A102838 -id:A102838 - cp's OEIS frontend

A374459 Nonsquarefree exponentially odd numbers.

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536

Offset: 1

Amiram Eldar, Jul 09 2024

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.
Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.
All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.
The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

			8 = 2^3 is a term since 3 is odd and larger than 1.

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

Intersection of A013929 (or A046099) and A268335.
Subsequence of A301517.
Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.
Cf. A005117, A374460.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]

is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

a(n) = A268335(A374460(n)).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s))) - zeta(s)/zeta(2*s) for s > 1.

A102836 Composite numbers whose exponents in their canonical factorization lie in the geometric progression 1, 2, 4, ...

Original entry on oeis.org

18, 50, 75, 98, 147, 242, 245, 338, 363, 507, 578, 605, 722, 845, 847, 867, 1058, 1083, 1183, 1445, 1587, 1682, 1805, 1859, 1922, 2023, 2523, 2527, 2645, 2738, 2883, 3179, 3362, 3698, 3703, 3757, 3971, 4107, 4205, 4418, 4693, 4805, 5043, 5547, 5618, 5819

Offset: 1

Cino Hilliard, Feb 27 2005

The first term not in A095990 is a(70) = 11250.

			Canonical factorization of a(70) = 11250 = 2^1 * 3^2 * 5*4 or 2,3,5 raised to powers 1,2,4 which is a geometric progression.

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

Cf. A095990, A102838.

q[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Length[e] > 1 && e == 2^Range[0, Length[e]-1]]; Select[Range[6000], q] (* Amiram Eldar, Jun 29 2024 *)

/* Numbers whose factors are primes to perfect powers in a geometric progression. */ geoprog(n,m) = { local(a,x,j,nf,fl=0); for(x=1,n, a=factor(x); nf=omega(x); for(j=1,nf, if(a[j,2]==2^(j-1),fl=1,fl=0;break); ); if(fl&nf>1,print1(x",")) ) }

is(n) = if(n == 1 || isprime(n), 0, my(e = factor(n)[, 2]); for(i = 1, #e, if(e[i] != 2^(i-1), return(0))); 1); \\ Amiram Eldar, Jun 29 2024

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A374459 Nonsquarefree exponentially odd numbers.

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A102836 Composite numbers whose exponents in their canonical factorization lie in the geometric progression 1, 2, 4, ...

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Mathematica

PARI

PARI