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.

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

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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

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

Views

Author

Matej Veselovac, Aug 14 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

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);
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

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

Views

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

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.

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