A120944 -id:A120944 - cp's OEIS frontend

A306330 Squarefree n with >= 3 factors that admit idempotent factorizations n = p*q.

30, 42, 66, 78, 102, 105, 114, 130, 138, 165, 170, 174, 182, 186, 195, 210, 222, 246, 255, 258, 266, 273, 282, 285, 290, 318, 330, 345, 354, 366, 370, 390, 399, 402, 410, 426, 434, 435, 438, 455, 462, 465, 474, 498, 510, 518, 530, 534, 546, 555, 570, 582, 602

Offset: 1

Barry Fagin, Feb 07 2019

nonn

An idempotent factorization of n is a way of writing n = p*q such that b^(k(p-1)(q-1)+1) is congruent to b mod n for any integer k >= 0 and any b in Z_n. For example, p = 19, q = 15 is an idempotent factorization of n = 285.  All factorizations of semiprimes are idempotent, so this sequence is restricted to n with >= 3 factors.  Idempotent factorizations have the property that p and q generate correctly functioning RSA keys, even if one or both are composite.
We show in the reference below that a bipartite factorization of a squarefree integer n = pq is idempotent if and only if lambda(pq) divides (p-1)(q-1).
(p and q are not required to be primes. - N. J. A. Sloane, Feb 08 2019)

			30 = 5 * 6, 42 = 7 * 6, 66 = 11 * 6, 78 = 13 * 6, 102 = 17 * 6, 105 = 7 * 15, 114 = 19 * 6, 130 = 13 * 10 are the idempotent factorizations for the first 8 terms in the sequence. 210 = 10 * 21 is the smallest n with a fully composite idempotent factorization, one in which both p and q are composite. The number n = p * 6 is idempotent for any prime p >= 5.

Barry Fagin, Table of n, a(n) for n = 1..5920 (all terms <= 2^16)
Barry Fagin, Idempotent factorizations for all n < 2^26
Barry Fagin, Idempotent Factorizations of Square-Free Integers, Information 2019, 10(7), 232.

Cf. A115957, A138636, A002322.

Subsequence of A120944 (composite squarefree numbers).

isok3(p, q, n) = frac((p-1)*(q-1)/lcm(znstar(n)[2])) == 0;
isok(n) = {if (issquarefree(n) && omega(n) >= 3, my(d = divisors(n)); for (k=1, #d\2, if ((d[k] != 1) && isok3(d[k], n/d[k], n), return (1););););} \\ Michel Marcus, Feb 22 2019

Edited by N. J. A. Sloane, Feb 08 2019

A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

Original entry on oeis.org

1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951

Offset: 0

Gus Wiseman, Nov 17 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
       1: {}
      13: {{1,2}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}

The smallest BII-number of a set of n sets is A000225(n).
BII-numbers of set-systems with no singletons are A326781.
MM-numbers of sets of nonempty sets are the odd terms of A302494.
MM-numbers of multisets of nonempty non-singleton sets are A320629.
The version with empty edges is A329556.
The version with singletons is A329557.
The version with empty edges and singletons is A329558.
Cf. A056239, A072639, A112798, A279952, A302242, A326031, A329552, A329556.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

sqvs=Select[Range[2,30],SquareFreeQ[#]&&!PrimeQ[#]&];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

a(n) = Product_{i = 1..n} prime(A120944(i)).

A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

Original entry on oeis.org

2, 6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 42, 46, 58, 62, 66, 70, 74, 77, 78, 82, 86, 94, 102, 105, 106, 110, 114, 118, 122, 130, 134, 138, 142, 143, 146, 154, 158, 165, 166, 170, 174, 178, 182, 186, 190, 194, 195, 202, 206, 210, 214, 218, 221, 222, 226, 230

Offset: 1

Gus Wiseman, Jul 10 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.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
   2: {1}
   6: {1,2}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  22: {1,5}
  26: {1,6}
  30: {1,2,3}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  42: {1,2,4}
  46: {1,9}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}
  70: {1,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For maximal instead of minimal difference we have A055932 or A066312.
Not prepending zero gives A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A056239 adds up prime indices.
A238352 counts partitions by fixed points, rank statistic A352822.
A279945 counts partitions by number of distinct differences.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 gives minimal difference if singletons go to 0, to index A355525.
Cf. A000005, A000040, A120944, A238354, A286469, A286470, A325160, A325161, A355526, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[Prepend[primeMS[#],0]]==1&]

Equals A005117 /\ (A005843 \/ A104210).

A365783 a(n) = squarefree kernel of A126706(n).

Original entry on oeis.org

6, 6, 10, 6, 14, 6, 10, 22, 15, 6, 10, 26, 6, 14, 30, 21, 34, 6, 15, 38, 10, 42, 22, 30, 46, 6, 14, 33, 10, 26, 6, 14, 58, 39, 30, 62, 42, 66, 15, 34, 70, 6, 21, 74, 30, 38, 51, 78, 10, 6, 82, 42, 57, 86, 35, 22, 30, 46, 94, 21, 6, 14, 66, 10, 102, 69, 26, 106

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Terms are squarefree and composite, i.e., in A120944.

			Let b(n) = A126706(n) and let squarefree kernel rad(n) = A007947(n).
a(1) = 6 = rad(b(1)) = rad(12).
a(2) = 6 = rad(b(2)) = rad(18).
a(3) = 10 = rad(b(3)) = rad(20), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A120944, A126706, A365784, A365785.

