A047983 -id:A047983 - cp's OEIS frontend

A007416 The minimal numbers: sequence A005179 arranged in increasing order.

1, 2, 4, 6, 12, 16, 24, 36, 48, 60, 64, 120, 144, 180, 192, 240, 360, 576, 720, 840, 900, 960, 1024, 1260, 1296, 1680, 2520, 2880, 3072, 3600, 4096, 5040, 5184, 6300, 6480, 6720, 7560, 9216, 10080, 12288, 14400, 15120, 15360, 20160, 25200, 25920, 27720, 32400, 36864, 44100

Offset: 1

N. J. A. Sloane

Numbers k such that there is no x < k such that A000005(x) = A000005(k). - Benoit Cloitre, Apr 28 2002
A047983(a(n)) = 0. - Reinhard Zumkeller, Nov 03 2015
Subsequence of A025487. If some m in A025487 is the first term in that sequence having its number of divisors, m is in this sequence. - David A. Corneth, Aug 31 2019

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..100000 (first 1000 from T. D. Noe and to 10000 from David A. Corneth)
Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116:1 (2005), pp. 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
J. Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.
Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.

Subsequence of A025487; A002182 is a subsequence.
Cf. A005179, A099317, A099312, A099314, A099316, A099318.
Cf. A000005, A047983, A166721 (subsequence of squares).
Cf. A053212 and A064787 (the sequence {A000005(a(n))} and its inverse permutation).

a007416 n = a007416_list !! (n-1)
a007416_list = f 1 [] where
   f x ts = if tau `elem` ts then f (x + 1) ts else x : f (x + 1) (tau:ts)
            where tau = a000005' x
-- Reinhard Zumkeller, Apr 18 2015

for n from 1 to 10^5 do
  t:= numtheory:-tau(n);
  if not assigned(B[t]) then B[t]:= n fi;
od:
sort(map(op,[entries(B)]));# Robert Israel, Nov 11 2015

A007416 = Reap[ For[ s = 1, s <= 10^5, s++, If[ Abs[ Product[ DivisorSigma[0, i] - DivisorSigma[0, s], {i, 1, s-1}]] > 0, Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Nov 19 2012, after Pari *)

for(s=1,10^6,if(abs(prod(i=1,s-1,numdiv(i)-numdiv(s)))>0,print1(s,",")))

is(n)=my(d=numdiv(n));for(i=1,n-1,if(numdiv(i)==d, return(0))); 1 \\ Charles R Greathouse IV, Feb 20 2013

A283980(n,f=factor(n))=prod(i=1, #f~, my(p=f[i, 1]); if(p==2, 6, nextprime(p+1))^f[i, 2])
A025487do(e) = my(v=List([1, 2]), i=2, u = 2^e, t); while(v[i] != u, if(2*v[i] <= u, listput(v, 2*v[i]); t = A283980(v[i]); if(t <= u, listput(v, t))); i++); Set(v)
winnow(v,lim=v[#v])=my(m=Map(),u=List()); for(i=1,#v, if(v[i]>lim, break); my(t=numdiv(v[i])); if(!mapisdefined(m,t), mapput(m,t,0); listput(u,v[i]))); m=0; Vec(u)
list(lim)=winnow(A025487do(logint(lim\1-1,2)+1),lim) \\ Charles R Greathouse IV, Nov 17 2022

A067004 Number of numbers <= n with same number of divisors as n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 5, 1, 6, 4, 5, 1, 7, 2, 8, 3, 6, 7, 9, 1, 3, 8, 9, 4, 10, 2, 11, 5, 10, 11, 12, 1, 12, 13, 14, 3, 13, 4, 14, 6, 7, 15, 15, 1, 4, 8, 16, 9, 16, 5, 17, 6, 18, 19, 17, 1, 18, 20, 10, 1, 21, 7, 19, 11, 22, 8, 20, 2, 21, 23, 12, 13, 24, 9, 22, 2, 2, 25, 23, 3, 26, 27

Offset: 1

Henry Bottomley, Dec 21 2001

nonn

			a(10)=3 since 6,8,10 each have four divisors. a(11)=5 since 2,3,5,7,11 each have two divisors.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000005, A008479, A058933, A067003. Inverse of A000040, A001248, A030513, A030514, A030515, A030516, A030626, A030627, A030628, A030629, A030630, A030631, A030632, A030633, A030634, A030635, A030636, A030637, A030638 etc.
Cf. also A079788, A138009.
A047983(n) = a(n)-1.

