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.
The a(900) = 12 matrix-arrangements of (3,3,2,2,1,1): [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1] [3 2 1] [3 1 2] [2 3 1] [2 1 3] [1 3 2] [1 2 3] . [1 3] [1 3] [2 2] [2 2] [3 1] [3 1] [2 2] [3 1] [1 3] [3 1] [1 3] [2 2] [3 1] [2 2] [3 1] [1 3] [2 2] [1 3]
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[ptnmats[n],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,100}]
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[2,1000],Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]
In general, no prime p is a term since they are a power of base p. Numbers having a single digit are not terms: 1 is not a term since 1 is a power of all bases b; Composites 4, 6, and 9 are not in the sequence since 4 = 2^2, 6 = 2*3, and 9 = 3^2. 10 is not a term since it ends in a single zero, and zero is not a power of another number. a(1) = 12 since it is not a perfect power, 2 | 12, and 12 mod 10 is a power of 2. a(2) = 14 since it is not a perfect power, 2 | 14, and 14 mod 10 is a power of 2. 20 is not a term since it ends with a zero, and zero is not a power of another number. 26 is not a term since 6 does not divide 26. 1100 is a term since it is not a perfect power, 100 = 100^1, and 100 | 1100.
nn = 130; t = Union@ Flatten@ Table[m = 10^IntegerLength[b] + b; If[m > nn, Nothing, s = b^Range[0, Floor@ Log[b, nn]]; Flatten@ Reap[Map[(w = IntegerDigits[#]; i = 0; While[Set[k, FromDigits@ Join[IntegerDigits[i], w]] <= nn, If[And[Divisible[k, b], FreeQ[s, k]], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]
21 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21. 50 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 2, 5, 10, 25, and 50 sum to 93, which is coprime to 50.
perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ Range@ 350, !PrimeQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 && !perfectPowerQ[ #] &] cpQ[n_]:=CoprimeQ[n,DivisorSigma[1,n]]&&!PrimeQ[n]&&GCD@@ FactorInteger[ n][[All,2]]<2; Select[Range[2,400],cpQ] (* Harvey P. Dale, Oct 05 2020 *)
forcomposite(n=1, 1e3, if(gcd(n, sigma(n))==1, if(!ispower(n), print1(n, ", ")))) \\ Felix Fröhlich, Oct 25 2014
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1; per=Select[Range[1000],perpowQ]; per-NestList[NestWhile[#+1&,#+1,!PrimePowerQ[#]&]&,1,Length[per]-1]
from sympy import mobius, primepi, integer_nthroot def A377043(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length()))) def g(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))) return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1; per=Select[Range[1000],perpowQ]; per-NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,2,Length[per]-1]
from sympy import mobius, primepi, integer_nthroot def A377044(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length()))) def g(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))) return bisection(f,n,n)-bisection(g,n,n) # Chai Wah Wu, Oct 27 2024
The composite numbers counted by a(n) cover A106543 with the following disjoint sets: . 6 . 10 12 14 15 18 20 21 22 24 26 28 30 33 34 35 38 39 40 42 44 45 46 48 50 51 52 54 55 56 57 58 60 62 63
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
v=Select[Range[100],perpowQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]
from sympy import mobius, integer_nthroot, primepi def A378614(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return int(n+x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length()))) return -(a:=bisection(f,n,n))+(b:=bisection(lambda x:f(x)+1,a+1,a+1))-primepi(b)+primepi(a)-1 # Chai Wah Wu, Dec 03 2024
50 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 2, 5, 10, 25, and 50 sum to 93, which is coprime to 50.
perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ 2 Range@ 7500, !PrimeQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 && !perfectPowerQ[ #] &]
lista(nn) = {forstep(n=4, nn, 2, if (!ispower(n) && (gcd(n, sigma(n))==1), print1(n, ", ")););} \\ Michel Marcus, Oct 02 2014
a(0) = 0 because 0[+oo]n = 0. a(1) = 0 because 1[+oo]n = 1. a(4) = 0 because 2[+oo]2 = 4. a(2) = 1 because 2 is prime. a(6) = 2 because 6 is composite but not a power. a(9) = 3 because 9 is a power but not a hyper-4 power. a(27) = 4 because 27 is a hyper-4 power but not a hyper-5 power. a(65536) = 5 because 65536 is a hyper-5 power but not a hyper-6 power. ...
Comments