A181800 -id:A181800 - cp's OEIS frontend

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

A085629 Let b(n) equal the product of the exponents in the prime factorization of n. Then a(n) gives the least k such that b(k) = n.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 2048, 432, 8192, 1152, 864, 1296, 131072, 1728, 524288, 2592, 3456, 18432, 8388608, 5184, 7776, 73728, 13824, 10368, 536870912, 15552, 2147483648, 20736, 55296, 1179648, 31104, 41472, 137438953472, 4718592

Offset: 1

Jason Earls, Jul 10 2003

nonn

a(n) <= 2^n. - Robert G. Wilson v, Jul 14 2014

a(n) = 2^n iff n is a prime or n equals 4 or 6. - Robert G. Wilson v, Jul 19 2014

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A005361, A005934.
Cf. A005179.
Subsequence of A181800.

f[n_, i_] := f[n, i] = Block[{d, b, p, x}, p = Prime[i]; b = p^n; d = Divisors[n]; For[j = Length[d], j > 1, j--, x = d[[j]]; b = Min[b, p^x*f[n/x, i + 1]]]; b]; f[1, 1] = 1; Array[ f[#, 1] &, 42] (* Robert G. Wilson v, Jul 17 2014, after David Wasserman's PARI program below *)

f(n, i) = local(d, best, p, x); p = prime(i); best = p^n; d = divisors(n); for (j = 2, length(d) - 1, x = d[j]; best = min(best, p^x*f(n/x, i + 1))); best; a(n) = f(n, 1) \\ David Wasserman, Feb 07 2005

More terms from David Wasserman, Feb 07 2005

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

Offset: 1

Gus Wiseman, May 17 2022

nonn

All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.

			The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Highly powerful number.

These are the positions of first appearances in A005361, counted by A266477.
This is the sorted version of A085629.
The version for shadows instead of exponents is A353397, firsts in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime exponents, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Subsequence of A181800.

nn=1000;
d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)

A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 6, 5, 7, 7, 6, 8, 9, 9, 9, 11, 12, 4, 10, 13, 10, 15, 7, 11, 15, 14, 18, 10, 12, 17, 18, 9, 21, 13, 15, 13, 19, 22, 14, 24, 16, 20, 14, 21, 26, 19, 10, 27, 19, 25, 16, 15, 23, 30, 24, 5, 21, 16, 30, 22, 30, 23, 16, 25, 34, 29, 9, 27, 22, 33

Offset: 1

Matthew Vandermast, Jun 07 2012

nonn

Also, number of divisors of A181800(n) that are members of A181800.

Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures (cf. A212642). Let m be any integer with second signature {S}. Then A212180(m) = k and A085082(m) is congruent to j modulo k. If {S} is the second signature of A181800(n), then A085082(m) is congruent to A212643(n) modulo a(n).

			The divisors of 72 represent 5 distinct second signatures (cf. A212172), as can be seen from the exponents >=2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Since 72 = A181800(8), a(8) = 5.

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

Cf. A181800, A085082, A212172, A212176, A212642, A212643.

a(n) = A212180(A181800(n)).

Data corrected by Amiram Eldar, Jul 14 2019

A212173 First integer with same second signature as n (cf. A212172).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 4, 1, 1, 4, 1, 1, 1, 16, 1, 4, 1, 4, 1, 1, 1, 8, 4, 1, 8, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 4, 1, 1, 16, 4, 4, 1, 4, 1, 8, 1, 8, 1, 1, 1, 4, 1, 1, 4, 64, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 4, 4, 1, 1, 1, 16, 16, 1, 1, 4

Offset: 1

Matthew Vandermast, Jun 03 2012

Two integers have the same second signature iff the same exponents >= 2 occur in the canonical prime factorization of each integer, regardless of the order in which they occur in each factorization.

			12 = 2^2*3 has 1 exponent >= 2 in its prime factorization, namely, 2. Hence, its second signature is {2}.  The smallest number with second signature {2} is 4; hence, a(12) = 4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Jason Kimberley, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

Cf. A212172, A046523. All terms belong to A181800.

A212173 := func; [A212173(n):n in[1..85]]; // Jason Kimberley, Jun 14 2012

f:= proc(n) local E,i;
E:= sort(select(`>`, map(t -> t[2], ifactors(n)[2]), 1),`>`);
mul(ithprime(i)^E[i],i=1..nops(E))
end proc:
map(f, [$1..100]); # Robert Israel, Jul 19 2017

Function[s, Sort[Apply[Join, Map[Function[k, Map[{#, First@ k} &, k]], Values@ s]]][[All, -1]]]@ KeySort@ PositionIndex@ Table[Sort@ DeleteCases[FactorInteger[n][[All, -1]], e_ /; e < 2] /. {} -> {1}, {n, 84}] (* Michael De Vlieger, Jul 19 2017 *)

a(n) = {my(sn = vecsort(select(x->(x>=2), factor(n)[,2]))); for (i=1, n, if (vecsort(select(x->(x>=2),factor(i)[,2])) == sn, return(i)););} \\ Michel Marcus, Jul 19 2017

from functools import reduce
from sympy import factorint
from operator import mul
def P(n): return sorted(factorint(n).values())
def a046523(n):
    x=1
    while True:
        if P(n)==P(x): return x
        else: x+=1
def a057521(n): return 1 if n==1 else reduce(mul, [1 if e==1 else p**e for p, e in factorint(n).items()])
def a(n): return a046523(a057521(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 19 2017

a(n) = A046523(A057521(n)) = A057521(A046523(n)).

