A027746 -id:A027746 - cp's OEIS frontend

A037271 Number of steps to reach a prime under "replace n with concatenation of its prime factors" when applied to n-th composite number, or -1 if no such number exists.

2, 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, 1, 2, 3, 4, 4, 1, 2, 2, 1, 5, 3, 2, 2, 1, 9, 2, 9, 6, 1, 15

Offset: 1

a(33) is presently unknown: starting with 49, no prime has been reached after 110 steps. See A037274 for the latest information.

			Starting with 14 (the seventh composite number) we get 14=2*7, 27=3*3*3, 333=3*3*37, 3337=47*71, 4771=13*367, 13367 is prime; so a(7)=5.

Patrick De Geest, Home Primes
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
Eric Weisstein's World of Mathematics, Home Prime

Cf. A037272, A037273, A037274, A056938, A002808, A037276, A027746, A230305.

a037271 = length . takeWhile ((== 0) . a010051'') .
                             iterate a037276 . a002808
-- Reinhard Zumkeller, Apr 03 2012

maxComposite = 49; maxIter = 40; concat[n_] := FromDigits[ Flatten[ IntegerDigits /@ Flatten[ Apply[ Table, {#[[1]], {#[[2]]}} & /@ FactorInteger[n], {1}]]]]; composites = Select[ Range[2, maxComposite], ! PrimeQ[#] &]; a[n_] := ( lst = NestWhileList[ concat, composites[[n]], ! PrimeQ[#] &, 1, maxIter]; If[PrimeQ[ Last[lst]], Length[lst] - 1, - 1]); Table[a[n], {n, 1, Length[composites]}] (* Jean-François Alcover, Jul 10 2012 *)

A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 01 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.

The version for standard compositions is A209281(n+1) (even: A346633).
Subtracting the even version gives A316524 (reverse: A344616).
The even version is A346698.
The reverse version is A346699.
The even reverse version is A346700.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A000302 counts compositions with odd alternating sum, ranked by A053738.
A001414 adds up prime factors, row sums of A027746.
A029837 adds up parts of standard compositions (alternating: A124754).
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Partition[Append[primeMS[n],0],2]],{n,100}]

a(n) = A056239(n) - A346698(n).
a(n) = A316524(n) + A346698(n).
a(n odd omega) = A346699(n).
a(n even omega) = A346700(n).
A344616(n) = A346699(n) - A346700(n).

A370582 Number of subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 20, 40, 52, 72, 116, 232, 320, 640, 1020, 1528, 1792, 3584, 4552, 9104, 12240, 17840, 27896, 55792, 67584, 83968, 130656, 150240, 198528, 397056, 507984, 1015968, 1115616, 1579168, 2438544, 3259680, 3730368, 7460736, 11494656, 16145952, 19078464, 38156928

Offset: 0

Gus Wiseman, Feb 25 2024

nonn

			The a(0) = 1 through a(6) = 20 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {6}
                             {2,5}    {2,3}
                             {3,4}    {2,5}
                             {3,5}    {2,6}
                             {4,5}    {3,4}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

The version for set-systems is A367902, ranks A367906, unlabeled A368095.
The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368098, complement A368097.
Multisets of this type are ranked by A368100, complement A355529.
For divisors instead of factors we have A368110, complement A355740.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370583.
For a unique choice we have A370584.
The maximal case is A370585.
Partial sums of A370586, complement A370587.
The version for partitions is A370592, complement A370593.
For binary indices instead of factors we have A370636, complement A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A001055, A001414, A003963, A005117, A045778, A133686, A355739, A355744, A355745, A367905.

Table[Length[Select[Subsets[Range[n]],Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]>0&]],{n,0,10}]

a(p) = 2 * a(p-1) for prime p. - David A. Corneth, Feb 25 2024

a(n) = 2^n - A370583(n).

a(19) from David A. Corneth, Feb 25 2024

a(20)-a(41) from Alois P. Heinz, Feb 25 2024

A099542 Rhonda numbers to base 10.

