A187039 -id:A187039 - cp's OEIS frontend

A187042 Numbers the expansion of which over distinct terms of A050376 contains the same number of primes and perfect squares.

8, 12, 18, 20, 27, 28, 32, 44, 45, 48, 50, 52, 63, 68, 75, 76, 80, 92, 98, 99, 112, 116, 117, 124, 125, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 207, 208, 212, 216, 236, 242, 243, 244, 245, 261, 268, 272, 275, 279, 284, 292, 304, 316, 325, 332, 333, 338

Offset: 1

Vladimir Shevelev, Mar 02 2011

nonn

The sequence does not contain squarefree numbers or perfect squares.

Initially the sequence matches A378494 (the intersection of A000379 and A026424). The first differences are the absence here of 120 and 168 and the inclusion here of 216. - Peter Munn, Jul 13 2024 (edited by Paolo Xausa, Nov 29 2024).

			147 and 216 are in the sequence, since their expansions over distinct terms of A050376 are 3*49 and 2*3*4*9 respectively. Thus the expansion of 147 contains one prime and one perfect square, while the expansion of 216 contains two primes and two perfect squares.

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

Cf. A050376, A187039, A378494.

aQ[n_] := Total @ (d = IntegerDigits[Last /@ FactorInteger[n], 2])[[;; , -1]] == Total @ Flatten @d / 2; Select[Range[350], aQ] (* Amiram Eldar, Oct 01 2019 *)

a(28)=153 inserted and more terms added by Amiram Eldar, Oct 01 2019

A371599 Numbers of least prime signature (A025487) whose prime factorization has equal number of even and odd exponents.

Original entry on oeis.org

1, 12, 48, 72, 192, 288, 432, 768, 1152, 1260, 1728, 2592, 3072, 4608, 5040, 6912, 10368, 12288, 12600, 15552, 18432, 20160, 27648, 41472, 45360, 49152, 50400, 62208, 73728, 75600, 80640, 93312, 110592, 165888, 181440, 196608, 201600, 248832, 264600, 294912, 302400

Offset: 1

Amiram Eldar, Mar 29 2024

nonn

			The prime signatures of the first 12 terms are:
   n     a(n)     signature  A162641(a(n)) = A162642(a(n))
  --  -------  ------------  -----------------------------
   1        1            {}                              0
   2       12         {2,1}                              1
   3       48         {4,1}                              1
   4       72         {3,2}                              1
   5      192         {6,1}                              1
   6      288         {5,2}                              1
   7      432         {4,3}                              1
   8      768         {8,1}                              1
   9     1152         {7,2}                              1
  10     1260     {2,2,1,1}                              2
  11     1728         {6,3}                              1
  12     2592         {5,4}                              1

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

Intersection of A025487 and A187039.

Cf. A162641, A162642, A371600.

fun[p_, e_] := (-1)^e; q[n_] := Module[{f = FactorInteger[n]}, n == 1 || (f[[-1, 1]] == Prime[Length[f]] && Max@ Differences[f[[;; , 2]]] < 1 && Plus @@ fun @@@ f == 0)]; Select[Range[3*10^5], q]

is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); n == 1 || (prime(#p) == p[#p] && e == vecsort(e, , 4) && sum(i = 1, #e, (-1)^e[i]) == 0);}

A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

Original entry on oeis.org

126, 468, 624, 792, 880, 1056, 1150, 2900, 3264, 4606, 5824, 6375, 6624, 8320, 9856, 10388, 11375, 12798, 13650, 16400, 16704, 19250, 20925, 30135, 32625, 36720, 39150, 39900, 53784, 56446, 56925, 57000, 59500, 63455, 65520, 71400, 71500, 72471

Offset: 1

Vladimir Shevelev, Apr 18 2014

nonn

The corresponding sequence of the sum over the primes, which equals the sum over the nonprimes, is 9, 13, 16, 13, 16, 16, 25, 29, 20, 49, 20, 25, 25, 20, 20, 53, 25, 81, 25, 41, 29, 25, 34, 49, 34, 25, 34, 29, 85, 169, 34, 29, 29, 49, 25, 29, 29, 49, ... - Wolfdieter Lang, Apr 25 2014

			126 and 468 are in the sequence since the factorizations are 2*7*9 and 4*9*13 respectively, and 2+7=9, 4+9=13.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A187039, A187042, A177329, A177333, A177334.

More terms from Peter J. C. Moses, Apr 18 2014

New extension from Wolfdieter Lang, Apr 25 2014

cp's OEIS Frontend

A187042 Numbers the expansion of which over distinct terms of A050376 contains the same number of primes and perfect squares.

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A371599 Numbers of least prime signature (A025487) whose prime factorization has equal number of even and odd exponents.

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PARI

A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

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