A350352 -id:A350352 - cp's OEIS frontend

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 420, 426, 429

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A007304 and A093599 in having 210.
First differs from A287483 in having 222.
First differs from A350352 in having 420.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			Selected terms together with their prime indices:
   660: {1,1,2,3,5}
   798: {1,2,4,8}
   840: {1,1,1,2,3,4}
  3120: {1,1,1,1,2,3,6}
  9900: {1,1,2,2,3,3,5}

A336500 has zeros at these positions.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 is the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A001055, A002033, A098859, A124010, A167865, A253249, A336420.

strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]
Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]

A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 15, 23, 37, 52, 80, 109, 156, 208, 289, 378, 509, 654, 865, 1098, 1425, 1789, 2290, 2852, 3603, 4450, 5569, 6830, 8467, 10321, 12701, 15393, 18805, 22678, 27535, 33057, 39908, 47701, 57304, 68226, 81572, 96766, 115212, 136201

Offset: 0

Gus Wiseman, Dec 25 2021

nonn

Also the number of non-weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The a(7) = 1 through a(11) = 15 partitions:
  (3211)  (4211)   (3321)    (5311)     (4322)
          (32111)  (4311)    (6211)     (4421)
                   (5211)    (32221)    (5411)
                   (42111)   (33211)    (6311)
                   (321111)  (43111)    (7211)
                             (52111)    (42221)
                             (421111)   (43211)
                             (3211111)  (53111)
                                        (62111)
                                        (322211)
                                        (332111)
                                        (431111)
                                        (521111)
                                        (4211111)
                                        (32111111)

Counting all non-strict partitions gives A047967.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A347548, ranked by A350352.
The version for compositions allowing strict is A349053, ranked by A349057.
Allowing strict partitions gives A349061, complement A349060.
The complement in non-strict partitions is A349795.
These partitions are ranked by A350140 = A349794 \ A005117.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047 = alternating compositions, ranked by A345167.
A025048/A025049 = directed alternating compositions.
A096441 = weakly alternating 0-appended partitions.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating permutations of prime indices.
Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]

a(n) = A349061(n) - A347548(n).

A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1...pr such that phi(N)=(p1-1)...(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).

Original entry on oeis.org

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 11305, 15841, 29341, 39865, 41041, 46657, 52633, 62745, 63973, 75361, 96985, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 401401, 410041

Offset: 1

Max Alekseyev, Feb 25 2005

nonn

A.K. Devaraj conjectured that these numbers are exactly Carmichael numbers. It was proved (see Alekseyev link) that every Carmichael number is indeed a Devaraj number, but the converse is not true. Devaraj numbers that are not Carmichael are given by A104017.

These numbers can't be even, since phi(N) is always even (N>2) but p1=2 implies that gcd{pi-1}=1 and N-1 is odd. - M. F. Hasler, Apr 03 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Max Alekseyev, Pomerance's proof, June 2005.

Subsequence of A350352 and hence of A033942.

Cf. A104017, A002997.

Devaraj() = for(n=2,10^8, f=factorint(n); if(vecmax(f[,2])>1,next); f=f[,1]; r=length(f); if(r==1,next); d=f[1]-1; p=f[1]-1; for(i=2,r,d=gcd(d,f[i]-1); p*=f[i]-1); if( ((n-1)^(r-2)*d^2)%p==0, print1(" ",n)) )

isA104016(n)= local(f=factor(n)); vecmax(f[,2])==1 && #(f*=[1,-1]~)>1 && gcd(f)^2*(n-1)^(#f-2)%prod(i=1,#f,f[i])==0
/* To print the list: */ forstep( n=3, 10^6, 2, vecmax((f=factor(n))[,2])>1 && next; #(f*=[1,-1]~)>1 || next; gcd(f)^2*(n-1)^(#f-2)%prod(i=1,#f,f[i]) || print1(n","))
/* The following version could be efficient for large omega(n) */
isA104016(n) = issquarefree(n) && !isprime(n) && Mod(n-1,prod(i=1,#n=factor(n)*[1,-1]~,n[i]))^(#n-2)*gcd(n)^2==0 \\ M. F. Hasler, Apr 03 2009

A353917 a(1) = 4. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 1, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)), but neither j | k nor k | j.

