A112798 -id:A112798 - cp's OEIS frontend

A002982 Numbers k such that k! - 1 is prime.

3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003

Offset: 1

N. J. A. Sloane

The corresponding primes n!-1 are often called factorial primes.

			From _Gus Wiseman_, Jul 04 2019: (Start)
The sequence of numbers n! - 1 together with their prime indices begins:
                    1: {}
                    5: {3}
                   23: {9}
                  119: {4,7}
                  719: {128}
                 5039: {675}
                40319: {9,273}
               362879: {5,5,430}
              3628799: {10,11746}
             39916799: {6,7,9,992}
            479001599: {25306287}
           6227020799: {270,256263}
          87178291199: {3610490805}
        1307674367999: {7,11,11,16,114905}
       20922789887999: {436,318519035}
      355687428095999: {8,21,10165484947}
     6402373705727999: {17,20157,25293727}
   121645100408831999: {119,175195,4567455}
  2432902008176639999: {11715,659539127675}
(End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 719 at p. 160.

A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26:118 (1972), pp. 567-570.
J. P. Buhler et al., Primes of the form n!+-1 and 2.3.5....p+-1, Math. Comp., 38:158 (1982), pp. 639-643.
Chris K. Caldwell, Factorial Primes.
C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71:237 (2002), pp. 441-448.
P. Carmody, Factorial Prime Search Progress Pages.
Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
R. K. Guy and N. J. A. Sloane, Correspondence, 1985.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99 (2015), pp 213-219. doi:10.1017/mag.2015.28.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
PrimeGrid, World Record Factorial Prime!!!.
PrimeGrid, Announcement of 94550, (2010). - _Felix Fröhlich_, Jul 11 2014
PrimeGrid, Announcement of 103040, (2010). - _Felix Fröhlich_, Jul 11 2014
PrimeGrid, Announcement of 147855, (2013). - _Felix Fröhlich_, Jul 11 2014
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Factorial Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989.
Index entries for sequences related to factorial numbers.

Cf. A002981 (numbers n such that n!+1 is prime).
Cf. A055490 (primes of form n!-1).
Cf. A088332 (primes of form n!+1).
Cf. A000142, A001221, A001222, A046051, A054991, A112798, A325272.

[n: n in [0..500] | IsPrime(Factorial(n)-1)]; // Vincenzo Librandi, Sep 07 2017

Select[Range[10^3], PrimeQ[ #!-1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=ispseudoprime(n!-1) \\ Charles R Greathouse IV, Mar 21 2013

from sympy import factorial, isprime
A002982_list = [n for n in range(1,10**2) if isprime(factorial(n)-1)] # Chai Wah Wu, Apr 04 2021

21480 sent in by Ken Davis (ken.davis(AT)softwareag.com), Oct 29 2001
Updated Feb 26 2007 by Max Alekseyev, based on progress reported in the Carmody web site.
Inserted missing 21480 and 34790 (see Caldwell). Added 94550, discovered Oct 05 2010. Eric W. Weisstein, Oct 06 2010
103040 was discovered by James Winskill, Dec 14 2010. It has 471794 digits. Corrected by Jens Kruse Andersen, Mar 22 2011
a(26) = 147855 from Felix Fröhlich, Sep 02 2013
a(27) = 208003 from Sou Fukui, Jul 27 2016

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 20, 1, 21, 1, 22, 6, 23, 1, 24, 2, 25, 13, 26, 1, 27, 5, 28, 17, 29, 1, 30, 1, 31, 10, 32, 7, 33, 1, 34, 19, 35, 1, 36, 1, 37, 9, 38, 3, 39, 1, 40, 8, 41, 1, 42

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a(n): return 0 if n==1 else n//2 if n%2==0 else a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A355529 Numbers of which it is not possible to choose a different prime factor of each prime index (with multiplicity).

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 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, 105, 106

Offset: 1

Gus Wiseman, Jul 24 2022

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.

Includes all even numbers.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}

The odd case is A355535.
The case of all divisors (not just primes) is A355740, zeros of A355739.
These choices are variously counted by A355741, A355744, A355745.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A000720, A076610, A318979, A335433, A335448, A355733.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]

A305078 Heinz numbers of connected integer partitions.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, May 24 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence lists all Heinz numbers of multisets S such that G(S) is a connected graph.

			The sequence of all connected multiset multisystems (see A302242, A112798) begins:
   2: {{}}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  59: {{7}}
  61: {{1,2,2}}
  63: {{1},{1},{1,1}}
  65: {{2},{1,2}}
  67: {{8}}
  71: {{1,1,3}}
  73: {{2,4}}
  79: {{1,5}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  97: {{3,3}}

Madeline Locus Dawsey, Tyler Russell and Dannie Urban, Polynomials Associated to Integer Partitions, arXiv:2108.00943 [math.NT], 2021.

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305079.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[300],Length[zsm[primeMS[#]]]==1&]

A355740 Numbers of which it is not possible to choose a different divisor of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 50, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188

Offset: 1

Gus Wiseman, Jul 22 2022

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.

