Herman Beeksma - cp's OEIS frontend

A212875 Primonacci numbers: composite numbers that appear in the Fibonacci-like sequence generated by their own prime factors.

4, 9, 12, 25, 27, 169, 1102, 7921, 22287, 54289, 103823, 777627, 876897, 2550409, 20854593, 34652571, 144237401, 144342653, 167901581, 267911895, 792504416, 821223649, 1103528482, 2040412557, 2852002829, 3493254541, 6033671841, 15658859018, 116085000401

Offset: 1

Herman Beeksma, May 29 2012

nonn

Given n, form a sequence that starts with the k prime factors of n in ascending order. After that, each term is the sum of the preceding k terms. If n eventually appears in the sequence, it is a primonacci number. Primes possess this property trivially and are therefore excluded.
Similar to A007629 (repfigit or Keith numbers), but base-independent. If n is in A005478 (Fibonacci primes), then n^2 is a primonacci number.
The only  entries that are semiprimes (A001358) are the squares of A005478. - Robert Israel, Mar 08 2016

			Fibonacci-like sequences for selected values of n:
n=12: 2, 2, 3, 7, 12, ...
n=25: 5, 5, 10, 15, 25, ...
n=1102: 2, 19, 29, 50, 98, 177, 325, 600, 1102, ...

Herman Beeksma, Table of n, a(n) for n = 1..42

Cf. A000045, A001358, A005478, A007629.

with(numtheory): P:=proc(q,h) local a,b,j,k,n,t,v; v:=array(1..h);
for n from 2 to q do if not isprime(n) then b:=ifactors(n)[2]; a:=[];
for k from 1 to nops(b) do for j from 1 to b[k][2] do a:=[op(a),b[k][1]]; od; od; a:=sort([op(a)]);
b:=nops(a);  for k from 1 to b do v[k]:=a[k]; od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]Paolo P. Lava, Mar 08 2016

PrimonacciQ[n_]:=Module[{k,seq},
  seq=FactorInteger[n];
  seq=Map[Table[#[[1]],{#[[2]]}]&, seq];
  seq=Flatten[seq];
  k=Length[seq];
  If[k==1,Return[False]];
  seq=Append[seq,Apply[Plus,seq]];
  While[seq[[-1]]Michael De Vlieger, Mar 08 2016 *)

from sympy import isprime, factorint
from itertools import chain
A212875_list = []
for n in range(2,10**6):
    if not isprime(n):
        x = sorted(chain.from_iterable([p]*e for p,e in factorint(n).items()))
        y = sum(x)
        while y < n:
            x, y = x[1:]+[y], 2*y-x[0]
        if y == n:
            A212875_list.append(n) # Chai Wah Wu, Sep 12 2014

A181061 a(n) is the smallest positive number such that the decimal digits of n*a(n) are all 0, 1 or 2.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 2, 3, 14, 1358, 1, 1, 1, 17, 8, 8, 7, 6, 679, 58, 1, 1, 1, 44, 5, 4, 47, 786, 4, 38, 4, 71, 35, 34, 3, 6, 617, 3, 29, 259, 3, 271, 5, 47, 5, 2716, 22, 26, 25, 229, 2, 2, 231, 4, 393, 2, 2, 193, 19, 19, 2, 2, 181, 194, 33, 34, 17, 3, 15, 29, 3, 31, 1696, 14, 3, 16, 145

Offset: 0

Herman Beeksma, Oct 01 2010

Records occur for n: 1, 3, 8, 9, 45, 89, 99, 495, 998, ..., . [Robert G. Wilson v, Oct 04 2010]

			a(9)=1358 because 9*1358=12222 is the smallest multiple of 9 whose decimal digits are all 0, 1 or 2.