Map[Times @@ FactorInteger[#][[All, 1]] &, Select[Range[12, 212], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] ]

apply(x->(x/factorback(factorint(x)[, 1])), select(x->(!issquarefree(x) && !isprimepower(x)), [1..1000])) \\ Michel Marcus, Sep 19 2023

a(n) = A007947(A126706(n)).

A365786 a(n) = squarefree kernel of A286708(n).

Original entry on oeis.org

6, 6, 10, 6, 6, 14, 10, 6, 15, 6, 6, 14, 10, 6, 21, 22, 10, 6, 6, 15, 26, 14, 10, 6, 30, 22, 6, 10, 33, 15, 6, 34, 35, 6, 21, 26, 14, 38, 39, 14, 10, 6, 42, 30, 22, 6, 10, 15, 46, 6, 34, 10, 6, 51, 30, 26, 14, 38, 6, 55, 21, 14, 10, 57, 33, 58, 15, 6, 42, 30, 62

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Terms are squarefree and composite, i.e., in A120944.

			Let b(n) = A286708(n) and let squarefree kernel rad(n) = A007947(n).
a(1) = 6 = rad(b(1)) = rad(36).
a(2) = 6 = rad(b(2)) = rad(72).
a(3) = 10 = rad(b(3)) = rad(100), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A084371, A120944, A126706, A286708.

With[{nn = 2^12}, Map[Times @@ FactorInteger[#][[All, 1]] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not @* PrimePowerQ]] ]

apply(x->factorback(factorint(x)[, 1]), select(x->((x>1) && ispowerful(x) && !isprimepower(x)), [1..5000])) \\ Michel Marcus, Sep 20 2023

a(n) = A007947(A286708(n)).

A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2

Offset: 1

Jaroslav Krizek, Mar 05 2009

nonn

a(n) for n >= 2 equals LCM of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 deviates from A072411, first different term is a(360), a(360) = 3, A072411(360) = 6.

			For n = 12 = 2^2 * 3^1 we have a(12) = lcm(2,1) = 2.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(4,2) = 4.

Cf. A000040, A003990, A006881, A033150, A120944, A000961, A000027, A072411, A051904, A051903, A158378.

Table[LCM @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); lcm(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150. - Amiram Eldar, Sep 11 2024

A158378 a(1) = 0, a(n) = gcd(A051904(n), A051903(n)) for n >= 2.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Jaroslav Krizek, Mar 17 2009

nonn

a(n) for n >= 2 equals GCD of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 it deviates from A052409(n), first different term is a(10800) = a(2^4*3^3*5^2), a(10800) = gcd(2,4) = 2, A052409(10800) = gcd(2,3,4) = 1.

			For n = 12 = 2^2 * 3^1 we have a(12) = gcd(2,1) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65536
Index entries for sequences computed from exponents in factorization of n.

Cf. A157754, A051904, A051903, A052409.

Table[GCD @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A158378(n) = gcd(A051903(n),A051904(n)); \\ Antti Karttunen, Jul 12 2017

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); gcd(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

For n >= 2 holds: a(n)*A157754(n) = A051904(n)*A051903(n).
a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Sep 11 2024

A306999 Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21^e for e >= 0.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 28, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, 66, 69, 70, 72, 75, 77, 78, 84, 87, 90, 91, 93, 96, 98, 99, 102, 105, 108, 111, 112, 114, 117, 119, 120, 123, 126, 129, 132, 133, 135, 138, 140, 141, 144, 150, 153, 154, 156, 159, 161

Offset: 1

Michael De Vlieger, Aug 22 2019

Complement of the union of A003594 and A160545.

Analogous to A081062 and A105115 regarding terms 1 and 2 of A120944, respectively. This sequence applies to A120944(5) = 21.

			6 is in the sequence since gcd(6, 21) = 3 and 6 does not divide 21^e with integer e >= 0.
5 is not in the sequence since it is coprime to 21.
3 is not in the sequence since 3 | 21.
9 is not in the sequence since 9 | 21^2.

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

Cf. A003594, A081062, A105115, A120944, A160545.

filter:= proc(n) local g;
  g:= igcd(n,21);
  if g = 1 or g = n then return false fi;
  3^padic:-ordp(n,3)*7^padic:-ordp(n,7) < n
end proc:
select(filter, [$1..200]); # Robert Israel, Aug 28 2019

With[{nn = 161, k = 21}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A316991 Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 22, 24, 26, 30, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 100, 102, 104, 105, 106, 108, 110, 114, 116, 118, 119, 120, 122, 124, 126, 130, 132

Offset: 1

Michael De Vlieger, Aug 02 2018

Complement of A000027 and union of A003591 and A162699.
Analogous to A081062 and A105115 that apply to A120944(1) and A120944(2), respectively.
This sequence applies to A120944(3).

			6 is in the sequence since gcd(6, 14) = 2 and 6 does not divide 14^e with integer e >= 0.
2 is not in the sequence since 2 | 14.
4 is not in the sequence since 4 | 14^2.
3 and 5 are not in the sequence since they are coprime to 14.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A003591, A081062, A105115, A120944, A162699.

With[{nn = 132, k = 14}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

Numbers k for which A001221(k) = A331591(k).
Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.
If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.
k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

			There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Cf. A000120, A046523, A225546, A267116, A329332.
Sequences with related definitions: A001221, A331591, A331592.
Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.
Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

cp's OEIS Frontend

A306330 Squarefree n with >= 3 factors that admit idempotent factorizations n = p*q.

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A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

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A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

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A365783 a(n) = squarefree kernel of A126706(n).

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A365786 a(n) = squarefree kernel of A286708(n).

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A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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A158378 a(1) = 0, a(n) = gcd(A051904(n), A051903(n)) for n >= 2.

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