A033942 -id:A033942 - cp's OEIS frontend

A259172 Numbers in A259145 that are neither prime nor semiprime.

561, 595, 1105, 1235, 1245, 1495, 1547, 1885, 2405, 2555, 2717, 2849, 3115, 3495, 3655, 3657, 3689, 3815, 4521, 4795, 4945, 5035, 5385, 5395, 5453, 5457, 5709, 5865, 6083, 6141, 6251, 6285, 6365, 6391, 6501, 6695, 6755, 6969, 7021, 7887, 8113, 8255, 8355

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Regarding the distribution: Let K be the union of primes and semiprimes in A259145. Let S be the set of other terms. The growth rate of the cardinality of S with respect to the cardinality of K is significantly slower. For instance, if we take the first 50000 terms of A259145, about 32.5 percent are contained in S. If we take the first 350000 terms, about 38.2 percent are contained in S.
a(n) that are in A002997 (Carmichael numbers) for a(n) <= 10^6 are 561, 1105, 8911, 10585, 29341, 825265.
a(n) that are in A051015 (Zeisel numbers) for a(n) <= 3*10^6 are 1885, 353977, 2953711.

Carlos Eduardo Olivieri, Table of n, a(n) for n = 1..8906
Eric Naslund, Integers with a predetermined prime factorization, arXiv:1203.2363 [math.NT], 2012.

Cf. A001221, A001358, A002997, A007053, A051015, A125527, A259145.

Subsequence of A000469, A033942, A050384 (conjuctered).

Select[Range[25000], PrimeQ[#^2 - EulerPhi[#]] && PrimeNu[#] > 2 &]

A001221(a(n)) > 2.

A000005(a(n)) = 2^k, k >= 3.

A321517 Numbers k such that A294902(k) != A321516(k).

Original entry on oeis.org

50, 75, 100, 125, 150, 175, 190, 200, 222, 225, 238, 242, 246, 250, 275, 285, 300, 325, 333, 338, 350, 357, 363, 369, 374, 375, 380, 400, 425, 438, 442, 444, 450, 475, 476, 484, 492, 494, 500, 507, 525, 550, 555, 561, 570, 575, 578, 595, 600, 605, 615, 625

Offset: 1

Felix Fröhlich, Nov 12 2018

nonn

Cf. A033942, A294902, A321516.

A321516[n_] := Length[Select[Most[Divisors[n]], CompositeQ]]; abQ[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[#==n]&]>0; A294902[n_] := Length[Select[Most[Divisors[n]], abQ[#] &]]; Select[Range[650], A321516[#] != A294902 [#] &] (* Amiram Eldar, Nov 12 2018 after Jean-François Alcover at A175526 *)

a292257(n) = sumdiv(n, d, (dAntti Karttunen in A292257
a294905(n) = (a292257(n) <= hammingweight(n)) \\ after Antti Karttunen in A294905
a294902(n) = sumdiv(n, d, (dAntti Karttunen in A294902
a321516(n) = my(d=divisors(n), i=0); for(k=2, #d-1, if(!ispseudoprime(d[k]), i++)); i
is(n) = a294902(n)!=a321516(n)

A357077 The lesser of two consecutive numbers with at least 3 prime factors (counted with multiplicity).

Original entry on oeis.org

27, 44, 63, 75, 80, 98, 99, 104, 116, 124, 125, 135, 147, 152, 153, 164, 170, 171, 174, 175, 188, 189, 195, 207, 224, 230, 231, 242, 243, 244, 245, 255, 260, 272, 275, 279, 284, 285, 296, 315, 324, 332, 342, 343, 344, 350, 351, 356, 363, 368, 369, 374, 375, 384, 387, 399

Offset: 1

Tanya Khovanova, Sep 10 2022

nonn

The first of two consecutive numbers in A033942.

			27 = 3^3 and 28 = 2^2 * 7. Thus, 27 and 28 both have at least three prime factors. Thus, 27 is in this sequence.

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

Cf. A033942, A344843.

