A162247 -id:A162247 - cp's OEIS frontend

A246063 First occurrence of n in sequence A112329.

2, 1, 3, 9, 15, 64, 45, 256, 96, 144, 192, 4096, 240, 16384, 768, 576, 480, 262144, 720, 1048576, 960, 2304, 12288, 16777216, 1440, 5184, 49152, 3600, 3840, 1073741824, 2880, 4294967296, 3360, 36864, 786432, 20736, 5040, 274877906944, 3145728, 147456, 6720

Offset: 0

Ray Chandler, Aug 24 2014

nonn

Inspired by a comment from Robert G. Wilson v in sequence A112329.

Ray Chandler, Table of n, a(n) for n = 0..3322 [terms <= 1000 digits]

Cf. A094572, A112329.

g[lst_,p_]:=Module[{t,i,j},Union[Flatten[Table[t=lst[[i]];t[[j]]=p*t[[j]];Sort[t],{i,Length[lst]},{j,Length[lst[[i]]]}],1],Table[Sort[Append[lst[[i]],p]],{i,Length[lst]}]]];f[n_]:=Module[{i,j,p,e,lst={{}}},{p,e}=Transpose[FactorInteger[n]];Do[lst=g[lst,p[[i]]],{i,Length[p]},{j,e[[i]]}];lst];
(* above factor functions from T. D. Noe in A162247 *)
nmax=100;
a1={2,1,3};
Do[
least=Infinity;
fn=f[n];
Do[
exps=Reverse[fnitem]-1;
odd=even=1;
cnt=0;
Do[
cnt++;
odd*=(Prime[cnt+1]^exp);
even*=(Prime[cnt]^exp);
,{exp,exps}];
least=Min[least,odd,4even];
,{fnitem,fn}];
AppendTo[a1,least];
,{n,3,nmax}];
a1

d(n) = if (denominator(n)==1, numdiv(n), 0);
f(n) = numdiv(n) - 2*d(n/2) + 2*d(n/4);
a(n) = {my(k = 1); while (f(k) != n, k++); k;} \\ Michel Marcus, Jul 30 2017

a(p) = 2^(p+1) for prime p >= 5.

A304649 Number of divisors d|n such that neither d nor n/d is a perfect power greater than 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 0, 1, 4, 2, 4, 2, 4, 4, 0, 2, 4, 2, 4, 4, 4, 2, 4, 1, 4, 0, 4, 2, 8, 2, 0, 4, 4, 4, 5, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 1, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 10, 2, 4, 4, 0, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 4, 0, 4, 2, 10, 4, 4, 4, 4

Offset: 1

Gus Wiseman, May 15 2018

nonn

			The a(36) = 5 ways to write 36 as a product of two numbers that are not perfect powers greater than 1 are 2*18, 3*12, 6*6, 12*3, 18*2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001055, A007427, A007916, A034444, A045778, A162247, A183096, A281116, A301700, A303386, A303707, A304650.

nn=1000;
sradQ[n_]:=GCD@@FactorInteger[n][[All,2]]===1;
Table[Length@Select[Divisors[n],sradQ[n/#]&&sradQ[#]&],{n,nn}]

a(n) = sumdiv(n, d, !ispower(d) && !ispower(n/d)); \\ Michel Marcus, May 17 2018

A318954 Minimum shifted Heinz number of a strict factorization of n into factors > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 6, 13, 10, 19, 14, 29, 15, 37, 26, 21, 34, 53, 33, 61, 35, 39, 58, 79, 30, 89, 74, 57, 65, 107, 42, 113, 85, 87, 106, 91, 66, 151, 122, 111, 70, 173, 78, 181, 145, 129, 158, 199, 102, 223, 161, 159, 185, 239, 114, 203, 130, 183, 214, 271, 105

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1).

			The strict factorizations of 60 are (2*3*10), (2*5*6), (2*30), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60), with shifted Heinz numbers 138, 154, 218, 105, 201, 215, 217, 253, 277 respectively, so a(60) = 105.