Original entry on oeis.org

4, 6, 8, 10, 16, 12, 9, 15, 25, 20, 30, 18, 27, 21, 49, 14, 32, 22, 64, 24, 42, 28, 70, 40, 60, 36, 66, 44, 110, 50, 90, 48, 78, 52, 128, 26, 169, 39, 81, 33, 121, 55, 125, 35, 343, 56, 84, 54, 102, 68, 170, 80, 120, 45, 105, 63, 168, 72, 114, 76, 190, 100, 130

Offset: 1

Michael De Vlieger, May 10 2022

nonn

The sequence exhibits phases involving alternating composite prime powers and squarefree semiprimes. These manifest in log-log scatterplot in a caustic fashion, where the composite prime power is very much larger than the squarefree semiprime for sufficiently large n.
Let P = the set of distinct prime divisors of j = a(n-1), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(j, k) > 1 and let G = {P intersect Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa, yet neither j | k nor k | j.
Theorem: primes are prohibited. Proof: since we have gcd(j, k) > 1 and do not allow divisibility, and since primes must either divide or be coprime to another number m, primes do not appear in this sequence.
Theorem: squarefree semiprimes j = pq are followed by k = p^2 or k = q^2. Proof: since omega(j) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with gcd(j, k) > 1.
1.) |P| == |G| implies |Q| > |G| and |Q| > |P|.
2.) |Q| == |G| implies |P| > |G| and |P| > |Q|.
The first case implies some prime r | k yet gcd(j, r) = 1. But this would require j | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = p^e, e > 1, or k = q^m, m > 1.
Corollary: powers of the same prime appear in natural order in this sequence.
There is a weaker alternation between numbers in A120944 and A350352 as n is sufficiently large. This alternation exhibits prime power factor features akin to the composite prime power-squarefree semiprime alternation.
Conjecture: permutation of composite numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..1192, showing records in red and local minima in blue.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16, showing records in red, local minima in blue, accentuating composite prime powers in green and squarefree semiprimes in gold.
Michael De Vlieger, Prime power factor study of a(n), n = 1..10^4 where n increases from left to right, and pi(p) from bottom to top in the top image. Multiplicity e of p^e is represented by black for e = 1, red for e = 2, and a color function to the maximum multiplicity in the range thereafter. On the lower bar we show composite prime powers in yellow, squarefree semiprimes in orange, numbers in A350352 in green, and those in A126706 in blue.
Index entries for sequences related to EKG sequence

Cf. A006881, A064413, A126706, A246547, A337687, A353916.

nn = 2^7; c[_] = 0; j = a[1] = 4; c[4] = 1; u = 6; Do[Set[k, u]; Set[m, PrimeNu[j]]; While[Nand[c[k] == 0, ! Divisible[#2, #1] & @@ Sort[{j, k}], And[#2 > #3, #1 == #3] & @@ Append[Sort[{m, PrimeNu[k]}], PrimeNu[GCD[j, k]]]], k++]; Set[{a[i], c[k]}, {k, i}]; j = k; If[k == u, While[Nand[c[u] == 0, CompositeQ@ u], u++]], {i, 2, nn}]; Array[a, nn]

A365829 Squarefree non-semiprimes.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 181, 182, 186

Offset: 1

Gus Wiseman, Oct 05 2023

nonn

First differs from A030059 in having 210.

			The terms together with their prime indices begin:
     1: {}          43: {14}       102: {1,2,7}
     2: {1}         47: {15}       103: {27}
     3: {2}         53: {16}       105: {2,3,4}
     5: {3}         59: {17}       107: {28}
     7: {4}         61: {18}       109: {29}
    11: {5}         66: {1,2,5}    110: {1,3,5}
    13: {6}         67: {19}       113: {30}
    17: {7}         70: {1,3,4}    114: {1,2,8}
    19: {8}         71: {20}       127: {31}
    23: {9}         73: {21}       130: {1,3,6}
    29: {10}        78: {1,2,6}    131: {32}
    30: {1,2,3}     79: {22}       137: {33}
    31: {11}        83: {23}       138: {1,2,9}
    37: {12}        89: {24}       139: {34}
    41: {13}        97: {25}       149: {35}
    42: {1,2,4}    101: {26}       151: {36}

First condition alone is A005117 (squarefree).
Second condition alone is A100959 (non-semiprime).
The nonprime case is 1 followed by A350352.
Partitions of this type are counted by A365827, non-strict A058984.
A001358 lists semiprimes, squarefree A006881.
Cf. A000009, A004526, A008967, A078408, A365659.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]!=2&]

isok(k) = my(f=factor(k)); issquarefree(f) && (bigomega(f) != 2); \\ Michel Marcus, Oct 07 2023

Intersection of A005117 and A100959.

Complement of A001358 in A005117.

cp's OEIS Frontend

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

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A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1*...*pr such that phi(N)=(p1-1)*...*(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).

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PARI

A353917 a(1) = 4. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 1, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)), but neither j | k nor k | j.

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A365829 Squarefree non-semiprimes.

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A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1...pr such that phi(N)=(p1-1)...(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).