R:= NULL: count:= 0: state:= 0:
for n from 1 while count < 100 do
  if numtheory:-bigomega(n) >= 3 then
     if state = 1 then R:= R, n-1; count:= count+1
     else state:= 1
     fi
  else state := 0
  fi
od:
R; # Robert Israel, Sep 16 2022

Select[Range[1000], Total[Transpose[FactorInteger[#]][[2]]] >= 3 && Total[Transpose[FactorInteger[# + 1]][[2]]] >= 3 &]

from sympy import factorint
def is033942(n): return sum(factorint(n).values()) > 2
def ok(n): return is033942(n) and is033942(n+1)
print([k for k in range(400) if ok(k)]) # Michael S. Branicky, Sep 10 2022

A374138 Numbers k that have more than two prime factors (with multiplicity) and A075255(k) is a multiple of A374127(k).

Original entry on oeis.org

8, 28, 496, 8128, 627013, 3713337, 33550336, 64488069, 3919528877

Offset: 1

Antti Karttunen, Jul 06 2024

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Intersection of A033942 and A374129.
Cf. A000203, A001222, A001414, A075255, A374127.
Cf. A000396 (after its initial 6, is a subsequence).

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A374128(n) = { my(s=A001414(n), x=n-s, u=(sigma(n)-s)-n); (gcd(x, u)==abs(u)); };
isA374138(n) = ((bigomega(n) > 2) && A374128(n));

{k | A001222(k) > 2 and gcd(A075255(k), A374127(k)) = abs(A374127(k))}.

A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 4, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 4, 2, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 23 2025

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 prime indices of 180 are {1,1,2,2,3}, conjugate {5,3,1}, and we have choices:
  {{5},{3},{1}}
  {{5},{2,1},{1}}
  {{4,1},{3},{1}}
  {{4,1},{2,1},{1}}
  {{3,2},{3},{1}}
  {{3,2},{2,1},{1}}
so a(180) = 6.

Positions of 1 are A037143, complement A033942.
For multiplicities instead of indices we have A050361.
Adding up over all integer partitions gives A270995, disjoint A279790, strict A279375.
The conjugate version is A357982, disjoint A383706.
The disjoint case is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.
Cf. A122111, A130091, A179009, A382913, A383707, A383710, A384010, A384011.

fop[y_]:=Join@@@Tuples[strptns/@y];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[fop[conj[prix[n]]]],{n,100}]

A009185 Long leg of more than one Pythagorean triangle.

Original entry on oeis.org

12, 24, 36, 40, 45, 48, 56, 60, 63, 72, 80, 84, 90, 96, 105, 108, 112, 120, 126, 132, 135, 140, 144, 156, 160, 165, 168, 176, 180, 189, 192, 195, 200, 204, 208, 210, 216, 220, 224, 225, 228, 231, 240, 252, 255, 260, 264, 270, 272, 273, 275, 276, 280, 285, 288, 300, 304

Offset: 1

David W. Wilson

nonn

If n is in the sequence, k*n is in the sequence for all k > 1. So sequence is union of arithmetic progressions such as numbers of the form 12*k, 40*k, 45*k, ... - Altug Alkan, Nov 30 2015

Numbers appearing more than once in A009012. - Sean A. Irvine, Apr 20 2018

Cf. A009012, A009188, A020883, A033942.

A387711 Numbers k for which A003959(k) > 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 100, 102, 104, 108, 112, 114, 116, 120, 124, 126, 128, 132, 135, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 174, 176, 180, 184, 186, 188, 189, 192, 196

Offset: 1

Antti Karttunen, Sep 06 2025

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

Cf. A003959, A387710.
Disjoint union of A387712 and A387713. Positions of nonzero terms in A387715.
Subsequence of A005101, and of A246282.
After the initial 4 also a subsequence of A033942.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387711(n) = (A003959(n)>(2*n));

cp's OEIS Frontend

A259172 Numbers in A259145 that are neither prime nor semiprime.

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A321517 Numbers k such that A294902(k) != A321516(k).

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A357077 The lesser of two consecutive numbers with at least 3 prime factors (counted with multiplicity).

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A374138 Numbers k that have more than two prime factors (with multiplicity) and A075255(k) is a multiple of A374127(k).

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A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

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Mathematica

A009185 Long leg of more than one Pythagorean triangle.

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A387711 Numbers k for which A003959(k) > 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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PARI