cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

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Author

Matej Veselovac, Aug 16 2020

Keywords

Comments

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).

Crossrefs

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).

Extensions

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

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

Views

Author

Matej Veselovac, Aug 14 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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).

Programs

  • PARI
    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

Extensions

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

Views

Author

Matej Veselovac, Aug 12 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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).

Programs

  • PARI
    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

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

Views

Author

Matej Veselovac, Aug 16 2020

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A337112 (products of rows).
Cf. A337037, A337080, A337081.
Cf. A056472 (all factorizations of n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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