N:= 1000: # to get a(1) to a(N)
R:= Vector(N):
for n from 1 to N do
  v:= numtheory:-tau(n);
  R[v]:= R[v]+1;
  A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, May 04 2015

b[_] = 0;
a[n_] := a[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

a(n)=my(d=numdiv(n)); sum(k=1,n,numdiv(k)==d) \\ Charles R Greathouse IV, Sep 02 2015

Ordinal transform of A000005. - Franklin T. Adams-Watters, Aug 28 2006

a(A000040(n)^(p-1)) = n if p is prime. - Robert Israel, May 04 2015

A335097 Number of integers less than n with the same number of prime factors (counted with multiplicity) as n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 2, 3, 4, 1, 5, 4, 5, 0, 6, 2, 7, 3, 6, 7, 8, 1, 8, 9, 4, 5, 9, 6, 10, 0, 10, 11, 12, 2, 11, 13, 14, 3, 12, 7, 13, 8, 9, 15, 14, 1, 16, 10, 17, 11, 15, 4, 18, 5, 19, 20, 16, 6, 17, 21, 12, 0, 22, 13, 18, 14, 23, 15, 19, 2, 20, 24, 16, 17, 25, 18, 21, 3

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

			a(10) = 3 because bigomega(10) = 2 and also bigomega(4) = bigomega(6) = bigomega(9) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000079 (positions of 0's), A001222, A047983, A058933, A067004, A322838, A334655.

A:= NULL:
for n from 1 to 100 do
  t:= numtheory:-bigomega(n);
  if not assigned(R[t]) then
    A:= A,0;
    R[t]:= 1;
   else
    A:= A, R[t];
    R[t]:= R[t]+1;
   fi
od:
A; # Robert Israel, Oct 24 2021

Table[Length[Select[Range[n - 1], PrimeOmega[#] == PrimeOmega[n] &]], {n, 80}]

a(n)={my(t=bigomega(n)); sum(k=1, n-1, bigomega(k)==t)} \\ Andrew Howroyd, Oct 31 2020

from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A335097(n):
    if n==1: return 0
    if isprime(n): return primepi(n)-1
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))-1) # Chai Wah Wu, Aug 28 2024

a(n) = |{j < n : bigomega(j) = bigomega(n)}|.

a(n) = A058933(n) - 1.

A334655 Number of integers less than n with the same number of distinct prime factors as n.

Original entry on oeis.org

0, 0, 1, 2, 3, 0, 4, 5, 6, 1, 7, 2, 8, 3, 4, 9, 10, 5, 11, 6, 7, 8, 12, 9, 13, 10, 14, 11, 15, 0, 16, 17, 12, 13, 14, 15, 18, 16, 17, 18, 19, 1, 20, 19, 20, 21, 21, 22, 22, 23, 24, 25, 23, 26, 27, 28, 29, 30, 24, 2, 25, 31, 32, 26, 33, 3, 27, 34, 35, 4, 28, 36, 29, 37, 38, 39, 40, 5, 30, 41

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

			a(12) = 2 because omega(12) = 2 and also omega(6) = omega(10) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001221, A002110 (positions of 0's), A047983, A067003, A067004, A322837, A322841, A335097.

R:= NULL:
for n from 1 to 100 do
  w:= nops(numtheory:-factorset(n));
  if assigned(V[w]) then V[w]:= V[w]+1 else V[w]:= 1 fi;
  R:= R, V[w]-1
od:
R; # Robert Israel, Feb 25 2024

Table[Length[Select[Range[n - 1], PrimeNu[#] == PrimeNu[n] &]], {n, 80}]

a(n)={my(t=omega(n)); sum(k=1, n-1, omega(k)==t)} \\ Andrew Howroyd, Oct 31 2020

a(n) = |{j < n : omega(j) = omega(n)}|.

a(n) = A067003(n) - 1.

A339514 Number of subsets of {1..n} whose elements have the same number of divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 19, 21, 23, 27, 43, 44, 76, 84, 100, 101, 165, 167, 295, 299, 331, 395, 651, 652, 656, 784, 1040, 1048, 1560, 1562, 2586, 2602, 3114, 4138, 6186, 6187, 8235, 12331, 20523, 20527, 24623, 24631, 32823, 32855, 32919, 49303, 65687, 65688

Offset: 0

Ilya Gutkovskiy, Dec 07 2020

nonn

			a(8) = 21 subsets: {}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}, {6, 8}, {2, 3, 5}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7} and {2, 3, 5, 7}.

