A182318 -id:A182318 - cp's OEIS frontend

A338669 The prime tower factorization of a(n) is obtained by replacing the rightmost prime number by 1 in the prime tower factorization of n; a(1) = 1.

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 3, 4, 1, 6, 1, 2, 3, 2, 5, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 7, 10, 3, 4, 1, 6, 5, 8, 3, 2, 1, 12, 1, 2, 9, 4, 5, 6, 1, 4, 3, 10, 1, 24, 1, 2, 15, 4, 7, 6, 1, 16, 9, 2, 1, 12

Offset: 1

Rémy Sigrist, Apr 23 2021

nonn

The prime tower factorization of a number is defined in A182318.

Sequence A338668 gives the rightmost prime number.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of initial terms

Cf. A106490, A182318, A338668.

a(n) = { if (n==1, 1, my (f=factor(n), w=#f~, p=f[w,1], e=f[w,2]); if (e==1, n/p, n*p^(a(e)-e))) }

a(n) = 1 iff n = 1 or n is a prime number.
A106490(a(n)) = 1 + A106490(n) for any n > 1.
a^k(n) = 1 for k = A106490(n) (where a^k denotes the k-th iterate of a).

A182338 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 3.

Original entry on oeis.org

3, 6, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 40, 42, 45, 48, 51, 54, 56, 57, 60, 63, 64, 66, 69, 72, 75, 78, 81, 84, 87, 88, 90, 93, 96, 99, 102, 104, 105, 108, 111, 114, 117, 120, 123, 125, 126, 129, 132, 135, 136, 138, 141, 144, 147, 150, 152, 153

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

This set is the complement of A182337.
The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251v1 [math.NT], 2012-2014.

Complement of A182337. Cf. A182318.

# The integer n is in this sequence if and only if
# containsPrimeInTower(3, n) returns true
containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:
select(x-> containsPrimeInTower(3,x), [$1..160])[];

indic[1] = 1; indic[n_] := indic[n] = Switch[f = FactorInteger[n], {{3, }}, 0, {{, }}, indic[f[[1, 2]]], , Times @@ (indic /@ (Power @@@ f))];
Select[Range[200], indic[#] != 1&] (* Jean-François Alcover, Jul 11 2018 *)

A182339 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 2.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

This set is the complement of A182318.
The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

Complement of A182318.

# The integer n is in this sequence if and only if
# conatinsPrimeInTower(2, n) returns true
conatinsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:

Select[Range[120], MemberQ[Flatten@ FixedPoint[Map[If[PrimeQ@ Last@# || Last@# == 1, #, {First@#, FactorInteger@Last@#}]&, #, {Depth@# - 2}]&, FactorInteger@#], 2]&] (* Jean-François Alcover, Mar 27 2018, using Michael De Vlieger's program for A182318 *)

A182340 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 5.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 32, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 224, 225, 230, 235, 240, 243, 245, 250, 255, 260

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

Cf. A182318.

# The integer n is in this sequence if and only if
# conatinsPrimeInTower(5, n) returns true
conatinsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:

containsPrimeInTower[q_, n_] := containsPrimeInTower[q, n] = Module[{i, L, currentExponent}, If[n <= 1, Return[False]]; If[IntegerQ[n/q], Return[True]]; L = FactorInteger[n]; For[i = 1, i <= Length[L], i++, currentExponent = L[[i, 2]]; If[containsPrimeInTower[q, currentExponent], Return[True]]]; Return[False]];
Select[Range[300], containsPrimeInTower[5, #]&] (* Jean-François Alcover, Jan 22 2019, from Maple *)

A182341 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 7.

Original entry on oeis.org

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 128, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.
Observation: the union of 128 and the first 54 nonzero multiples of 7 (cf. A008589) gives the first 55 terms of this sequence. - Omar E. Pol, Feb 01 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

Cf. A008589, A182318.

# The integer n is in this sequence if and only if
# containsPrimeInTower(7, n) returns true
containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:

containsPrimeInTower[q_, n_] := containsPrimeInTower[q, n] = Module[{i, L, currentExponent}, If[n <= 1, Return[False]]; If[IntegerQ[n/q], Return[True]]; L = FactorInteger[n]; For[i = 1, i <= Length[L] , i++, currentExponent = L[[i, 2]]; If[containsPrimeInTower[q, currentExponent], Return[True]]]; Return[False]];
Select[Range[400], containsPrimeInTower[7, #]&] (* Jean-François Alcover, Jan 22 2019, from Maple *)