Original entry on oeis.org

1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284, 25662, 33475, 34935, 35581, 45951, 47265, 47594, 52374, 53176, 53742, 54479, 55272, 56356, 56718, 95232, 118465, 133857, 148653, 154462, 161785

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 21 2004

An integer m is a Rhonda number to base b if the product of its digits in base b equals b*(sum of prime factors of m (taken with multiplicity)).
Does every Rhonda number to base 10 contain at least one 5? - Howard Berman (howard_berman(AT)hotmail.com), Oct 22 2008
Yes, every Rhonda number m to base 10 contains at least one 5 and also one even digit, otherwise A007954(m) mod 10 > 0. - Reinhard Zumkeller, Dec 01 2012

			1568 has prime factorization 2^5 * 7^2. Sum of prime factors = 2*5 + 7*2 = 24. Product of digits of 1568 = 1*5*6*8 = 240 = 10*24, hence 1568 is a Rhonda number to base 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (first 180 terms from Harvey P. Dale)
Kevin Brown, Infinitely Many Rhondas.
Giovanni Resta, Rhonda numbers the 64507 base-10 Rhonda numbers up to 10^12.
Walter Schneider, Rhonda Numbers.
Eric Weisstein's World of Mathematics, Rhonda Number.

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), A100970 (base 8), A100973 (base 9), A100971 (base 12), A100972 (base 14), A100974 (base 15), A100975 (base 16), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60), see also A255880.
Cf. A001414, A027746, A007954.
Column k=5 of A291925.

import Data.List (unfoldr); import Data.Tuple (swap)
a099542 n = a099542_list !! (n-1)
a099542_list = filter (rhonda 10) [1..]
rhonda b x = a001414 x * b == product (unfoldr
       (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b) x)
-- Reinhard Zumkeller, Mar 05 2015, Dec 01 2012

Select[Range[200000],10Total[Times@@@FactorInteger[#]]==Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Oct 16 2011 *)

A007954(a(n)) = 10 * A001414(a(n)).

A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 9, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 15, 31, 8, 11, 17, 7, 9, 37, 19, 13, 10, 41, 21, 43, 22, 15, 23, 47, 12, 7, 25, 17, 26, 53, 9, 11, 14, 19, 29, 59, 15, 61, 31, 21, 8, 13, 33, 67, 34, 23, 35, 71

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

Appears to contain each positive integer at least once, but only a finite number of times.

			The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 7716.

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

The greatest divisor without the condition is A006530 (smallest: A020639).
Divisors of this type are counted by A096825 (exact: A345957).
The case of powers of 2 is A163403.
The smallest divisor of this type is given by A347043 (exact: A347045).
The exact version is A347046.
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A038548 counts inferior (or superior) divisors (strict: A056924).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A060775, A106529, A217581, A244990, A335433, A335448.

Table[Max[Select[Divisors[n],PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; Times @@ p[[Floor[np/2] + 1;; np]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[::-1]:
        if len(factorint(d, multiple=True)) == (npf+1)//2: return d
    return 1
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347044(n):
    fs = factorint(n,multiple=True)
    l = len(fs)
    return prod(fs[l//2:]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=floor(A001222(n)/2)+1..A001222(n)} A027746(n,k). - Amiram Eldar, Nov 02 2024

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 6, 9, 10, 22, 25, 28, 30, 34, 42, 45, 62, 63, 66, 75, 82, 92, 98, 99, 102, 104, 110, 118, 121, 134, 140, 147, 152, 153, 156, 166, 170, 186, 210, 218, 228, 230, 232, 234, 246, 254, 260, 275, 276, 279, 289, 308, 310, 314, 315, 330, 342, 343, 344, 348, 350

Offset: 1

Gus Wiseman, Mar 14 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

All squarefree terms are even.

