A366251 -id:A366251 - cp's OEIS frontend

A366219 Smallest positive integer whose smallest coprime divisor shift is n.

1, 3, 91, 325, 2093, 1001, 12025, 1045, 4945, 6391, 189, 455, 245, 11825, 128843, 368809, 273, 1295, 14495, 37961, 252263, 91375, 595, 13013, 46189, 104951, 63875, 136345, 42237, 22253, 192647, 18655, 8225, 194545, 200629, 192907, 27625, 1911, 464783, 27797

Offset: 0

M. Farrokhi D. G., Oct 05 2023

nonn

A nonnegative number s is a coprime divisor shift of n if GCD(d + s, n) = 1 for all divisors d of n. The coprime divisor shift of n is the infimum of the set of all nonnegative coprime divisor shifts of n.

Conjecture. Every positive integer s is the coprime divisor shift of a positive integer.

			a(0) = 1 for GCD(1 + 0, 1) = 1.
a(1) = 3 for GCD(1 + 1, 3) = GCD(3 + 1, 3) = 1 but GCD(1 + 1, 2) > 1.
a(2) = 91 for GCD(d + 2, 91) = 1 for all divisors d = 1, 7, 13, 91 of 91, GCD(13 + 1, 91) > 1, and 91 is the smallest number with this property.

M. Farrokhi D. G., Table of n, a(n) for n = 0..1000

Cf. A366251, A366330.

isds(k,s)={fordiv(k,d,if(gcd(d+s, k)<>1, return(0))); 1}
findds(k)={for(s=0, k-1, if(isds(k,s), return(s))); -1}
a(n)={for(k=1, oo, if(isds(k,n) && findds(k)==n, return(k)))} \\ Andrew Howroyd, Oct 05 2023

A366330 Minimal numbers (with respect to division) with no coprime divisor shift.

Original entry on oeis.org

2, 15, 33, 51, 69, 87, 123, 141, 159, 177, 213, 249, 267, 303, 321, 339, 393, 411, 447, 501, 519, 537, 573, 591, 665, 681, 699, 717, 753, 771, 789, 807, 819, 843, 879, 933, 951, 1015, 1041, 1059, 1077, 1149, 1167, 1203, 1235, 1257, 1293, 1329, 1347, 1383

Offset: 1

M. Farrokhi D. G., Oct 07 2023

nonn

A number k has a coprime divisor shift s if GCD(d + s, n) = 1 for all divisors d of k.
A number k has a coprime divisor shift iff it is not divisible by any number in the sequence.
If k has no coprime divisor shift, then so is any multiple of k.

a(1) = 2 for GCD(2 + 0, 2) > 1 and GCD(1 + 1, 2) > 1.
a(2) = 15 for GCD(3 + 0, 15) > 1, GCD(5 + 1, 15) > 1, GCD(1 + 2, 15) > 1, and any odd number between 2 and 15 has a coprime divisor shift.

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000

Cf. A044135, A044516, A366219, A366251.

cp's OEIS Frontend

A366219 Smallest positive integer whose smallest coprime divisor shift is n.

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