A189987 -id:A189987 - cp's OEIS frontend

A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134

Offset: 1

Joseph L. Pe, Dec 10 2001

nonn

Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - Amiram Eldar, Sep 09 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A118914.
Numbers with exactly 2*p divisors: A030513 (p=2), A030515 (p=3), A030628 \ {1} (p=5), A030632 (p=7), A137485 (p=11), A137489 (p=13), A175744 (p=17), A175747 (p=19).
Subsequences: A006881, A030078, A050997, A054753, A095990, A096156, A138031, A178739, A179665, A189987.

Select[Range[1, 1000], PrimeQ[DivisorSigma[0, # ] / 2] == True &]

n=0; for (m=1, 10^9, f=numdiv(m)/2; if (frac(f)==0 && isprime(f), write("b065985.txt", n++, " ", m); if (n==1000, return))) \\ Harry J. Smith, Nov 05 2009

is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ Charles R Greathouse IV, Oct 15 2015

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378

Offset: 1

Amiram Eldar, Feb 19 2025

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

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

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));

A349931 Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 121, 124, 147, 148, 153, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 240, 242, 244, 245, 252, 261, 268, 272, 275, 279, 284, 289, 292, 300

Offset: 1

Tian Vlasic, Dec 05 2021

Intersection of A319240 and A013929, i.e., terms of A319240 that are not squarefree.
A319240 lists the numbers that have an equal number of factorizations of even and odd length in an unordered manner.
A013929 lists the numbers that have an equal number of factorizations of even and odd length in an ordered manner.
There are infinitely many terms in this sequence since p^2 is always such a number for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.
Wilf's conjecture is equivalent to the statement that this sequence is the set difference of A319240 and A006881.

			12 = 2*6 = 3*4 = 2*2*3 (unordered) has an equal number (2) of even-length factorizations and odd-length factorizations, and 12 = 2*6 = 6*2 = 3*4 = 4*3 = 2*2*3 = 2*3*2 = 3*2*2 (ordered) has an equal number (4) of even-length factorizations and odd-length factorizations.

Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005.

Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly 123.6 (2016): 557-573.
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.

Cf. A006881, A013929, A074206, A319240, A316441, A000587.

f(n, m=n, k=0) = if(1==n, (-1)^k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += f(n/d, d, k+1))); (s)); \\ A316441
isok(m) = (f(m) == 0) && ! issquarefree(m); \\ Michel Marcus, Dec 09 2021

A350486 Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 180, 183, 185, 187, 192, 194

Offset: 1

Tian Vlasic, Jan 01 2022

First differs from A006881 at a(53) = 180.
By length, we mean the number of factors in a particular factorization.
Intersection of A319240 (factors are not necessarily distinct) and A319238 (factors are distinct).
Numbers k such that A316441(k) = A114592(k) = 0.
There are infinitely many terms in this sequence since all squarefree semiprimes (listed in A006881) are always such numbers.
There are no terms of the form p^k with p prime (listed in A000961).
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.

			6=2*3 (unrestricted) has an equal number (1) of even-length factorizations and odd-length factorizations, and 6=2*3 (distinct) has an equal number (1) of even-length factorizations and odd-length factorizations.

Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A006881, A114592, A189987, A316441, A319240, A319238.

facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Intersection @@ First@Flatten[Position[#, 0] & /@ Transpose@Table[Sum[(-1)^Length[f], {f, #}] & /@ {facs[n], Select[facs[n], UnsameQ @@ # &]}, {n, #1, #2}], {3}]&[1,194] (* Robert P. P. McKone, Jan 05 2022, from Gus Wiseman in A319238 and A319240 *)

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A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

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A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

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A349931 Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.

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A350486 Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.

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