The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A354532 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A354532

Numbers k that are not Mersenne exponents (A000043) such that 2*(2^k-1) is in A354525.
(history; published version)

#24 by Michael De Vlieger at Tue Jan 21 09:01:45 EST 2025

STATUS

reviewed

approved

#23 by Joerg Arndt at Tue Jan 21 08:50:15 EST 2025

STATUS

proposed

reviewed

#22 by Jinyuan Wang at Tue Jan 21 05:59:58 EST 2025

STATUS

editing

proposed

#21 by Jinyuan Wang at Tue Jan 21 05:59:28 EST 2025

DATA

1, 9, 67, 137, 727

COMMENTS

2^a(n) - 1 is a semiprime for n = 2,3,4,5.

CROSSREFS

Cf. A354525, A000043, A354525, A354531, A354534, A354537.

EXTENSIONS

a(5) from Jinyuan Wang, Jan 21 2025

STATUS

approved

editing

#20 by Michael De Vlieger at Wed Aug 17 10:14:59 EDT 2022

STATUS

proposed

approved

#19 by Jianing Song at Wed Aug 17 06:18:53 EDT 2022

STATUS

editing

proposed

#18 by Jianing Song at Wed Aug 17 06:17:46 EDT 2022

PROG

isA354532(n, {lim=256}, {p_lim=1<<32}) = {

my(N=2^n-1);

if(isprime(N), return(0));

isA354532 if(n, {lim=1<<256}, {p_>lim=1<<32}) = if(n==1, return(1)); my(N=2^n-1); if(isprime(N), return(0)); , forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p, p), return(0))); my(d=divisors(n)); for(i=1, #d, my(f=factor(2^d[i]-1)[, 1]); for(j=1, #f, if(!ispsmooth(2*N+f[j], f[j], lim), return(0)))); 1 \\ first check if there is a prime factor p <= p_lim of 2^n-1 such that 2*(2^n-1)+p is not p-smooth, then check if 2* (2^n-1)+p is p-smooth for p|2^d-1, d|Nlarge n)

my(d=divisors(n));

for(i=1, #d, my(f=factor(2^d[i]-1)[, 1]); for(j=1, #f, if(!ispsmooth(2*N+f[j], f[j], 1<<lim), return(0)))); 1 \\ then check if 2*(2^n-1)+p is p-smooth for p|2^d-1, d|N

}

STATUS

proposed

editing

#17 by Jianing Song at Wed Aug 17 06:06:12 EDT 2022

STATUS

editing

proposed

#16 by Jianing Song at Wed Aug 17 06:04:20 EDT 2022

PROG

isA354532(n, {lim=1<<256}, {p_lim=1<<32}) = if(n==1, return(1)); my(N=2^n-1); if(isprime(N), return(0)); forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p, p), return(0))); my(d=divisors(n)); for(i=1, #d, my(f=factor(2^d[i]-1)[, 1]); for(j=1, #f, if(!ispsmooth(2*N+f[j], f[j], lim), return(0)))); 1 \\ first check if there is a prime factor p < = p_lim of 2^n-1 such that 2*(2^n-1)+p is not p-smooth, then check if 2*(2^n-1)+p is p-smooth for p|2^d-1, d|N

#15 by Jianing Song at Wed Aug 17 06:01:18 EDT 2022

PROG

ispsmooth(n, p, {lim=1<<256}) = if(n<=lim, n==1 || gpf(n) <= p, my(N=n/p^valuation(n, p)); forprime(q=2, p, N=N/q^valuation(N, q)); if((N<=lim && isprime(N)) || N==1, return(N<=p))); 0); \\ check if n is p-smooth, using brute force if n is too large

isA354532(n, {lim_=1<<256}, {p_lim=10001<<32}) = if(n==1, return(1)); my(N=2^n-1); if(isprime(N), return(0)); forprime(p=3, p_lim_p, , if(N%p==0 && !ispsmooth(2*N+p, p), return(0))); my(d=divisors(n)); for(i=1, #d, my(f=factor(N)2^d[, i]-1]); if(#f==[, 1, return(0)]); for(ij=1, #f, if(gpf!ispsmooth(2*N+f[ij])!=, f[ij], lim), return(0)))); 1 \\ first check if there is a small prime factor p <= p_lim_p of 2^n-1 such that 2*(2^n-1)+p is not p-smooth, then check if 2*(2^n-1)+p is p-smooth for p|2^d-1, d|N