A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

Original entry on oeis.org

3, 5, 27, 7, 10, 11, 9, 25, 14, 13, 20, 17, 15, 21, 7625597484987, 19, 24, 23, 28, 22, 26, 29, 50, 32, 33, 3125, 44, 31, 42, 37, 49, 34, 35, 38, 100, 41, 39, 46, 45, 43, 66, 47, 52, 56, 51, 53, 80, 121, 98, 55, 54, 59, 68, 57, 63, 58, 62, 61, 84, 67, 65, 75

Offset: 2

Rémy Sigrist, Feb 07 2017

nonn

The prime tower factorization of a number is defined in A182318.
The prime tower factorization equivalence classes are described in A279686.
For any n>1, a(n)=least k>n such that A279690(n)=A279690(k).
This sequence is a permutation of the complement of A279686.
This sequence is to prime tower factorization what A081761 is to prime signature.

Rémy Sigrist, Table of n, a(n) for n = 2..2500
Rémy Sigrist, Illustration of the first terms

Cf. A000040, A006881, A007304, A046386, A046387, A051674, A067885, A081761, A182318, A279686, A279690

a(n) = my (c=a279690(n)); my (k=n+1); while (c!=a279690(k), k++); k

a(A000040(n)) = A000040(n+1) for any n>0.
a(A006881(n)) = A006881(n+1) for any n>0.
a(A051674(n)) = A051674(n+1) for any n>0.
a(A007304(n)) = A007304(n+1) for any n>0.
a(A046386(n)) = A046386(n+1) for any n>0.
a(A046387(n)) = A046387(n+1) for any n>0.
a(A067885(n)) = A067885(n+1) for any n>0.

A284889 Numbers n such that A279513(n) is a primorial number (A002110).

Original entry on oeis.org

1, 2, 6, 8, 9, 30, 40, 45, 75, 96, 210, 250, 280, 315, 486, 525, 672, 735, 1750, 1920, 2310, 3080, 3402, 3430, 3465, 5775, 6125, 7392, 8085, 8575, 10976, 11907, 12705, 15625, 16000, 19250, 21120, 21870, 30030, 31104, 32768, 37422, 37730, 40040, 45045, 54675

Offset: 1

Rémy Sigrist, Apr 05 2017

nonn

Also numbers with the k first prime numbers in their prime tower factorization, without duplicate, for some k (see A182318 for the definition of the prime tower factorization of a number).
This sequence contains the primorial numbers (A002110); 8 = 2^3 is the first term in this sequence that is not a primorial number.
This sequence contains A260548.
All terms belong to A284763.
If a(n) <= p# for some prime p, then a(n) is p-smooth (p# denotes the product of the primes <= p, see A002110).
There are A000272(k+1) terms with k prime numbers in their prime tower factorization:
- for k=0: 1,
- for k=1: 2,
- for k=2: 2*3, 2^3, 3^2,
- for k=3: 2*3*5, 2^3*5, 2^5*3, 3^2*5, 3^5*2, 5^2*3, 5^3*2, 2^(3*5), 3^(2*5), 5^(2*3), 2^3^5, 2^5^3, 3^2^5, 3^5^2, 5^2^3, 5^3^2.

			1626625 = 5^3*7*11*13^2 appears in this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..500
Rémy Sigrist, Illustration of the first terms

Cf. A000272, A002110, A182318, A260548, A279513, A284763.

isprimorial(n) = if (n==1, 1, my (f=factor(n)); (#f~ == primepi(vecmax(f[,1]))) && (vecmax(f[,2]) == 1));
a279513(n) =  my (f=factor(n)); prod(i=1, #f~, f[i, 1]*a279513(f[i, 2]));
isok(n) = isprimorial(a279513(n)); \\ Michel Marcus, Apr 08 2017

A286068 a(n) = least k such that the prime tower factorizations of k and k+1 both contain the n-th prime.

Original entry on oeis.org

8, 8, 95, 384, 10240, 57343, 1179647, 4718592, 92274688, 8053063679, 32212254720, 2611340115967, 46179488366591, 184717953466368, 3236962232172544, 243194379878006783, 16717361816799281152, 71481133285624512511, 4869940435459321626624, 82641413450218791239680

Offset: 1

Rémy Sigrist, Jun 13 2017

nonn

The prime tower factorization of a number is defined in A182318.
Two consecutive numbers cannot have a common prime factor; however, their prime tower factorizations can share a prime number.
For example, the prime tower factorizations of 8 and 9, that is, 2^3 and 3^2, share the prime numbers 2 and 3.
We can also find triples of consecutive numbers whose prime tower factorizations share a prime number:
- if n is an odd squarefree number > 1, then the prime tower factorizations of n^2-1, n^2 and n^2+1 share the prime number 2,
- the prime tower factorizations of 5344, 5345 and 5346 share the prime number 5.
Also, the prime tower factorizations of:
- 342, 343, 344 and 345 share the prime number 3,
- 99125, 99126, 99127, 99128 and 99129 share the prime number 3,
- 72470 ... 72480 share the prime number 2,
- 1674274 ... 1674288 share the prime number 2.
Are there tuples of more than 15 consecutive numbers with such a property?

