizations of n have equal sums of factors).

b:= proc(n, k, t) option remember; expand(`if`(t=1, `if`(kk, 0, b(n/d, d, t-1)*x^d), d=numtheory[divisors](n))))
    end:
q:= n-> (p-> degree(p)=2 and coeff(p, x, 2)=1)(
             add(x^i, i=[coeffs(b(n$2, 3))])):
select(q, [$1..2000])[];  # Alois P. Heinz, May 16 2020

b[n_, k_, t_] := b[n, k, t] = If[t==1, If[kk, 0, b[n/d, d, t-1] x^d], {d, Divisors[n]}]];
q[n_] := Exponent[#, x]==2 && Coefficient[#, x, 2]==1& @ Total[ x^CoefficientList[b[n, n, 3], x]];
Select[Range[2000], q] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)

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

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).

A337093 Difference between the number of unordered factorizations and the number of distinct sums of terms in these unordered factorizations for those integers where this difference is positive.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 6, 1, 4, 3, 2, 1, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 13, 1, 2, 2, 8, 1, 6, 1, 4, 5, 2, 1, 12, 2, 4, 2, 4, 1, 12, 2, 7, 2, 2, 1, 15, 1, 2, 5, 11, 3, 5, 1, 2, 4, 2, 5, 1, 20, 1, 2, 5, 4, 2, 5, 1, 13, 6, 2, 1

Offset: 1

Michel Marcus, Aug 15 2020

nonn

Cf. A001055, A069016, A337080.

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);
lista(nn) = {for (n=1, nn, my(vf = factorz(n)); my(vs = apply(x->vecsum(x), vf)); my(d = #vs - #Set(vs)); if (d>0, print1(d, ", ")););}

a(n) = A001055(A337080(n)) - A069016(A337080(n)).

cp's OEIS Frontend

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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A334911 Census-taker numbers: Numbers k such that exactly two unordered triples of positive numbers have product k and equal sums.

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

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A337093 Difference between the number of unordered factorizations and the number of distinct sums of terms in these unordered factorizations for those integers where this difference is positive.

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