Matthew Vandermast - cp's OEIS frontend

A253141 If n is a prime power, then a(n) = lambda(tau(n)) = A014963(A000005(n)); otherwise, a(n) = 1.

1, 2, 2, 3, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 5, 1, 2, 1, 1, 1, 1

Offset: 1

Matthew Vandermast, Dec 27 2014

For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence. Since A253139(n) = Product_{d|n} a(d), A253139 is a divisibility sequence.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			2 is a prime number, i.e., a prime power with 2 divisors; a(2) = A014963(2) = 2.
6 = 2*3 is not a prime power; a(6) = 1.
8 = 2^3 is a prime power with 4 divisors; a(8) = A014963(4) = 2.
32 = 2^5 is a prime power with 6 divisors; a(32) = A014963(6) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A014963, A253139.

Table[If[PrimePowerQ[n], Exp[MangoldtLambda[DivisorSigma[0, n]]], 1], {n, 1, 100}] (* Indranil Ghosh, Jul 19 2017 *)

A014963(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ This function from Charles R Greathouse IV, Jun 10 2011
A253141(n) = if(1==omega(n), A014963(numdiv(n)), 1); \\ Antti Karttunen, Jul 19 2017

A253139 a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 12, 6, 4, 2, 12, 2, 4, 4, 60, 2, 12, 2, 12, 4, 4, 2, 24, 6, 4, 12, 12, 2, 8, 2, 60, 4, 4, 4, 36, 2, 4, 4, 24, 2, 8, 2, 12, 12, 4, 2, 120, 6, 12, 4, 12, 2, 24, 4, 24, 4, 4, 2, 24, 2, 4, 12, 420, 4, 8, 2, 12, 4, 8, 2, 72, 2, 4, 12, 12, 4, 8

Offset: 1

Matthew Vandermast, Dec 27 2014

A divisibility sequence (cf. Ward link and second formula).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			The divisors of 20 are 1, 2, 4, 5, 10 and 20, which have 1, 2, 3, 2, 4 and 6 divisors respectively. The least common multiple of 1, 2, 3, 2, 4 and 6 is 12; therefore, a(20) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.

A250270 gives range of values. A141586 lists numbers n such that a(n) divides n.

Cf. A003418, A253141.

Table[LCM@@DivisorSigma[0,Divisors[n]],{n,100}] (* Harvey P. Dale, Sep 01 2017 *)
lcm[n_] := lcm[n] = LCM @@ Range[n]; a[1] = 1; a[n_] := Times @@ (lcm [Last[#] + 1] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ Michel Marcus, Jan 23 2015

If n = Product_ prime(i)^e(i), then a(n) = Product_ A003418(e(i)+1).

a(n) = Product_{d|n} A253141(d).

A250270 Products of terms of A003418.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 960, 1024, 1152, 1296, 1440, 1536, 1680, 1728, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456

Offset: 1

Matthew Vandermast, Dec 16 2014

nonn

Includes all factorials and Jordan-Polya numbers, since n! = Product_{i = 1..n} A003418(floor(n/i)) for positive n.

			720 = 2*6*60 = 12*60. Since 2, 6, 12 and 60 are all terms of A003418, 720 is a term of this sequence.

Michel Marcus, Table of n, a(n) for n = 1..5000

Range of values of A253139. Subsequences include A000142, A001013, A001813, A025527, A064350, A166338, A250569.

Subsequence of A025487.

f(n) = lcm(vector(n, i, i)); \\ A003418
mul(x,y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok,  nv = select(x->(xMichel Marcus, May 09 2021

A250269 Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).

Original entry on oeis.org

1, 2, 6, 12, 120, 60, 5040, 1680, 60480, 15120, 39916800, 55440, 6227020800, 8648640, 1816214400, 518918400, 355687428096000, 147026880, 121645100408832000, 55870214400, 1689515283456000, 14079294028800, 25852016738884976640000, 771008958720

Offset: 1

Matthew Vandermast, Dec 16 2014

nonn

The title is analogous to the title of A061446.
For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence.  Not every divisibility sequence b corresponds to some integer sequence a such that b(n) = Product_{d|n} a(d), however.
This sequence is not itself a divisibility sequence; a(15) does not divide a(30), for example.

			The divisors of 10 are 1, 2, 5 and 10. 10! = a(1) * a(2) * a(5) * a(10) = 1 * 2 * 120 * 15120 = 3628800.
Between 1 and 10 inclusive, 4 integers are coprime to 10: 1, 3, 7 and 9. Let b(n) = lcm (1...n) = A003418(n), and let [x] denote the floor function. Then:
a(10) = b[10/1] * b[10/3] * b[10/7] * b[10/9]
"   "   = b(10) * b(3) * b(1) * b(1)
"   "   = 2520 * 6 * 1 * 1
"   "   = 15120.

Michael De Vlieger, Table of n, a(n) for n = 1..456
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.

