A085629 -id:A085629 - cp's OEIS frontend

A340233 a(n) is the least number with exactly n exponential divisors.

1, 4, 16, 36, 65536, 144, 18446744073709551616, 576, 1296, 589824

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

a(11) = 2^(2^10) has 309 digits and is too large to be included in the data section.

See the link for more values of this sequence.

			a(2) = 4 since 4 is the least number with 2 exponential divisors, 2 and 4.

Amiram Eldar, Table of n, a(n) for n = 1..12
Amiram Eldar, Table of n, a(n) for n = 1..100 (given by prime factorizations)

Subsequence of A025487.
Cf. A049419, A318278, A322791.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340232 (bi-unitary).

f[p_, e_] := DivisorSigma[0, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 6; s = Table[0, {max}]; c = 0; n = 1;  While[c < max, i = d[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* ineffective for n > 6 *)

A049419(a(n)) = n and A049419(k) != n for all k < a(n).

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 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, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Dec 26 2023

The odd terms of A005361.

The first position of 2*k-1, for k = 1, 2, ..., is 1, 7, 24, 91, 154, 1444, 5777, 610, 92349, ..., which is the position of A085629(2*k-1) in A268335.

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

Cf. A005361, A013661, A065463, A085629, A098198, A268335.
Cf. A366438, A366439.
Similar sequences: A322327, A368473, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[OddQ[p], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p%2, print1(p, ", ")));}

a(n) = A005361(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/d) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^5) = 1.38446562720473484463..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A036965 Record values of the product of the exponents in the prime factorization of highly powerful numbers (A005934).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, 72, 75, 84, 90, 96, 105, 108, 112, 120, 126, 128, 135, 140, 144, 150, 160, 162, 168, 180, 192, 200, 210, 216, 220, 224, 225, 240, 252, 256, 270

Offset: 1

N. J. A. Sloane

Amiram Eldar, Table of n, a(n) for n = 1..609 (terms 1..300 from T. D. Noe)
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A005361, A005934, A085629.

Union@ FoldList[Max, Array[Times @@ FactorInteger[#][[All, -1]] &, 10^7]] (* Michael De Vlieger, Oct 15 2017 *)

a(n) = A005361(A005934(n)). - Amiram Eldar, May 13 2019

Reference gives an extensive table.

More terms from Naohiro Nomoto, Jul 25 2001

A358252 a(n) is the least number with exactly n non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 13824, 10368, 20736, 28800, 41472, 64800, 279936, 115200, 331776, 345600, 663552, 259200, 1679616, 518400, 1620000, 1166400, 4860000, 1036800, 17915904, 2073600, 15552000, 6998400, 26873856, 4147200, 53747712, 8294400

Offset: 0

Amiram Eldar, Nov 05 2022

nonn

a(n) is the least number k such that A056626(k) = n.

Since A056626(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

			a(1) = 8 since 8 is the least number that has exactly one non-unitary square divisor, 4.

Amiram Eldar, Table of n, a(n) for n = 0..176

Cf. A056626, A025487, A358253.

Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[21, 10^6]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + floor(f[i,2]/2)) - 2^sum(i = 1, #f~, 1 - f[i,2]%2);}
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A368474 Product of exponents of prime factorization of the numbers whose exponents in their prime power factorization are squares (A197680).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 26 2023

All the terms are squares (A000290).

The first position of k^2, for k = 1, 2, ..., is 1, 12, 331, 834, 21512290, 26588, ..., which is the position of A085629(k^2) in A197680.

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

Cf. A000290, A005361, A085629, A197680, A357016.

Similar sequences: A322327, A368472, A368473.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, IntegerQ[Sqrt[#]] &], Times @@ e, Nothing]]; Array[f, 150]

lista(kmax) = {my(e, ok); for(k = 1, kmax, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(!issquare(e[i]), ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}

a(n) = A005361(A197680(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=1} k^2/p^(k^2)) = 1.16776748073813763932..., where d = A357016 is the asymptotic density of A197680.

A353745 Number of runs in the ordered prime signature of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Gus Wiseman, May 20 2022

nonn

First differs from A071625 at a(90) = 3.
First differs from A331592 at a(90) = 3.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The prime indices of 630 are {1,2,2,3,4}, with multiplicities {1,2,1,1}, with runs {{1},{2},{1,1}}, so a(630) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A354233.
A001222 counts prime factors, distinct A001221.
A005361 gives product of prime signature, firsts A353500/A085629.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A182850/A323014 give frequency depth, counted by A225485/A325280.
Cf. A005811, A097318, A130091, A304678, A333755, A353503, A353507, A353742.
Cf. also A329747.