Robert G. Wilson v, Table of n, a(n) for n = 0..100000 (first 20000 terms and some terms up to 100000 from Victor Maslov)
Victor Maslov, Log-log plot for n = 0..20000
Project Euler, Problem 303: Multiples with small digits

n*a(n) yields sequence A181060.

f[n_] := Block[{id = {0, 1, 2}, k = 1}, While[ Union@ Join[id, IntegerDigits[k*n]] != id, k++ ]; k]; Array[f, 76] (* Robert G. Wilson v, Oct 04 2010 *)

a(n)=my(k=n,t);while(1,t=vecsort(eval(Vec(Str(k))),,8);if(t[#t]<3,return(k/n),k+=n)) \\ Charles R Greathouse IV, Sep 05 2011

anydiggt2(n)=if(n<2,0,while(n>0,if(n%10>2,return(1));n\=10);0)
a(n)={local(prd,ms=[],rems=[],msn,remsn,pow=10,maxm);
  if(n<=0,return(0));
  while(n%10==0,n\=10);
  maxm=if(n%10==5,5,if(n%2==1,3,4));
  for(d=1,9,if(!anydiggt2(prd=d*n),return(d));
    if(prd%10<=2,ms=concat(ms,[d]);rems=concat(rems,[prd\10])));
  while(1,
    msn=listcreate(maxm*#ms);remsn=listcreate(maxm*#ms);
    for(d=0,9,
      for(k=1,#ms,
        if(!anydiggt2(prd=d*n+rems[k]),return(d*pow+ms[k]));
        if(prd%10<=2,listput(msn,d*pow+ms[k]);listput(remsn,prd\10))));
    ms=Vec(msn);rems=Vec(remsn);pow*=10;print1("."))}

If a(n)=k, then a(10n)=k. [Robert G. Wilson v, Oct 04 2010]

A181060 a(n) is the smallest positive multiple of n whose decimal digits are all 0, 1 or 2.

Original entry on oeis.org

1, 2, 12, 12, 10, 12, 21, 112, 12222, 10, 11, 12, 221, 112, 120, 112, 102, 12222, 1102, 20, 21, 22, 1012, 120, 100, 1222, 21222, 112, 1102, 120, 2201, 1120, 1122, 102, 210, 22212, 111, 1102, 10101, 120, 11111, 210, 2021, 220, 122220, 1012, 1222, 1200

Offset: 1

Herman Beeksma, Oct 01 2010

			a(9)=12222 because 12222 is the smallest multiple of 9 whose decimal digits are all 0, 1 or 2.

David A. Corneth, Table of n, a(n) for n = 1..10000
Inspired by Project Euler, Problem 303: Multiples with small digits.

a(n)/n yields sequence A181061.

With[{pms=Rest[Flatten[FromDigits/@Tuples[{0,1,2},6]]]},Table[ SelectFirst[ pms, Divisible[ #,n]&],{n,50}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2017 *)

a(n) = my(k=1); while(vecmax(digits(k*n))>2, k++); k*n; \\ Michel Marcus, May 17 2020

A134857 Numbers that can be written as (a^2-1)(b^2-1) in four or more distinct ways.

Original entry on oeis.org

241920, 1048320, 10200960, 25724160, 37255680, 93139200, 123963840, 245044800, 588107520, 819786240, 1407893760, 1871251200, 3758169600, 5886558720, 8553283200, 10783342080, 13470367680, 19769460480, 30791819520, 40446806400

Offset: 1

Herman Beeksma, Nov 13 2007

Subsequence of A134856. Contains A134858 as a subsequence.

			241920 = (4^2-1)(127^2-1) = (7^2-1)(71^2-1) = (9^2-1)(55^2-1) = (17^2-1)(29^2-1).

MathPages, Numbers Expressible As (a^2-1)(b^2-1).

Cf. A134856, A134858.

Cf. A057535, A063066, A063067, A063068.

A134856 Numbers that can be written as (a^2-1)(b^2-1) in three or more distinct ways.

Original entry on oeis.org

2880, 27720, 40320, 49920, 63360, 98280, 241920, 282744, 491400, 547200, 604800, 950400, 970200, 1048320, 1370880, 1614600, 1774080, 2489760, 2608320, 2882880, 2923200, 3931200, 4817400, 6126120, 7338240, 7673400, 8426880, 10200960

Offset: 1

Herman Beeksma, Nov 13 2007

Contains A134857 and A134858 as subsequences.

			2880 = (2^2-1)(31^2-1) = (3^2-1)(19^2-1) = (5^2-1)(11^2-1).

MathPages, Numbers Expressible As (a^2-1)(b^2-1).

