A354525 -id:A354525 - cp's OEIS frontend

A354532 Numbers k that are not Mersenne exponents (A000043) such that 2*(2^k-1) is in A354525.

1, 9, 67, 137, 727

Offset: 1

Jianing Song, Aug 16 2022

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

Conjecture: all terms beyond a(2) = 9 are primes.

			k = 9: 2^9 - 1 = 7*73 (not a prime), and we have 2*(2^9-1) + 7 = 7^3 is 7-smooth and 2*(2^9-1) + 73 = 3*5*73 is 73-smooth, so 9 is a term.
k = 67: 2^67 - 1 = 193707721*761838257287 (not a prime), and we have 2*(2^67-1) + 193707721 = 3*5^2*16033*1267117*193707721 is 193707721-smooth and 2*(2^67-1) + 761838257287 = 3*5011*25771*761838257287 is 761838257287-smooth, so 67 is a term.
k = 137: 2^137 - 1 = 32032215596496435569*5439042183600204290159 (not a prime), and we have 2*(2^137-1) + 32032215596496435569 = 379*28702069570449626861*32032215596496435569 is 32032215596496435569-smooth and 2*(2^137-1) + 5439042183600204290159 = 9007*7112738002996877*5439042183600204290159 is 5439042183600204290159-smooth, so 137 is a term.

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

gpf(n) = vecmax(factor(n)[, 1]);
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=256},{p_lim=1<<32}) = {
  my(N=2^n-1);
  if(isprime(N), return(0));
  if(n>lim, forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p,p), return(0)))); \\ 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 (for large 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<

By definition, equals A354531 \ A000043.

a(5) from Jinyuan Wang, Jan 21 2025

A354526 Numbers k such that A354512(k) < omega(k); complement of A354525.

Original entry on oeis.org

4, 8, 10, 12, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 56, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117

Offset: 1

Jianing Song, Aug 16 2022

Numbers k such that there is a prime factor p of k such that gpf(k+p) != p.

Numbers k such that there is a prime factor p of k such that k+p is not p-smooth.

			57 is a term since the prime factors of 57 are 3,19, and we have gpf(57+3) != 3.

Jianing Song, Table of n, a(n) for n = 1..7946 (all terms <= 10000)

Cf. A001221, A354512, A006530. Indices of positive terms in A354527.

gpf(n) = vecmax(factor(n)[, 1]);
isA354526(n) = my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])!=f[i], return(1))); 0

A354531 Numbers k such that 2*(2^k-1) is in A354525.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 17, 19, 31, 61, 67, 89, 107, 127, 137, 521, 607, 727

Offset: 1

Jianing Song, Aug 16 2022

Numbers k such that for every prime factor p of 2^k-1 we have gpf(2*(2^k-1)+p) = p.
Numbers k such that for every prime factor p of 2^k-1, 2*(2^k-1)+p is p-smooth.
All terms except 2 are odd: if k is even, then 3 is a factor of 2^k-1, so 3^m = 2*(2^k-1)+3 = 2^(k+1) + 1 => k+1 >= 3^(m-1). The only possible case is (k,m) = (2,2).
Clearly A000043 is a subsequence. The exceptional terms (1, 9, 67, 137, ...) are listed in A354532.
The next term is >= 349. The next composite term, if it exists, is >= 7921 = 89^2.

			See A354532.

Cf. A000043, A354525, A354532, A354533, A354536.

gpf(n) = vecmax(factor(n)[, 1]);
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
isA354531(n,{lim=256},{p_lim=1<<32}) = {
  my(N=2^n-1);
  if(isprime(N), return(1));
  if(n>lim, forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p,p), return(0)))); \\ 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 (for large 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<

a(17)-a(19) from Jinyuan Wang, Jan 21 2025

A354533 Even terms in A354525.

Original entry on oeis.org

2, 6, 14, 62, 254, 1022, 16382, 262142, 1048574, 4294967294, 4611686018427387902, 295147905179352825854, 1237940039285380274899124222, 324518553658426726783156020576254, 340282366920938463463374607431768211454, 348449143727040986586495598010130648530942

Offset: 1

Jianing Song, Aug 16 2022

Even numbers k such that for every prime factor p of k we have gpf(k+p) = p, gpf = A006530.

Even numbers k such that for every prime factor p of k, k+p is p-smooth.

			See A354532.

Cf. A006530, A354525, A354531, A354532, A354534, A354536.

lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354531(n,lim,lim_p), print1(2*(2^n-1), ", "))) \\ See A354531 for the function isA354531

a(n) = 2*(2^A354531(n) - 1).

A354534 Even terms in A354525 that are not twice the Mersenne primes (A000668).

Original entry on oeis.org

2, 1022, 295147905179352825854, 348449143727040986586495598010130648530942

Offset: 1

Jianing Song, Aug 16 2022

Terms in A354533 that are not twice the Mersenne primes. Note that all twice the Mersenne primes are in A354533.

			See A354532.

Cf. A006530, A354525, A354532, A354533, A354537.

lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354532(n,lim,lim_p), print1(2*(2^n-1), ", "))) \\ See A354532 for the function isA354532

a(n) = 2*(2^A354532(n) - 1) = 2*A354537(n).

A354536 Numbers k such that 2*k is in A354525.