Table[Length[Split[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i < #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
A353745(n) = #runlengths(runlengths(pis_to_runs(n))); \\ Antti Karttunen, Jan 20 2025

A358262 a(n) is the least number with exactly n noninfinitary square divisors.

Original entry on oeis.org

1, 16, 144, 256, 3600, 1296, 2304, 65536, 129600, 16777216, 32400, 20736, 57600, 331776, 589824, 4294967296, 6350400, 1099511627776, 150994944, 810000, 1587600, 1679616, 518400, 5308416, 2822400, 84934656, 8294400, 26873856, 14745600, 21743271936, 38654705664

Offset: 0

Amiram Eldar, Nov 06 2022

nonn

a(n) is the least number k such that A358261(k) = n.

Since A358261(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

			a(1) = 16 since 16 is the least number with exactly one noninfinitary divisor, 4.

Amiram Eldar, Table of n, a(n) for n = 0..50

Cf. A025487, A358261, A358263.

Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square), A358252 (non-unitary square).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[15, 2*10^7]

s(n) = {my(f = factor(n));  prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 9, 11, 12, 14, 14, 18, 21, 23, 26, 29, 29, 33, 36, 39, 40, 43, 44, 50, 53, 55, 59, 65, 69, 72, 78, 79, 81, 85, 92, 95, 97, 100, 103, 108, 109, 112, 118, 124, 129, 137, 139, 142, 149, 155, 159, 165, 166, 173, 178, 181, 187

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(2) = 1 through a(9) = 6 partitions:
  11   111   1111   2111    21111    22111     221111     222111
                    11111   111111   31111     311111     411111
                                     211111    2111111    2211111
                                     1111111   11111111   3111111
                                                          21111111
                                                          111111111

LHS (product of parts) is counted by A339095, ranked by A003963.
RHS (product of multiplicities) is counted by A266477, ranked by A005361.
The version for greater instead of less is A353505.
The version for equal instead of less is A353506, ranked by A353503.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same product of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A085629, A097318, A098859, A114640, A116608, A118914, A124010, A304678, A353394, A353500, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#

A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 11, 17, 24, 35, 47, 66, 89, 121, 162, 214, 276, 362, 464, 599, 763, 971, 1219, 1537, 1918, 2393, 2966, 3668, 4512, 5549, 6784, 8287, 10076, 12238, 14807, 17898, 21556, 25931, 31094, 37243, 44486, 53075, 63158, 75069, 89025, 105447, 124636

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(0) = 0 through a(7) = 11 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)    (7)
             (21)  (22)  (32)   (33)   (43)
                   (31)  (41)   (42)   (52)
                         (221)  (51)   (61)
                         (311)  (222)  (322)
                                (321)  (331)
                                (411)  (421)
                                       (511)
                                       (2221)
                                       (3211)
                                       (4111)

RHS (product of multiplicities) is counted by A266477, ranked by A005361.
LHS (product of parts) is counted by A339095, ranked by A003963.
The version for less instead of greater is A353504.
The version for equality is A353506, ranked by A353503.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same products of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A008619, A085629, A097318, A098859, A114640, A116608, A130091, A304678, A353394, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#>Times@@Length/@Split[#]&]],{n,0,30}]

A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 17 2025

First differs from A386259 at n = 36.
First differs from A370078 at n = 64.
The first occurrence of k = 0, 1, 2, ... is at n = A085629(2^k) = 1, 4, 16, 144, 1296, 20736, 518400, ... .
The asymptotic density of the occurrences of 1 in this sequence is the asymptotic density of numbers whose prime factorization has only odd exponents except for one exponent that is of the form 4*k+2 (k >= 0) which equals A065463 * Sum_{p prime} p^2/(p^4+p^3+p-1) = 0.22670657681840536721... .

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

Cf. A005361, A007814, A065463, A085629, A268335, A295664, A370078, A386259.

a[n_] := IntegerExponent[Times @@ FactorInteger[n][[;; , 2]], 2]; Array[a, 100]

a(n) = vecsum(apply(x -> valuation(x, 2), factor(n)[, 2]));

a(n) = A007814(A005361(n)).
Additive with a(p^e) = A007814(e).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.37572872586497617473..., where f(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)).

cp's OEIS Frontend

A340233 a(n) is the least number with exactly n exponential divisors.

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A368472 Product of exponents of prime factorization of the exponentially odd numbers.

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A036965 Record values of the product of the exponents in the prime factorization of highly powerful numbers (A005934).

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A358252 a(n) is the least number with exactly n non-unitary square divisors.

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A368474 Product of exponents of prime factorization of the numbers whose exponents in their prime power factorization are squares (A197680).

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A353745 Number of runs in the ordered prime signature of n.

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A358262 a(n) is the least number with exactly n noninfinitary square divisors.

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A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

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