Cf. A001055, A007716, A045778, A056239, A080688, A162247, A215366, A246868, A318871, A318953.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@(#-1)&/@Select[facs[n],UnsameQ@@#&]],{n,100}]

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 4, 1, 4, 2, 4, 3, 4, 1, 2, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 2, 5, 3, 5, 4, 5, 1, 2, 5, 1, 3, 5, 1, 4, 5, 2, 3, 5, 2, 4, 5, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A355032 a(n) is the maximum number of prime signatures of numbers with n divisors that have the same number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

			a(2) = 1 since the numbers with 2 divisors are all primes and thus have only 1 prime signature.
a(36) = 2 since numbers with 36 divisors have 2 prime signatures, p1^5 * p2^5 and p1 * p2 * p3^8, that correspond to numbers with 10 prime divisors (counted with multiplicity).
a(72) = 3 since numbers with 72 divisors have 3 prime signatures, p1 * p2^5 * p3^5, p1^2 * p2^2 * p3^7 and p1 * p2 * p3 * p4^8, that correspond to numbers with 11 prime divisors (counted with multiplicity).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A118914, A162247, A355028, A355030, A355031, A355033.

Table[Max[Tally[Total[#-1]& /@ f[n]][[;;,2]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(A355033(n)) = n.

A355033 a(n) is the least number k such that A355032(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 36, 72, 288, 960, 720, 1440, 3456, 2880, 6912, 5760, 10080, 11520, 8640, 24192, 21600, 47520, 17280, 28800, 20160, 62208, 46080, 82944, 34560, 50400, 40320, 57600, 51840, 110880, 126720, 141120, 69120, 60480, 248832, 86400, 80640, 233280, 237600, 103680, 100800

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A000005, A001222, A162247, A355030, A355031, A355032.

m[n_] := Max[Tally[Total[# - 1] & /@ f[n]][[;; , 2]]]; seq[len_, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, k}, While[c < len && n < max, k = m[n]; If[k <= len && s[[k]] == 0, c++; s[[k]] = n]; n++]; s]; seq[30, 10^6] (* using the function f by T. D. Noe at A162247 *)

A380760 Integers k with at least one proper factorization for which the sum of the same fixed integer power >= 2 of the factors equals k.

Original entry on oeis.org

16, 27, 48, 54, 256, 270, 528, 1134, 1755, 2916, 3125, 7216, 7830, 11520, 11934, 15360, 19683, 22464, 30000, 31752, 40095, 40960, 46656, 65536, 69168, 81702, 86436, 93555, 100368, 146880, 200000, 212400, 264654, 273600, 291060, 303030, 317520, 340470, 362880

Offset: 1

Charles L. Hohn, Feb 02 2025

nonn

Superset of A381538 for values >= 16, and it is conjectured that the terms that match multiple nonequivalent factorizations here, such as a(5) = 256 (see Example), are exactly the terms of A381538 that can be produced as m^(m^e) by multiple m.

			a(1) = 16: 2 * 2 * 2 * 2 = 2^2 + 2^2 + 2^2 + 2^2 = 16.
a(5) = 256: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 = 256, and also 4 * 4 * 4 * 4 = 4^3 + 4^3 + 4^3 + 4^3 = 256.
a(8) = 1134: 2 * 3 * 3 * 7 * 9 = 2^3 + 3^3 + 3^3 + 7^3 + 9^3 = 1134.

Sums of squares only: A380902.

Cf. A162247, A372053, A381538.

a380760_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, my(c=0); for(p=2, oo, my(t=sum(i=1, #f, f[i]^p)); if(tCharles L. Hohn, Mar 09 2025

Edited by Peter Munn, Mar 25 2025

A086556 Lexicographically earliest sequence of pairwise coprime numbers such that tau(a(n)) = n, where tau(k) = number of divisors of k.

Original entry on oeis.org

1, 2, 9, 35, 14641, 2873, 47045881, 20677, 2301289, 160683647, 174887470365513049, 14226847, 16409682740640811134241, 11955403876831, 375917360641, 107972737, 397030588105939686862303733490241, 26979607331, 289048159795659680188475502761294616529, 8398900121459

Offset: 1

Amarnath Murthy, Aug 29 2003

nonn

			a(5) = 11^4, tau(14641) = 5 and a(6) = 2873 = 13^2*17 and tau(2873) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..270

Cf. A000005, A162247.

s[n_, m_] := Module[{e = f[n] - 1}, Min[Times @@@ ((Prime[m + Range[Length[#]]]^#) & /@ Reverse /@ e)]]; a[1] = 1; a[n_] := a[n] = s[n, PrimePi[FactorInteger[a[n-1]][[-1, 1]]]]; Array[a, 20] (* Amiram Eldar, Jul 26 2025 using the function f by T. D. Noe at A162247 *)

More terms from David Wasserman, Mar 25 2005

Name clarified and a(19)-a(20) added by Amiram Eldar, Jul 26 2025

A174524 The total number of factors in unordered factorizations of A025487(n).