			The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   42: {1,2,4}
   45: {2,2,3}
   62: {1,11}
   63: {2,2,4}
   66: {1,2,5}
   75: {2,3,3}
   82: {1,13}
   92: {1,1,9}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  104: {1,1,1,6}

For factors instead of divisors on the RHS we have A319899.
A version for binary indices is A367917.
For (greater than) instead of (equal) we have A370348, counted by A371171.
The RHS is A370820, for prime factors instead of divisors A303975.
Partitions of this type are counted by A371130, strict A371128.
For divisors instead of factors on LHS we have A371165, counted by A371172.
For only distinct prime factors on LHS we have A371177, counted by A371178.
Other inequalities: A371166, A371167, A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.

Select[Range[100],PrimeOmega[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001222(a(n)) = A370820(a(n)).

A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 4, 3, 2, 0, 2, 0, 1, 4, 5, 0, 3, 3, 6, 2, 1, 0, 2, 0, 2, 5, 7, 4, 3, 0, 8, 6, 4, 0, 2, 0, 1, 2, 9, 0, 2, 4, 3, 7, 1, 0, 4, 5, 5, 8, 10, 0, 4, 0, 11, 2, 3, 6, 2, 0, 1, 9, 3, 0, 3, 0, 12, 3, 1, 5, 2, 0, 2, 4, 13, 0, 5, 7, 14, 10, 6, 0, 5, 6, 1, 11, 15, 8, 4, 0, 4, 2, 4, 0, 2, 0, 7, 3

Offset: 1

Gus Wiseman, Aug 01 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 = 4.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 3 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Subtracting from the odd version gives A316524 (reverse: A344616).
The version for standard compositions is A346633 (odd: A209281(n+1)).
The odd version is A346697.
The even reverse version is A346699.
The reverse version is A346700.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A001414 adds up prime factors, row-sums of A027746.
A029837 adds up parts of standard compositions (alternating: A124754).
A056239 adds up prime indices, row-sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A000097, A025047, A035363, A120452, A344617, A344653, A344654, A345957, A345958.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Last/@Partition[Append[primeMS[n],0],2]],{n,100}]

A346698(n) = if(1==n,0,my(f=factor(n),s=0,p=0); for(k=1,#f~,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ Antti Karttunen, Nov 30 2021

a(n) = A056239(n) - A346697(n).
a(n) = A346697(n) - A316524(n).
a(n even omega) = A346699(n).
a(n odd omega) = A346700(n).
A344616(n) = A346699(n) - A346700(n).

Data section extended up to 105 terms by Antti Karttunen, Nov 30 2021

A347043 Smallest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 4, 13, 2, 3, 4, 17, 6, 19, 4, 3, 2, 23, 4, 5, 2, 9, 4, 29, 6, 31, 8, 3, 2, 5, 4, 37, 2, 3, 4, 41, 6, 43, 4, 9, 2, 47, 8, 7, 10, 3, 4, 53, 6, 5, 4, 3, 2, 59, 4, 61, 2, 9, 8, 5, 6, 67, 4, 3, 10, 71, 8, 73, 2, 15, 4, 7, 6, 79, 8

Offset: 1

Gus Wiseman, Aug 15 2021

nonn

Appears to contain every positive integer at least once.

This is correct. For any integer m, let p be any prime > m. Then a(m*p^A001222(m)) = m. - Sebastian Karlsson, Oct 11 2022

			The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 16.

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

Positions of 2's are A001747.
Positions of odd terms are A005408.
Positions of even terms are A005843.
The case of powers of 2 is A016116.
The smallest divisor without the condition is A020639 (greatest: A006530).
These divisors are counted by A096825 (exact: A345957).
The greatest of these divisors is A347044 (exact: A347046).
The exact version is A347045.
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A027746, A038548, A106529, A140271, A244991, A324522, A335433, A335448.

