A025487 -id:A025487 - cp's OEIS frontend

A329897 Numbers k for which the 2-adic and 3-adic valuations of A025487(k) are equal, where A025487(k) is the k-th number which is a product of primorial numbers.

1, 4, 9, 11, 20, 22, 23, 38, 41, 43, 44, 54, 69, 72, 73, 77, 79, 93, 110, 114, 118, 123, 124, 128, 129, 131, 147, 154, 181, 186, 190, 191, 199, 201, 208, 209, 212, 232, 242, 245, 246, 272, 279, 286, 294, 299, 300, 307, 310, 312, 321, 324, 327, 345, 359, 371, 374, 376, 416, 424, 425, 430, 434, 442, 446, 451, 454, 466, 469

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

Numbers k for which A007814(A025487(k)) = A007949(A025487(k)).

Numbers k for which A181815(k) is odd.

Antti Karttunen, Table of n, a(n) for n = 1..7806

Cf. A007814, A007949, A025487, A329898 (complement), A330682 (characteristic function).
Sequence A330683 sorted into ascending order.
Positions of odd terms in A181815.

s = {1}; k = 1; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], k++; If[Equal @@ IntegerExponent[n, {2, 3}], AppendTo[s, k]]], {n, 2, 10^5}]; s (* Amiram Eldar, Jul 28 2023 *)

A146288 Number of divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 8, 6, 9, 10, 12, 7, 12, 12, 16, 8, 15, 18, 14, 16, 16, 20, 9, 18, 24, 16, 24, 20, 24, 10, 21, 30, 18, 32, 24, 27, 28, 11, 32, 24, 36, 25, 36, 20, 40, 28, 36, 32, 12, 40, 27, 32, 48, 30, 42, 22, 48, 32, 45, 36, 13, 48, 30, 48, 60, 35, 48, 48, 24, 54, 50, 56

Offset: 1

Matthew Vandermast, Nov 11 2008

nonn

			a(4) = 4 because 4 positive integers divide evenly into A025487(4) = 6: 1, 2, 3 and 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

a(n) = sum of the n-th row of A146290, A146292.
A rearrangement of A080444.
Cf. A000005 (number of divisors), A025487.

a146288 = a000005 . a025487  -- Reinhard Zumkeller, Sep 17 2014

s = {1}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], AppendTo[s, DivisorSigma[0, n]]], {n, 2, 10000}]; s (* Amiram Eldar, Aug 05 2024 *)

a(n) = A000005(A025487(n)).

A181817 a(n) is the smallest integer that, when divided by any divisor of A025487(n), yields a member of A025487.

Original entry on oeis.org

1, 2, 4, 12, 8, 24, 16, 48, 360, 32, 144, 96, 720, 64, 288, 192, 1440, 128, 576, 4320, 384, 75600, 1728, 2880, 256, 1152, 8640, 768, 151200, 3456, 5760, 512, 2304, 17280, 1536, 302400, 6912, 129600, 11520, 1024, 51840, 4608, 907200, 20736, 34560, 3072, 604800, 13824, 259200, 23040, 2048

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

A permutation of A181818.

			For any divisor d of 6 (d = 1, 2, 3, 6), 12/d (12, 6, 4, 2) is always a member of A025487. 12 is the smallest number with this relationship to 6; therefore, since 6 = A025487(4), a(4) = 12.

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

Cf. A025487, A108951, A181816, A181818.

If A025487(n) = Product prime(i)^e(i), then a(n) = Product A002110(i)^e(i). I.e., a(n) = A108951(A025487(n)).
If A025487(n) = Product A002110(i)^e(i), then a(n) = Product A006939(i)^e(i).
a(n) = A025487(n) * A181816(n).

A181827 Members of A025487 such that A025487(n) > A181822(n).

Original entry on oeis.org

6, 30, 60, 180, 210, 420, 840, 900, 1260, 1800, 2310, 2520, 4620, 6300, 7560, 9240, 12600, 13860, 18480, 25200, 27720, 30030, 37800, 44100, 55440, 60060, 69300, 83160, 88200, 120120, 138600, 166320, 176400, 180180, 189000, 240240, 264600, 277200

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

			A025487(9) = 30 and A181822(9) = 8 have the prime signatures (1,1,1) and (3) respectively. 30 is the larger member of the pair and is therefore included in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition.

Cf. A025487, A181822, A181823, A181824, A181825, A181826.

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

A329904 Position of A329899(n) in A025487.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 4, 7, 3, 8, 9, 10, 11, 12, 13, 14, 15, 6, 16, 9, 5, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 11, 31, 32, 8, 33, 13, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 22, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 15, 57, 58, 20, 12, 59, 60, 61, 17, 62, 10, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

Restricted growth sequence transform of A329899.
Each positive natural number occurs exactly twice.
For all i, j > 1:
  a(i) = a(j) => A329907(i) = A329907(j).

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

Cf. A025487, A085089, A329899, A329905, A329907.

A329904(n) = { my(k=A329899(n)); for(i=1,oo,if(A025487(i)==k,return(i))); };

a(n) = A085089(A329899(n)).