Sebastian Karlsson, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Divisor

Cf. A000005, A339511, A339512, A047983.

from sympy import divisors
def test(n):
    if n<2: return n-1
    return len(divisors(n))
def a(n):
    tests = [test(i) for i in range(n+1)]
    return sum(2**tests.count(v)-1 for v in set(tests))
print([a(n) for n in range(49)]) # Michael S. Branicky, Dec 07 2020

a(0) = 1, a(n) = a(n-1) + 2^A047983(n). - Sebastian Karlsson, Dec 26 2020

a(25)-a(48) from Michael S. Branicky, Dec 07 2020

A337557 Number of integers less than n with the same number of odd divisors as n.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 3, 3, 0, 4, 5, 6, 7, 8, 0, 4, 9, 1, 10, 11, 1, 12, 13, 14, 2, 15, 2, 16, 17, 3, 18, 5, 4, 19, 5, 3, 20, 21, 6, 22, 23, 7, 24, 25, 0, 26, 27, 28, 4, 5, 8, 29, 30, 9, 10, 31, 11, 32, 33, 12, 34, 35, 1, 6, 13, 14, 36, 37, 15, 16, 38, 6, 39, 40, 2, 41, 17, 18, 42, 43

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

			a(10) = 4 because A001227(10) = 2 and also A001227(3) = A001227(5) = A001227(6) = A001227(7) = 2.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001227, A047983, A334655, A335097.

Table[Length[Select[Range[n - 1], Sum[Mod[d, 2], {d, Divisors[#]}] == Sum[Mod[d, 2], {d, Divisors[n]}] &]], {n, 80}]

a(n)={my(t=numdiv(n/2^valuation(n, 2))); sum(k=1, n-1, numdiv(k/2^valuation(k, 2))==t)} \\ Andrew Howroyd, Oct 31 2020

first(n) = { my(m = Map(), res = vector(n)); for(i = 1, n, q = numdiv(i >> valuation(i, 2)); if(mapisdefined(m, q), res[i] = mapget(m, q); mapput(m, q, res[i]+1); , mapput(m, q, 1) ) ); res } \\ David A. Corneth, Oct 31 2020

a(n) = |{j < n : A001227(j) = A001227(n)}|.

A338483 a(n) is the smallest number having n smaller numbers with the same number of divisors.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Ivan N. Ianakiev, Oct 30 2020

nonn

Inspired by A047983.

Are there prime terms greater than 31?

			The smallest number having two smaller numbers (2 and 3) with the same number of divisors is 5, so a(2) is 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A007422, A030513, A047983.

N:= 500: # for terms before the first term > N
T:= map(numtheory:-tau, [$1..N]):
M:= max(T):
V:= Vector(M):
for n from 1 to N do
  v:= T[n];
  V[v]:= V[v]+1;
  if not assigned(R[V[v]]) then R[V[v]]:= n fi
od:
for nn from 1 while assigned(R[nn]) do od:
seq(R[i],i=2..nn-1); # Robert Israel, Oct 30 2020

f[n_]:=With[{tau=DivisorSigma[0,n]},Length[Select[Range[n-1],DivisorSigma[0,#]==tau&]]];t=Table[f[n],{n,1,300}]; a[n_]:=FirstPosition[t,n]; Rest[a/@Range[0,65]]//Flatten (* f(n) by Jean-François Alcover at A047983 *)

f(n) = {my(d=numdiv(n)); sum(k=1, n-1, (numdiv(k)==d))} \\ A047983
a(n) = my(k=1); while (f(k)!= n, k++); k; \\ Michel Marcus, Oct 30 2020

A047983(a(n)) = n. - Rémy Sigrist, Dec 06 2020

A338569 Number of integers less than n with the same number of ordered factorizations as n.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 4, 0, 1, 1, 5, 0, 6, 2, 3, 1, 7, 2, 8, 3, 4, 5, 9, 0, 2, 6, 1, 4, 10, 0, 11, 0, 7, 8, 9, 0, 12, 10, 11, 1, 13, 1, 14, 5, 6, 12, 15, 0, 3, 7, 13, 8, 16, 2, 14, 3, 15, 16, 17, 0, 18, 17, 9, 0, 18, 2, 19, 10, 19, 3, 20, 0, 21, 20, 11, 12, 21, 4, 22, 1

Offset: 1

Ilya Gutkovskiy, Nov 02 2020

nonn

			a(14) = 2 because A074206(14) = 3 and also A074206(6) = A074206(10) = 3.

Eric Weisstein's World of Mathematics, Ordered Factorization

Cf. A047983, A074206, A338568.

A074206[1] = 1; A074206[n_] := A074206[n] = A074206 /@ Most[Divisors[n]] // Total; Table[Length[Select[Range[n - 1], A074206[#] == A074206[n] &]], {n, 80}]

a(n) = |{j < n : A074206(j) = A074206(n)}|.