Cf. A134857, A134858, A057535, A063066, A063067, A063068.

A134858 Numbers that can be written as (a^2 - 1)(b^2 - 1) in five or more distinct ways.

Original entry on oeis.org

588107520, 67270694400, 546939993600, 2128050512640, 37400697734400, 5566067918611200

Offset: 1

Herman Beeksma, Nov 13 2007

Subsequence of A134856 and A134857.
Depending on the interpretation of A057535, this is either the same or a supersequence of A057535. [R. J. Mathar, Oct 16 2009]
The next term (if it exists) is greater than 2^70.

			588107520 = (13^2 - 1)(1871^2 - 1) = (17^2 - 1)(1429^2 - 1) = (55^2 - 1)(441^2 - 1) = (79^2 - 1)(307^2 - 1) = (129^2 - 1)(188^2 - 1).

MathPages, Numbers Expressible As (a^2-1)(b^2-1).

Cf. A057535, A134857, A134856, A063067, A063066, A063068.

A108066 Number of distinct ways to dissect a square into n rectangles of equal area.

Original entry on oeis.org

1, 1, 2, 6, 18, 65, 281, 1343, 6953, 38023

Offset: 1

Hans Riesebos (hans.riesebos(AT)wanadoo.nl) and Herman Beeksma, Jun 03 2005

"Distinct" here means that dissections differing only by a rotation and/or reflection are not counted as different (see A189243).

The first time the pieces can be made to all have different shapes (but the same area) is at n=7 - see Descartes (1971) and the illustration; also Wells, Weisstein. - N. J. A. Sloane, Dec 05 2012

			There are six ways to dissect a square into four rectangles of equal area, so a(4)=6:
+-+-----+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+
| |     | | | |   | | |     | | | | | | | |   | | |   |   |
| |     | | | |   | | |     | | | | | | | |   | | |   |   |
| +--+--+ | | |   | | +-----+ | | | | | | |   | | |   |   |
| |  |  | | | +---+ | |     | | | | | | | +---+ | +---+---+
| |  |  | | | |   | | |_____| | | | | | | |   | | |   |   |
| |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
| |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
+-+--+--+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+

David Wells, Penguin Dictionary of Curious and Interesting Geometry, 1991, pp. 15-16.

Blanche Descartes, Divisions of a square into rectangles, Eureka, No. 34 (1971), 31-35.
R. Häggkvist, P.-O. Lindberg, and B. Lindström, Dissecting a square into rectangles of equal area, Discr. Math. 47 (1983), 321-323.
Johan Nilsson, Illustrations for terms up to a(6)..
N. J. A. Sloane, Blanche Descartes's dissection into 7 pieces with equal areas but different shapes.
Eric Weisstein, Blanche's Dissection.

Cf. A189243, A219861, A359146.

A100664 Number of inequivalent ways to dissect a square into n rectangles of equal perimeter.

Original entry on oeis.org

1, 1, 2, 6, 16, 50, 177, 664, 2532, 9785

Offset: 1

Herman Beeksma and Hans Riesebos (hans.riesebos(AT)wanadoo.nl), Aug 04 2005

Dissections differing only by rotations and/or reflections are not counted as distinct. - N. J. A. Sloane, Dec 06 2012

			There are six ways to dissect a square into four rectangles of equal perimeter, so a(4)=6.

Johan Nilsson, Illustrations of terms up to a(6).

Cf. A108066.

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User: Herman Beeksma

A212875 Primonacci numbers: composite numbers that appear in the Fibonacci-like sequence generated by their own prime factors.

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A181061 a(n) is the smallest positive number such that the decimal digits of n*a(n) are all 0, 1 or 2.

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A181060 a(n) is the smallest positive multiple of n whose decimal digits are all 0, 1 or 2.

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A134857 Numbers that can be written as (a^2-1)(b^2-1) in four or more distinct ways.

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A134856 Numbers that can be written as (a^2-1)(b^2-1) in three or more distinct ways.

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A134858 Numbers that can be written as (a^2 - 1)(b^2 - 1) in five or more distinct ways.

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A108066 Number of distinct ways to dissect a square into n rectangles of equal area.

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A100664 Number of inequivalent ways to dissect a square into n rectangles of equal perimeter.

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