A330683 a(n) is the position of A283980(A025487(n)) in A025487.

Original entry on oeis.org

1, 4, 11, 9, 23, 20, 44, 41, 22, 79, 38, 73, 43, 131, 69, 124, 77, 212, 118, 72, 201, 54, 110, 129, 327, 191, 123, 312, 93, 181, 209, 493, 300, 199, 474, 154, 286, 128, 324, 725, 190, 454, 147, 272, 310, 697, 245, 434, 208, 490, 1044, 299, 671, 114, 232, 416, 469, 1008, 374, 646, 321, 721, 1481, 451, 974, 186, 359

Offset: 1

Antti Karttunen, Dec 26 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12868

Permutation of A329897.

Cf. A025487, A085089, A101296, A181815, A283980, A329898 (positive integers not in this sequence), A329904 (a left inverse), A329906, A330681.

(* First, load the function f at A025487, then: *)
With[{s = Union@ Flatten@ f@ 10}, TakeWhile[#, # != 0 &] &@ Map[If[# > Max@ s, 0, FirstPosition[s, #][[1]] ] &[(Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2]] &, s]] (* Michael De Vlieger, Jan 11 2020 *)

upto_e = 101;
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A330683list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,A283980(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
v330683 = A330683list(upto_e);
A330683(n) = v330683[n];

a(n) = A085089(A330681(n)) = A101296(A330681(n)) = A101296(A283980(A025487(n))).

For all n >= 1, A329904(a(n)) = n.

A086141 Permutation of A025487 (least prime signatures) which, when values are factored, exhibit self-similarity (cf. A008687).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 36, 30, 16, 24, 72, 60, 216, 180, 900, 210, 32, 48, 144, 120, 432, 360, 1800, 420, 1296, 1080, 5400, 1260, 27000, 6300, 44100, 2310, 64, 96, 288, 240, 864, 720, 3600, 840, 2592, 2160, 10800, 2520, 54000, 12600, 88200, 4620, 7776, 6480

Offset: 1

Alford Arnold, Aug 24 2003

nonn

			Factored sequences are
1
0 2 0 3 0 3 0 5 0 3 0 5 0 5 0 7 ...
0 0 4 2 0 0 9 3 0 0 9 3 0 0 25 5 ...
0 0 0 0 8 4 4 2 0 0 0 0 27 9 9 3 ...
0 0 0 0 0 0 0 0 16 8 8 4 8 4 4 2 ...
yielding
1 2 4 6 8 12 36 30 16 24 72 60 216 180 900 210 ...

Cf. A008687, A080801.

A108464 a(n) = A057567(A025487(n)).

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 12, 21, 15, 19, 26, 38, 36, 30, 52, 64, 74, 45, 98, 92, 105, 52, 109, 141, 67, 171, 198, 165, 135, 212, 250, 97, 289, 392, 254, 296, 382, 249, 426, 139, 444, 467, 371, 424, 719, 381, 592, 662, 560, 696, 195, 907, 737, 203, 850, 783, 1261, 562, 1098

Offset: 1

Christian G. Bower, Jun 03 2005

nonn

Number of partitions of n where product of parts divides n by prime signature.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A025487, A057567.

Offset corrected by Amiram Eldar, Jul 23 2024

A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 4, 1, 2, 2, 4, 2, 4, 4, 4, 4, 6, 8, 6, 8, 1, 2, 2, 6, 6, 6, 10, 2, 4, 4, 6, 8, 6, 10, 4, 8, 6, 8, 12, 6, 10, 6, 8, 12, 10, 8, 12, 12, 10, 16, 12, 20, 1, 2, 6, 8, 10, 6, 10, 8, 10, 16, 14, 20, 2, 4, 12, 10, 10, 14, 10, 16, 12, 20, 6, 6, 10, 8, 10, 12, 20, 4, 8, 14, 14, 20, 14, 10, 16, 14, 24, 6, 12, 12

Offset: 1

Antti Karttunen, Feb 27 2019

nonn

A098719 gives the positions of ones in this sequence. See also comments in A324383.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (based on the b-file of A025487 as computed by Will Nicholes and Franklin T. Adams-Watters)
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Cf. A002110, A025487, A098719 (positions of ones), A276150, A324342.

Cf. A324382 for a subsequence, and A324383, A324386 for permutations of this sequence.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324387(n) = A276150(A025487(n));

a(n) = A276150(A025487(n)).

cp's OEIS Frontend

A329897 Numbers k for which the 2-adic and 3-adic valuations of A025487(k) are equal, where A025487(k) is the k-th number which is a product of primorial numbers.

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A146288 Number of divisors of the n-th prime signature number (A025487(n)).

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A181817 a(n) is the smallest integer that, when divided by any divisor of A025487(n), yields a member of A025487.

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A181827 Members of A025487 such that A025487(n) > A181822(n).

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A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

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A329904 Position of A329899(n) in A025487.

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A330683 a(n) is the position of A283980(A025487(n)) in A025487.

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A086141 Permutation of A025487 (least prime signatures) which, when values are factored, exhibit self-similarity (cf. A008687).

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A108464 a(n) = A057567(A025487(n)).

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