			See illustration of first terms in Links section.

Rémy Sigrist, Illustration of the first terms

Cf. A182318.

a(n) = my (p=prime(n)); if (p==2, return (8), my (k = p\4); if (p % 4 == 1, return (2^p*(2*k+1)-1), return (2^p*(2*k+1))))

a(1) = 8.
If prime(n) = 4*k+1, then a(n) = 2^(4*k+1)*(2*k+1)-1.
If prime(n) = 4*k+3, then a(n) = 2^(4*k+3)*(2*k+1).
To prove the formula for n > 1:
- we use Fermat's little theorem: 2^p = 2 mod p,
- we check that there are no lower values near a multiple of 2^p,
- we check that the given value is less than 3^p - 1.

A288532 Literal reading of the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 23, 32, 25, 11, 223, 13, 27, 35, 222, 17, 232, 19, 225, 37, 211, 23, 233, 52, 213, 33, 227, 29, 235, 31, 25, 311, 217, 57, 2232, 37, 219, 313, 235, 41, 237, 43, 2211, 325, 223, 47, 2223, 72, 252, 317, 2213, 53, 233, 511, 237, 319, 229

Offset: 1

Rémy Sigrist, Jun 11 2017

The prime tower factorization of a number is defined in A182318.
The sequence is similar to A080670; however here we recursively factorize prime exponents.
a(1) = 1 by convention.
a(p) = p for any prime p.
As for A080670, 13532385396179 is a composite fixed point.

			See illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A080670, A182318.

Array[FromDigits@ Flatten@ Map[IntegerDigits, DeleteCases[#, 1] /. {} -> {1}] &@ Flatten@ FixedPoint[Map[If[PrimeQ@ Last@ # || Last@ # == 1, #, {First@ #, FactorInteger@ Last@ #}] &, #, {Depth@ # - 2}] &, FactorInteger@ #] &, 58] (* or *)
Table[FromDigits@ Flatten@ Map[IntegerDigits, DeleteCases[ Flatten[ FactorInteger[n] //. {p_, e_} /; e > 1 :> {p, FactorInteger@ e}], 1] /. {} -> {1}], {n, 58}] (* Michael De Vlieger, Jun 11 2017 *)

a(n) = my (s="", f=factor(n)); for (i=1, #f~, s=concat(s,Str(f[i,1])); if (f[i,2]>1, s=concat(s,Str(a(f[i,2]))))); return (if(s=="", 1, eval(s)))

A309002 Multiplicative with a(p) = p^2 and a(p^e) = p^a(e) for any e > 1 and prime number p.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 512, 81, 100, 121, 144, 169, 196, 225, 65536, 289, 324, 361, 400, 441, 484, 529, 4608, 625, 676, 19683, 784, 841, 900, 961, 33554432, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 12800, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 589824

Offset: 1

Rémy Sigrist, Jul 05 2019

To compute a(n): square every prime number at leaf position in the prime tower factorization of n (the prime tower factorization of a number is defined in A182318).

For any n > 0, a(n) is the least k such that A308993(k) = n.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of first terms

Cf. A004709, A182318, A185102, A308993.

a(n) = my (f=factor(n)); prod (i=1, #f~, f[i, 1]^if (f[i, 2]==1, 2, a(f[i, 2])))

A308993(a(n)) = n.
A185102(a(n)) = 1 + A185102(n) for any n > 1.
a(n) >= n^2 with equality iff n is cubefree (A004709).

cp's OEIS Frontend

A338669 The prime tower factorization of a(n) is obtained by replacing the rightmost prime number by 1 in the prime tower factorization of n; a(1) = 1.

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A182338 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 3.

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Maple

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A182339 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 2.

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Maple

Mathematica

A182340 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 5.

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Maple

Mathematica

A182341 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 7.

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Maple

Mathematica

A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

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PARI

Formula

A284889 Numbers n such that A279513(n) is a primorial number (A002110).

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PARI

A286068 a(n) = least k such that the prime tower factorizations of k and k+1 both contain the n-th prime.

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