By Hall's marriage theorem, k is a term if and only if there is a sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, k is divisible by a member of A370348. - Robert Israel, Feb 15 2024

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   48: {1,1,1,1,2}
For example, the choices of a divisor of each prime index of 90 are: (1,1,1,1), (1,1,1,3), (1,1,2,1), (1,1,2,3), (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3). But none of these has all distinct elements, so 90 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Cartesian product.

Positions of 0's in A355739.
The case of just prime factors (not all divisors) is A355529, odd A355535.
The unordered case is counted by A355733, firsts A355734.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A000720, A076610, A335433, A335448, A340827, A355737, A355749, A370348.

filter:= proc(n) uses numtheory, GraphTheory; local B, S, F, D, E, G, t, d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]), t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i; seq([t[1], i], i=1..t[2]) end proc, F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t, d}, d=divisors(t[1])), t = F)};
  S:= map(t -> convert(t, name), [op(F), op(D)]);
  E:= map(e -> map(convert, e, name), E);
  G:= Graph(S, E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
remove(filter, [$1..200]); # Robert Israel, Feb 15 2024

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Divisors/@primeMS[#]],UnsameQ@@#&]=={}&]

We have A001221(a(n)) >= A303975(a(n)).

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A359908 Numbers whose prime indices have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 23 2023

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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 180 are {1,1,2,2,3}, with median 2, so 180 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with median 3/2, so 360 is not in the sequence.

The odd-length case is A027193.
For mean instead of median we have A316413.
These partitions are counted by A325347, strict A359907.
The complement is A359912, counted by A307683.
The median of prime indices is given by A360005/2.
The case of integer mean also is A360009.
A112798 lists prime indices, length A001222, sum A056239.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A359889, A359890, A359903, A359905, A360006.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],IntegerQ[Median[prix[#]]]&]

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

The first term that is not a factorial number is a(180) = 12.
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.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

			The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

Andrew Howroyd, Table of n, a(n) for n = 1..4096
Wikipedia, Permutation pattern

Separations are counted by A003242 and ranked by A333489.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535 and ranked by A335448.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[,2])} \\ Andrew Howroyd, Feb 03 2021

A324850 Numbers divisible by the product of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896

Offset: 1

Gus Wiseman, Mar 18 2019

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, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
  72: {1,1,1,2,2}
  96: {1,1,1,1,1,2}

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

Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.

Select[Range[100],Divisible[#,Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]]&]

isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k,1])^f[k,2])); \\ Michel Marcus, Mar 22 2019

n/A003963(n) = A324933(n)/A324934(n).

A333219 Heinz number of the n-th composition in standard order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 8, 7, 10, 9, 12, 10, 12, 12, 16, 11, 14, 15, 20, 15, 18, 18, 24, 14, 20, 18, 24, 20, 24, 24, 32, 13, 22, 21, 28, 25, 30, 30, 40, 21, 30, 27, 36, 30, 36, 36, 48, 22, 28, 30, 40, 30, 36, 36, 48, 28, 40, 36, 48, 40, 48, 48, 64, 17, 26, 33, 44

Offset: 1

Gus Wiseman, Mar 16 2020

nonn

Includes all positive integers.
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.
The Heinz number of a composition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}           15: {2,3}          25: {3,3}
   2: {1}          20: {1,1,3}        30: {1,2,3}
   3: {2}          15: {2,3}          30: {1,2,3}
   4: {1,1}        18: {1,2,2}        40: {1,1,1,3}
   5: {3}          18: {1,2,2}        21: {2,4}
   6: {1,2}        24: {1,1,1,2}      30: {1,2,3}
   6: {1,2}        14: {1,4}          27: {2,2,2}
   8: {1,1,1}      20: {1,1,3}        36: {1,1,2,2}
   7: {4}          18: {1,2,2}        30: {1,2,3}
  10: {1,3}        24: {1,1,1,2}      36: {1,1,2,2}
   9: {2,2}        20: {1,1,3}        36: {1,1,2,2}
  12: {1,1,2}      24: {1,1,1,2}      48: {1,1,1,1,2}
  10: {1,3}        24: {1,1,1,2}      22: {1,5}
  12: {1,1,2}      32: {1,1,1,1,1}    28: {1,1,4}
  12: {1,1,2}      13: {6}            30: {1,2,3}
  16: {1,1,1,1}    22: {1,5}          40: {1,1,1,3}
  11: {5}          21: {2,4}          30: {1,2,3}
  14: {1,4}        28: {1,1,4}        36: {1,1,2,2}

The length of the k-th composition in standard order is A000120(k).
The sum of the k-th composition in standard order is A070939(k).
The maximum of the k-th composition in standard order is A070939(k).
A partial inverse is A333220. See also A233249.
Cf. A048793, A056239, A066099, A112798, A114994, A124767, A213925, A225620, A228351, A233564, A272919, A333218.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Times@@Prime/@stc[n],{n,0,100}]

A056239(a(n)) = A070939(n).

cp's OEIS Frontend

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