Original entry on oeis.org

0, 1, 3, 3, 6, 8, 12, 17, 10, 20, 22, 34, 27, 35, 46, 61, 60, 54, 94, 75, 107, 37, 101, 123, 86, 170, 170, 176, 104, 207, 230, 128, 304, 356, 284, 242, 386, 217, 413, 192, 397, 506, 303, 434, 680, 442, 512, 698, 502, 703, 275, 847, 832, 151, 725, 818, 1244, 676, 993, 1190, 1079, 1162, 399, 1654, 1311, 446, 1572, 1501, 1207, 2158, 1007, 917, 1840, 1831, 1980, 2104, 1785, 1859, 1579, 556

Offset: 1

Alford Arnold, Mar 21 2010

nonn

A025487(n) has a certain number of unordered factorizations with between 1 and A001222(A025487(n)) factors.

a(n) counts these factors that are > 1. A025487(5) counts 8, 2*4, 2*2*2 with 1+2+3 = 6 = a(5) factors, for example.

			The table of factorizations in A162247 begins:
  1
  2
  3
  4 2 2
  5
  6 2 3
  7
  8 2 4 2 2 2
  9 3 3
  10 2 5
  11
  12 2 6 3 4 2 2 3
A025487(n) begins 1 2 4 6 8 12 ...
a(n) counts the elements larger than 1 in the A025487(n)-th row of A162247: 0 1 3 3 6 8 ...

Amiram Eldar, Table of n, a(n) for n = 1..10001

Cf. A001222, A025487, A066637, A162247.

a(n) = A066637(A025487(n)). - R. J. Mathar, Oct 03 2010

a(1) corrected and data extended by R. J. Mathar, Oct 03 2010

A304650 Number of ways to write n as a product of two positive integers, neither of which is a perfect power.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 0, 2, 0, 6, 0, 0, 2, 2, 2, 5, 0, 2, 2, 2, 0, 6, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 8, 0, 2, 2, 0, 2, 6, 0, 2, 2, 6, 0, 4, 0, 2, 2, 2, 2, 6, 0, 2, 0, 2, 0, 8, 2, 2, 2, 2, 0, 8, 2, 2, 2, 2, 2, 2, 0, 2

Offset: 1

Gus Wiseman, May 15 2018

nonn

			The a(60) = 8 ways to write 60 as a product of two numbers, neither of which is a perfect power, are 2*30, 3*20, 5*12, 6*10, 10*6, 12*5, 20*3, 30*2.

Cf. A000005, A001055, A007427, A007916, A034444, A045778, A162247, A183096, A281116, A301700, A303386, A303707, A304649.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
Table[Length[Select[Divisors[n],radQ[#]&&radQ[n/#]&]],{n,100}]

ispow(n) = (n==1) || ispower(n);
a(n) = sumdiv(n, d, !ispow(d) && !ispow(n/d)); \\ Michel Marcus, May 17 2018

cp's OEIS Frontend

A246063 First occurrence of n in sequence A112329.

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A304649 Number of divisors d|n such that neither d nor n/d is a perfect power greater than 1.

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A318954 Minimum shifted Heinz number of a strict factorization of n into factors > 1.

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A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

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A355032 a(n) is the maximum number of prime signatures of numbers with n divisors that have the same number of prime divisors (counted with multiplicity).

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A355033 a(n) is the least number k such that A355032(k) = n, or -1 if no such k exists.

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A380760 Integers k with at least one proper factorization for which the sum of the same fixed integer power >= 2 of the factors equals k.

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PARI

Extensions

A086556 Lexicographically earliest sequence of pairwise coprime numbers such that tau(a(n)) = n, where tau(k) = number of divisors of k.

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A174524 The total number of factors in unordered factorizations of A025487(n).

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