A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

Original entry on oeis.org

1, 16, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(47) = 2^35 * 3^10.
Indices of records in A188585.
All the terms are powerful numbers since A188585(1) = 1 and A188585(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..713
Amiram Eldar, Table of n, a(n), A188585(a(n)) for n = 1..713
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A025475, A046055, A188585, A384783, A384786 (cubefull analog).

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

Original entry on oeis.org

1, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(85) = 2^61 * 3^18.
Indices of records of the multiplicative function f(n) with f(p^e) = A008483(e).
All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
The corresponding record values are 1, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..567
Amiram Eldar, Table of n, a(n), and record values for n = 1..567
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A008483, A046055, A246549, A384784 (powerful analog), A384785.

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A384791 Numbers with a record number of ways in which they can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

Original entry on oeis.org

1, 256, 4096, 32768, 262144, 1048576, 8388608, 16777216, 134217728, 268435456, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 110075314176, 549755813888, 557256278016, 1761205026816, 4458050224128, 7044820107264, 8916100448256, 56358560858112, 71328803586048

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(16) = 2^24 * 3^8.	
Indices of records of the multiplicative function f(n) with f(p^e) = A008680(e).
All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, ... (see the link for more values).
Every exponent must be the index of the first occurrence of A008680(e) in A008680. So possible exponents of prime factors of terms are 0, 8, 12, 15, 18, 20, ... - David A. Corneth, Jun 30 2025

			256 in the sequence as 256 = 1^3 * 4^4 * 1^5 = 2^3 * 1^4 * 2^5 so there are two ways to write 256 as b^3 * c^4 * d^5, with b, c and d >= 1 and no smaller positive integer can be written in at least two ways like that. - _David A. Corneth_, Jun 30 2025

David A. Corneth, Table of n, a(n) for n = 1..810 (first 216 terms from Amiram Eldar, terms <= 10^120)
David A. Corneth, Table of n, a(n), exponents of prime factors and record values for n = 1..810
Index entries for sequences related to powerful numbers.

Subsequence of A025487, A036966 and A181800.

Cf. A008680, A046055, A384789, A384790 (powerful analog).

f[p_, e_] := Floor[(1+(-1)^e)*(-1)^Floor[e/2]/8 + (e^2 + 12*e + 90)/120]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A212641 Largest odd divisor of A212640(n) (number of divisors of n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

1, 3, 1, 5, 3, 9, 7, 3, 1, 15, 1, 9, 9, 5, 5, 21, 3, 27, 11, 3, 25, 7, 9, 3, 27, 15, 1, 45, 13, 15, 35, 3, 9, 27, 9, 7, 33, 5, 15, 5, 63, 21, 15, 9, 45, 9, 1, 11, 9, 3, 75, 1, 39, 25, 21, 81, 49, 5, 3, 81, 27, 45, 17, 21, 55, 3, 27, 7, 3, 13, 45, 15, 105, 9, 45

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

The odd part of d(n), or largest odd divisor of d(n) (A212181(n)), is a function of the second signature of n (cf. A212172).  Since A181800 gives the first integer of each second signature, this sequence gives the value of A212181 for each second signature in order of its first appearance.
Note: The odd part of d(n) is not the same as the number of odd divisors of n (A001227(n)).
Each odd integer appears an infinite number of times.

			A181800(5) = 32 has 6 divisors (1, 2, 4, 8, 16 and 32).  The largest odd divisor of 6 is 3. Hence, a(5) = 3.

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

Cf. A000005, A000265, A181800, A212172, A212181, A212640.

a(n) = A000265(A212640(n)) = A212181(A181800(n)).

A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

Original entry on oeis.org

1, 64, 4096, 46656, 2985984, 191102976, 2176782336, 12230590464, 46656000000, 2985984000000, 34012224000000, 191102976000000, 2176782336000000, 139314069504000000, 351298031616000000, 4001504141376000000, 22483074023424000000, 256096265048064000000, 16390160963076096000000

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

Indices of records of A370256.
All the terms are powerful numbers since A370256(1) = 1 and A370256(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..61.
Amiram Eldar, n, a(n), A370256(a(n)) for n = 1..61.
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A002182 (sixth root), A046055, A057523, A370256, A384791 (cubefull analog).

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

cp's OEIS Frontend

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A085629 Let b(n) equal the product of the exponents in the prime factorization of n. Then a(n) gives the least k such that b(k) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A212173 First integer with same second signature as n (cf. A212172).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A384791 Numbers with a record number of ways in which they can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

Views