Original entry on oeis.org

1, 3, 7, 31, 127, 511, 8191, 131071, 524287, 2147483647, 2305843009213693951, 147573952589676412927, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 174224571863520493293247799005065324265471

Offset: 1

Jianing Song, Aug 17 2022

Numbers k such that for every prime factor p of k we have gpf(2*k+p) = p, gpf = A006530.
Numbers k such that for every prime factor p of k, 2*k+p is p-smooth.
a(17) = 2^521 - 1 is too large to include here. - Jinyuan Wang, Jan 21 2025

			See A354532.

Cf. A006530, A354525, A354531, A354532, A354533, A354537.

lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354531(n,lim,lim_p), print1(2^n-1, ", "))) \\ See A354531 for the function isA354531

a(n) = 2^A354531(n) - 1 = A354533(n)/2.

A354537 Numbers k that are not Mersenne primes (A000668) such that 2*k is in A354525.

Original entry on oeis.org

1, 511, 147573952589676412927, 174224571863520493293247799005065324265471

Offset: 1

Jianing Song, Aug 17 2022

Terms in A354536 that are not Mersenne primes. Note that all Mersenne primes are in A354536.

a(5) = 2^727 - 1 is too large to include here. - Jinyuan Wang, Jan 21 2025

			See A354532.

Cf. A000668, A354525, A354532, A354534, A354536.

lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354532(n,lim,lim_p), print1(2^n-1, ", "))) \\ See A354532 for the function isA354532

By definition, equals A354536 \ A000668.

a(n) = 2^A354532(n) - 1 = A354534(n)/2.

A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 2, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2

Offset: 1

Jianing Song, Aug 16 2022

Number of primes p such that gpf(n+p) = p (such p must be prime factors of n).
Number of distinct prime factors p of n such that n+p is p-smooth.
Clearly we have a(n) <= omega(n) for all n, omega = A001221. The differences are given by A354527.
Is this sequence unbounded? Note that 4 does not appear until a(1660577).

			a(78) = 2 since the prime factors of 78 are 2,3,13, and we have gpf(78+3) = 3 and gpf(78+13) = 13, so the solutions to m - gpf(m) = 78 are m = 78+3 = 81 or m = 78+13 = 91. Note that gpf(78+2) != 2.
a(12) = 0 since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A006530, A076563, A001221, A354516 (indices of first occurrence of each number), A354527.

Cf. A354514 (0 together with indices of positive terms), A354515 (indices of 0), A354516, A354525 (indices n for which a(n) reaches omega(n)), A354526 (indices n for which a(n) is smaller than omega(n)).

gpf(n) = vecmax(factor(n)[, 1]);
a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i])

A354527 a(n) = A001221(n) - A354512(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 1, 0, 3, 0

Offset: 1

Jianing Song, Aug 16 2022

Number of distinct prime factors p of n such that gpf(n+p) != p, gpf = A006530.

Number of distinct prime factors p of n such that n+p is not p-smooth.

			a(30) = 2 since the prime factors of 30 are 2,3,5, and we have gpf(30+3) != 3 and gpf(30+5) != 5.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A001221, A354512, A006530.

Cf. A354525 (indices of 0), A354526 (indices of positive terms).

gpf(n) = vecmax(factor(n)[, 1]);
a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])!=f[i])

A354516 Smallest k such that m - gpf(m) = k has exactly n solutions m >= 2, gpf = A006530; or -1 if no such k exists.

Original entry on oeis.org

1, 2, 6, 483, 1660577

Offset: 0

Jianing Song, Aug 16 2022

Smallest k such that there are exactly n primes p such that gpf(k+p) = p (such p must be prime factors of k).
Smallest k having exactly n distinct prime factors p such that k+p is p-smooth.
Conjectures (if no term equals -1): (Start)
(1) Sequence is strictly increasing.
(2) All terms are squarefree.
(3) All terms are in A354525. (End)

			a(4) = 1660577: 1660577 = 17*23*31*127, and we have 1660577+17 = 2*13^2*17^3 is 17-smooth, 1660577+23 = 2^3*5^2*19^2*23 is 23-smooth, 1660577+31 = 2^6*3^3*31^2 is 31-smooth, 1660577+137 = 2*11*19*29*137, so m - gpf(m) = 1660577 has 4 solutions m = 1660577+17 = 1660594, 1660577+23 = 1660600, 1660577+31 = 1660608, and 1660577+137 = 1660714.

Cf. A006530, A076563, A354512, A354525.

gpf(n) = vecmax(factor(n)[, 1]);
A354512(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i]);
a(n) = my(k=1); while(omega(k)A354512(k) != n, k++); return(k)

cp's OEIS Frontend

A354532 Numbers k that are not Mersenne exponents (A000043) such that 2*(2^k-1) is in A354525.

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A354526 Numbers k such that A354512(k) < omega(k); complement of A354525.

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A354531 Numbers k such that 2*(2^k-1) is in A354525.

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A354533 Even terms in A354525.

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A354534 Even terms in A354525 that are not twice the Mersenne primes (A000668).

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A354536 Numbers k such that 2*k is in A354525.

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A354537 Numbers k that are not Mersenne primes (A000668) such that 2*k is in A354525.

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A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

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