Cf. A000142, A075071. Subsequence of A250270.

Cf. A000010 (comments on product formulas), A008683.

Array[Product[(d!)^MoebiusMu[#/d], {d, Divisors[#]}] &, 24] (* Michael De Vlieger, Nov 11 2021 *)

a(n)={my(r=1);fordiv(n,d,r*=d!^moebius(n/d));r} \\ Joerg Arndt, Jan 18 2015

a(n) = Product_{i = 1..n, gcd(n, i) = 1} lcm (1..floor(n/i)).
a(n) = Product_{i = 1..floor(n/2), gcd(n, i) = 1} lcm (1..floor(n/i)) (equivalent formula).
a(n) = n! iff n is prime.
a(n) = Product_{d|n} (d!)^moebius(n/d). - Joerg Arndt, Jan 18 2015
a(n) = Product_{k=1..n} (gcd(n,k)!)^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} ((n/gcd(n,k))!)^(mu(gcd(n,k))/phi(n/gcd(n,k))) where mu = A008683, phi = A000010. - Richard L. Ollerton, Nov 08 2021

A242453 (Conjectured) infinite nondecreasing sequence with a(1) = 1 such that the divisors of n appear a total of a(n) times in the sequence.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23

Offset: 1

Matthew Vandermast, Jul 24 2014

nonn

Can it be proved or disproved that this sequence is valid? It is invalid if, for some value of n, it is impossible for a(n) to be as great as the total number of appearances that the aliquot divisors of n have already made in the sequence.
A near-example: The combination of the sequence definition and the values of terms a(1)-a(29) eliminates all possible values of a(30) except for the number 13. If the aliquot divisors of 30 (1, 2, 3, 5, 6, 10 and 15) appeared in the sequence a total of 14 or more times, 13 would also be eliminated as a possible value, and this sequence would be invalid. Since 30's aliquot divisors appear in the sequence a total of only 12 times, no contradiction occurs.
The first 2500 terms appear to be free of any contradiction that would "annihilate" the sequence.

			a(2) cannot be 1, because then it would be the second term in the sequence to divide 1. (Since a(1) = 1, it is necessary that exactly 1 term in the sequence is a divisor of 1.)
Nor can a(2) be 3 or greater, because then a(2) would be greater than the number of terms in the sequence that divided 2 (there would be only 1 such term). Since, by definition, a(2) must equal the number of terms in the sequence that divide 2, this is a contradiction.
Since 2 is the only possible value for a(2) that does not create a contradiction, a(2) = 2.

Cf. A001462.

A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Matthew Vandermast, May 08 2014

nonn

Values of n for which all numbers in row A238747(n) are even. Also, numbers n such that A000005(n^m) is a perfect square for all nonnegative integers m; numbers n such that A181819(n) is a perfect square; numbers n such that A182860(n) is odd.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 33, 314, 3119, 31436, 315888, 3162042, 31626518, 316284320, 3162915907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3162... . - Amiram Eldar, Nov 28 2023

			The prime signature of 36 = 2^2 * 3^2 is {2,2}. One distinct integer (namely, 2) appears in the prime signature, and it appears an even number of times (2 times). Hence, 36 appears in the sequence.
The prime factorization of 1260 = 2^2 * 3^2 * 5^1 * 7^1. Exponent 2 occurs twice (an even number of times), as well as exponent 1, thus 1260 is included. It is also the first term k > 1 in this sequence for which A182850(k) = 4, not 3. - _Antti Karttunen_, Feb 06 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465, A367590.
Subsequence of A000379, A028260, A063774 and A268390.
Cf. A000005, A181819, A182850, A182860, A238747.

q[n_] := n == 1 || AllTrue[Tally[FactorInteger[n][[;; , 2]]][[;; , 2]], EvenQ]; Select[Range[200], q] (* Amiram Eldar, Nov 28 2023 *)

is(n) = {my(e = factor(n)[, 2], m = #e); if(m%2, return(0)); e = vecsort(e); forstep(i = 1, m, 2, if(e[i] != e[i+1], return(0))); 1;} \\ Amiram Eldar, Nov 28 2023

(define A238748 (MATCHING-POS 1 1 (lambda (n) (square? (A181819 n)))))
(define (square? n) (not (zero? (A010052 n))))
;; Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 06 2016

A238747 Row n of table gives prime metasignature of n: count total appearances of each distinct integer that appears in the prime signature of n, then arrange totals in nonincreasing order.

Original entry on oeis.org

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

Offset: 2

Matthew Vandermast, May 08 2014

A prime metasignature is analogous to the signature of a partition (cf. A115621); it is the signature of a prime signature.

Row n also gives prime signature of A181819(n).