A377734 Number of integers less than n that have the same smallest prime factor as n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 6, 2, 7, 0, 8, 0, 9, 3, 10, 0, 11, 1, 12, 4, 13, 0, 14, 0, 15, 5, 16, 2, 17, 0, 18, 6, 19, 0, 20, 0, 21, 7, 22, 0, 23, 1, 24, 8, 25, 0, 26, 3, 27, 9, 28, 0, 29, 0, 30, 10, 31, 4, 32, 0, 33, 11, 34, 0, 35, 0, 36, 12, 37, 2, 38, 0, 39

Offset: 1

Ilya Gutkovskiy, Nov 05 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor.

Cf. A020639, A038110, A038111, A047983, A078898, A334655, A335097, A377730.

Table[Length[Select[Range[n - 1], If[# == 1, 1, FactorInteger[#][[1, 1]]] == If[n == 1, 1, FactorInteger[n][[1, 1]]] &]], {n, 80}]
seq[len_] := Module[{t = Table[FactorInteger[n][[1,1]], {n, 1, len}], s = Table[0, {len}]}, Do[s[[i]] = Count[t[[1;;i-1]], t[[i]]], {i, 1, len}]; s]; seq[80] (* Amiram Eldar, Nov 21 2024 *)

a(n) = if (n>1, my(p=vecmin(factor(n)[,1])); sum(k=2, n-1, p == vecmin(factor(k)[,1])), 0); \\ Michel Marcus, Nov 16 2024

a(n) = |{j < n : lpf(j) = lpf(n)}|.
a(n) = A078898(n) - 1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Nov 21 2024

A347494 a(n) is the number of instances where tau(k)|tau(n); n>=1, kA000005.

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 4, 6, 2, 7, 5, 8, 6, 10, 11, 1, 7, 11, 8, 13, 14, 15, 9, 17, 3, 17, 18, 16, 10, 21, 11, 19, 21, 22, 23, 4, 12, 25, 26, 29, 13, 31, 14, 23, 24, 29, 15, 17, 4, 27, 31, 28, 16, 37, 33, 39, 34, 35, 17, 50, 18, 38, 32, 1, 39, 46, 19, 34, 41, 49, 20, 59

Offset: 1

David James Sycamore, Aug 31 2021

nonn

a(n) >= 1, for n >= 2; equality only when n = 2^(p-1) for any prime p. More generally, if p is prime(m), q any prime, and n=p^(q-1) then tau(n) = q, and the only numbers k < m such that tau(k)|tau(n) are 1 and q. Every prime < p contributes 1 to the count of a(n), and so does 1 itself, therefore a(n) = m-1+1 = m; see formula. Since for a given m, this holds for all primes q, it follows that every m > 0 appears in the sequence infinitely many times.

			a(1) = 0 because there is no k < 1 such that tau(k)|tau(1).
a(2) = 1, since there is only one instance of tau(k)|tau(2), namely k=1.
a(3) = 2, since there are two instances of tau(k)|tau(3), namely k=1 and k=2.
a(4) = 3, since there is only one instance of tau(k)|tau(4), namely k=1, etc.

David A. Corneth, Table of n, a(n) for n = 1..10001

Cf: A000005, A010553, A047983, A010553.

With[{s = DivisorSigma[0, Range[72]]}, Array[Count[Mod[#2, s[[Range[#1 - 1]]]], 0] & @@ {#, s[[#]]} &, Length[s] - 1, 2]] (* Michael De Vlieger, Sep 09 2021 *)

first(n) = {my(l = List(), res = vector(n)); for(i = 1, n, nd = numdiv(i); if(nd > #l, for(i = #l + 1, nd, listput(l, 0) ) ); d = divisors(nd); for(j = 1, #d, res[i] += l[d[j]] ); l[nd]++; ); res } \\ David A. Corneth, Sep 03 2021

a(prime(m)^(q-1)) = m for m >= 1 and any prime q.

cp's OEIS Frontend

A007416 The minimal numbers: sequence A005179 arranged in increasing order.

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A067004 Number of numbers <= n with same number of divisors as n.

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A335097 Number of integers less than n with the same number of prime factors (counted with multiplicity) as n.

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A334655 Number of integers less than n with the same number of distinct prime factors as n.

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A339514 Number of subsets of {1..n} whose elements have the same number of divisors.

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A337557 Number of integers less than n with the same number of odd divisors as n.

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A338483 a(n) is the smallest number having n smaller numbers with the same number of divisors.

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