Table[Min[Select[Divisors[n],PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]}, Times @@ p[[1 ;; Ceiling[Length[p]/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

a(n) = my(bn=ceil(bigomega(n)/2)); fordiv(n, d, if (bigomega(d)==bn, return (d))); \\ Michel Marcus, Aug 18 2021

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n):
        if len(factorint(d, multiple=True)) == (npf+1)//2: return d
    return 1
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347043(n):
    fs = factorint(n,multiple=True)
    l = len(fs)
    return prod(fs[:(l+1)//2]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=1..ceiling(A001222(n)/2)} A027746(n,k). - Amiram Eldar, Nov 02 2024

A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 18, 36, 48, 68, 104, 208, 284, 568, 888, 1296, 1548, 3096, 3968, 7936, 10736, 15440, 24008, 48016, 58848, 73680, 114368, 132608, 176240, 352480, 449824, 899648, 994976, 1399968, 2160720, 2859584, 3296048, 6592096, 10156672, 14214576, 16892352

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {2,3}
                             {2,5}    {2,5}
                             {3,4}    {2,6}
                             {3,5}    {3,4}
                             {4,5}    {3,5}
                             {2,3,5}  {3,6}
                             {3,4,5}  {4,5}
                                      {4,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

For divisors instead of factors we have A051026, cf. A368110, A355740.
The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
For existence we have A370582, differences A370586.
For nonexistence we have A370583, differences A370587.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, cf. A370592, A370593.
For binary indices instead of factors we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.

Table[Length[Select[Subsets[Range[n]], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 28 2025

A037019 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 2^(p_1 - 1)3^(p_2 - 1)...A000040(k)^(p_k - 1).

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 64, 30, 36, 48, 1024, 60, 4096, 192, 144, 210, 65536, 180, 262144, 240, 576, 3072, 4194304, 420, 1296, 12288, 900, 960, 268435456, 720, 1073741824, 2310, 9216, 196608, 5184, 1260, 68719476736, 786432, 36864, 1680, 1099511627776

Offset: 1

Wouter Meeussen

This is an easy way to produce a number with exactly n divisors and it usually produces the smallest such number (A005179(n)). The references call n "ordinary" if A005179(n) = a(n) and "exceptional" or "extraordinary" otherwise. - David Wasserman, Jun 12 2002

			12 = 3*2*2, so a(12) = 2^2*3*5 = 60.

T. D. Noe, Table of n, a(n) for n = 1..1000
R. Brown, The minimal number with a given number of divisors, Journal of Number Theory 116 (2006) 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
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.

Cf. A005179, A000040, A072066 (exceptional (or extraordinary) numbers).

Cf. A027746.

a037019 = product .
   zipWith (^) a000040_list . reverse . map (subtract 1) . a027746_row
-- Reinhard Zumkeller, Nov 25 2012

a:= n-> (l-> mul(ithprime(i)^(l[i]-1), i=1..nops(l)))(
        sort(map(i-> i[1]$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 28 2019

(Times@@(Prime[ Range[ Length[ # ] ] ]^Reverse[ #-1 ]))&@Flatten[ FactorInteger[ n ]/.{ a_Integer, b_}:>Table[ a, {b} ] ]

A037019(n,p=1)=prod(i=1,#f=Vecrev(factor(n)~),prod(j=1,f[i][2],(p=nextprime(p+1))^(f[i][1]-1))) \\ M. F. Hasler, Oct 14 2014

from math import prod
from sympy import factorint, prime
def a(n):
    pf = factorint(n, multiple=True)
    return prod(prime(i)**(pi-1) for i, pi in enumerate(pf[::-1], 1))
print([a(n) for n in range(1, 42)]) # Michael S. Branicky, Jul 24 2022

More terms from David Wasserman, Jun 12 2002

cp's OEIS Frontend

A037271 Number of steps to reach a prime under "replace n with concatenation of its prime factors" when applied to n-th composite number, or -1 if no such number exists.

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A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

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A370582 Number of subsets of {1..n} such that it is possible to choose a different prime factor of each element.

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A099542 Rhonda numbers to base 10.

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A347044 Greatest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.

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A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.

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A037019 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 2^(p_1 - 1)3^(p_2 - 1)...A000040(k)^(p_k - 1).