			The prime signature of 72 (2^3*3^2) is {3,2}. The numbers 3 and 2 each appear once; therefore, the prime metasignature of 72 is {1,1}.
The prime signature of 120 (2^3*3*5) is {3,1,1}. 3 appears 1 time and 1 appears 2 times; therefore, the prime metasignature of 120 is {2,1}.

Length of row n equals A071625(n); sum of numbers in row n is A001221(n).

Cf. A115621, A181819, A238748.

Row n is identical to row A181819(n) of table A212171.

A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Also the number of members of A025487 that divide A025487(n).

			5 members of A025487 divide A025487(6) = 12 (namely, 1, 2, 4, 6 and 12); therefore, a(6) = 5.

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

Cf. A085082, A025487, A181822, A322584.
Rearrangement of A115728, A115729 and A238690.
A116473(n) is the number of times n appears in the sequence.

lpsQ[n_] := n == 1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]); lps = Select[Range[6000], lpsQ]; c[n_] := Count[Divisors[n], ?(MemberQ[lps, #] &)]; c /@ lps  (* _Amiram Eldar, Jan 21 2024 *)

a(n) = A085082(A025487(n)) = A085082(A181822(n)).

a(n) = A322584(A025487(n)). - Amiram Eldar, Jan 21 2024

A238745 a(1) = 1; for n > 1, if the first integer with the same prime signature as n is factorized into primorials as Product A002110(i)^e(i), then a(n) = Product prime(i)^e(i).

Original entry on oeis.org

1, 2, 2, 4, 2, 3, 2, 8, 4, 3, 2, 6, 2, 3, 3, 16, 2, 6, 2, 6, 3, 3, 2, 12, 4, 3, 8, 6, 2, 5, 2, 32, 3, 3, 3, 9, 2, 3, 3, 12, 2, 5, 2, 6, 6, 3, 2, 24, 4, 6, 3, 6, 2, 12, 3, 12, 3, 3, 2, 10, 2, 3, 6, 64, 3, 5, 2, 6, 3, 5, 2, 18, 2, 3, 6, 6, 3, 5, 2, 24, 16, 3, 2

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Alternate definition: a(1) = 1; for n > 1, if row n of table A238744 is {k(1), k(2),...,k(A051903(n))}, then a(n) = Product {i = 1 to A051903(n)} prime(k(i)).
Since the first integer of each prime signature (A025487) is always a product of primorials (A002110), there is always a value for a(n). Every positive integer appears in the sequence.
a(m) = a(n) iff m and n have the same prime signature.  If the prime signatures of m and n are conjugate to each other when they are viewed as partitions, then a(n) = A181819(m) and a(m) = A181819(n).

			The first integer with the same prime signature as 40 is 24 = 2^3*3. Since the factorization of 24 into primorials is 24 = 2^2*6 = A002110(1)^2*A002110(2), a(24) = a(40) = prime(1)^2*prime(2) = 2^2*3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A181815, A181819, A238744.

f[n_] := Block[{k = 1, d, a}, While[n - Times @@ Prime@ Range[k + 1] >= 0, k++]; If[n == Product[Prime@ i, {i, k}], Prime@ k, d = Select[Reverse@ FoldList[#1 #2 &, Prime@ Range@ k], Divisible[n, #] &]; If[AllTrue[#, IntegerQ], Times @@ Map[(FactorInteger[#1][[-1, 1]])^#2 & @@ # &, Reverse@ Tally@ #], False] &@ Rest@ NestWhileList[Function[P, {#1/P, P}]@ SelectFirst[d, Function[k, Divisible[#1, k]]] & @@ # &, {n, 1}, First@ # > 1 &][[All, -1]]]]; Table[f@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]] - Boole[n == 1], {n, 83}] (* Michael De Vlieger, May 19 2017, Version 10.2 *)

a(n) = A181819(A124859(n)).

a(n) = A122111(A181819(n)).

A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

Original entry on oeis.org

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

Offset: 2

Matthew Vandermast, Apr 28 2014

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)

Row lengths are A051903(n); row sums are A001222(n).
Cf. A217171.
These partitions are ranked by A238745.
For prime indices A296150 instead of exponents we get A321649, rev A321650.
A000700 counts self-conjugate partitions, ranked by A088902.
A003963 gives product of prime indices, conjugate A329382.
A008480 gives number of permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221.
A352486-A352490 are sets related to the fixed points of A122111.
Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Row n is identical to row A124859(n) of table A212171.

cp's OEIS Frontend

User: Matthew Vandermast

A253141 If n is a prime power, then a(n) = lambda(tau(n)) = A014963(A000005(n)); otherwise, a(n) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A253139 a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A250270 Products of terms of A003418.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A250269 Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242453 (Conjectured) infinite nondecreasing sequence with a(1) = 1 such that the divisors of n appear a total of a(n) times in the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A238747 Row n of table gives prime metasignature of n: count total appearances of each distinct integer that appears in the prime signature of n, then arrange totals in nonincreasing order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

Views

Author

Keywords

Comments

Examples