Emmanuel Vantieghem - cp's OEIS frontend

A283418 Numbers n such that n and n+1 are primitive abundant.

82004, 158235, 326864, 442035, 516704, 1102724, 1606275, 2151435, 2697435, 2912084, 2921535, 2979675, 3002804, 3241755, 3647475, 4322835, 5801984, 5905844, 6069195, 7251075, 7387604, 7553924, 8272124, 8788724, 9292724, 9909584

Offset: 1

Emmanuel Vantieghem, May 02 2017

nonn

Intersection of A091191 and -1 + A091191.

			82004 is in the sequence because it is abundant (sum divisors = 164640, > 2*82004) and 82005 is also abundant (sum divisors = 165888, > 2*82005).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A091191, A005105, A096399.

fQ[m_] := DivisorSigma[1, m] > 2 m;
gQ[m_] := fQ[m] && Union[fQ /@ Rest[Most[Divisors[m]]]] == {False};
V = Select[Range[10^7], gQ]; Intersection[V, V - 1]

A283658 Numbers d > 1 such that the class number of Q(sqrt(d)) is strictly greater than the class number of Q(sqrt(m)) for all m < d.

Original entry on oeis.org

10, 79, 82, 226, 730, 1534, 2305, 3601, 4762, 5626, 11026, 21610, 23410, 27226, 38026, 50626, 116554, 164026, 176401, 189226, 342226, 345745, 411394, 518401, 540226, 613090, 804610, 893026, 1071226, 1199026, 1299601, 1334026, 1550026, 2205226, 2433601, 2873026, 3515626, 3920401

Offset: 1

Emmanuel Vantieghem, Mar 13 2017

nonn

Every element d of the sequence is squarefree because, if f is the squarefree part of d, then Q(sqrt(f)) = Q(sqrt(d)). If f would be < d, the class number of Q(sqrt(f)) would not be < the class number of Q(sqrt(d)). Thus, f = d.

			The sequence starts with 10 because the class number of Q(sqrt(10)) = 2 and all fields Q(sqrt(m)) with m < 10 have class number 1.
The next term is 79 because the class number of Q(sqrt(79)) is 3 and all fields Q(sqrt(m)) with m < 79 have class number 1 or 2.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Robin Visser, Table of n, a(n) for n = 1..50

Cf. A003172, A003649, A283659.

A={}; hx = 1; d = 2; While[hx<300, d++; If[SquareFreeQ[d], h = NumberFieldClassNumber[Sqrt[d]]; If[h > hx, AppendTo[A,d]; hx = h]]]; A

classn(n) = qfbclassno(if(n%4>1, 4, 1)*n);
isok(d) = {if (issquarefree(d), cld = classn(d); for (k=2, d-1, if (issquarefree(k) && (classn(k) >= cld), return (0))); 1;);} \\ Michel Marcus, Mar 13 2017

More terms from Robin Visser, May 25 2024

A283659 Class numbers of the fields Q(sqrt(A283658(n))).

Original entry on oeis.org

2, 3, 4, 8, 12, 14, 16, 20, 22, 28, 44, 48, 52, 58, 74, 96, 116, 130, 153, 154, 176, 180, 200, 230, 240, 256, 288, 296, 312, 316, 357, 394, 412, 452, 504, 540, 574, 575, 584, 616, 692, 924, 994, 1061, 1068, 1080, 1245, 1248, 1302, 1336

Offset: 1

Emmanuel Vantieghem, Mar 13 2017

			The sequence starts with 2 because the first number in A283658 is 10 and the class number of Q(sqrt(10)) equals 2.
The fifth term is 12 because A283658(5) = 226 and the class number of Q(sqrt(226)) is 12.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Cf. A283658, A003649, A003172.

H = {}; hx = 1; d = 2; While[hx < 5, d++;
If[SquareFreeQ[d], h = NumberFieldClassNumber[Sqrt[d]];
  If[h > hx, AppendTo[H, h]; hx = h]]]; H

a(30)-a(50) from Robin Visser, May 25 2024

A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

Original entry on oeis.org

17, 33, 37, 41, 57, 65, 73, 89, 97, 101, 105, 113, 129, 137, 141, 145, 161, 177, 185, 193, 197, 201, 209, 217, 233, 241, 249, 257, 265, 269, 273, 281, 305, 313, 321, 329, 337, 345, 349, 353, 373, 377, 381, 385, 389, 393, 401, 409, 417, 433, 449, 457, 465, 473, 481, 485, 489, 497, 505, 521, 537, 545, 553, 557, 561, 569, 573

Offset: 1

Emmanuel Vantieghem, Mar 07 2017

nonn

Squarefree integers m congruent to 1 modulo 4 such that the minimal solution of the Pell equation x^2 - d*y^2 = +-4 has both x and y even.
The sequence contains the squarefree numbers congruent to 5 modulo 8 that are not in A107997.
This sequence union A107997 = A039955.
This sequence contains all numbers of the form 4*k^2+1 (k > 1) that are squarefree.

			33 is in the sequence since the fundamental unit of the field Q(sqrt(33)) is 23+4*sqrt(33).
53 is not in the sequence since the fundamental unit of the field Q(sqrt(53)) is 3+omega, where omega = (1+sqrt(53))/2.

Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

Keith Matthews, Finding the fundamental unit of a real quadratic field

Cf. A107997, A039955, A107998, A197170.

A283105 Numbers that are an integer multiple of the mean of their smallest and largest nontrivial divisors.

Original entry on oeis.org

4, 9, 12, 25, 45, 49, 121, 169, 289, 361, 529, 637, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 13357, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521

Offset: 1

Emmanuel Vantieghem, Feb 28 2017

nonn

No prime is in the sequence since there are no nontrivial divisors of a prime.
The sequence includes every number that is the square of a prime.
It is easy to show that the other terms are of the form (2p-1)*p^2 where p and 2p-1 are prime. Therefore, the mean of the two divisors in question is always an integer.

			4 is in the sequence because its smallest nontrivial divisor is 2, its largest nontrivial divisor is 2, and their mean is 2.
45 is in the sequence because its smallest nontrivial divisor is 3, its largest nontrivial divisor is 15, and their mean is 9, a divisor of 45.
10 is not in the sequence because it is not an integral multiple of 7/2, the mean of 2 and 5.

Harvey P. Dale, Table of n, a(n) for n = 1..600

Cf. A005382, A088595.

mslndQ[n_]:=Module[{d=Divisors[n]},Divisible[n,Mean[{d[[2]],d[[-2]]}]]]; Select[Range[2,50000],mslndQ] (* Harvey P. Dale, Jul 24 2017 *)

is(n) = my(d=divisors(n), m=(d[2]+d[#d-1])/2); if(n%m==0, 1, 0) \\ Felix Fröhlich, Feb 28 2017

A280965 Nonsquares whose distances to the two nearest squares are squares.

Original entry on oeis.org

5, 8, 40, 45, 65, 80, 153, 160, 200, 221, 325, 360, 416, 425, 493, 520, 680, 725, 925, 936, 1025, 1040, 1073, 1088, 1305, 1360, 1768, 1800, 1813, 1845, 1961, 2000, 2320, 2385, 2501, 2600, 2925, 3016, 3185, 3200, 3400, 3445, 3848, 3869, 3944, 3965, 4640, 4745, 5185, 5248, 5265, 5328, 5525, 5576, 5785, 5920, 6120

Offset: 1

Emmanuel Vantieghem, Feb 27 2017

nonn

The sequence is infinite because there are terms of it between n^2 and (n+1)^2 whenever 2n+1 is a sum of two squares.

			a(3) = 40 because the two nearest squares are 36 and 49 and 40 - 36 = 4, 49 - 40 = 9 are both squares.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A057653, A234334.

Select[Range[6120], IntegerQ[Sqrt[# - (Floor[Sqrt[#]])^2]] && IntegerQ[Sqrt[(Ceiling[Sqrt[#]])^2 - #]] &]

is(n)=my(k=sqrtint(n)); issquare(n-k^2) && issquare((k+1)^2-n) && n>k^2 \\ Charles R Greathouse IV, Feb 27 2017

list(lim)=my(v=List(),k2,K2,n); for(k=2,sqrtint(lim\1)-1, k2=k^2; K2=(k+1)^2; for(s=1,sqrtint(K2-k2-1), n=k2+s^2; if(issquare(K2-n), listput(v,n)))); k2=sqrtint(lim\1)^2; K2=(sqrtint(lim\1)+1)^2; for(n=k2+1,lim, if(issquare(n-k2) && issquare(K2-n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Feb 27 2017

A282668 Numbers m whose greatest divisor <= sqrt(m) is prime.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 18, 21, 22, 25, 26, 27, 30, 33, 34, 35, 38, 39, 40, 45, 46, 49, 50, 51, 55, 56, 57, 58, 62, 63, 65, 69, 70, 74, 75, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 98, 105, 106, 111, 115, 118, 119, 121, 122, 123, 125, 129, 132, 133, 134

Offset: 1

Emmanuel Vantieghem, Feb 20 2017

nonn

The squares of the primes are in the sequence.

			15 is a term since its biggest divisor <= sqrt(15) is 3 (this is a not sqrt(15)-smooth example).
18 is a term since its biggest divisor <= sqrt(18) is 3 (this is a sqrt(18)-smooth example).
24 is not a term since its biggest divisor <= sqrt(24) is 4 (this is a sqrt(24)-smooth counterexample).
42 is not a term since its biggest divisor <= sqrt(42) is 6 (this is a not sqrt(42)-smooth counterexample).

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

Cf. A048098, A064052.

f[m_]:=Module[{A=Divisors[m],a},a=Length[A];A[[Floor[(a+1)/2]]]];
Select[Range[176],PrimeQ[f[#]]&]

{n: A033676(n) in A000040}. - R. J. Mathar, Feb 23 2017

A282246 Primes p such that the sum of all primes <= p has no prime divisor > p.

Original entry on oeis.org

2, 5, 11, 19, 23, 31, 41, 47, 59, 71, 83, 97, 101, 103, 109, 113, 127, 137, 157, 163, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 241, 263, 269, 271, 317, 337, 349, 353, 367, 389, 401, 409, 433, 439, 449, 457, 461, 463, 467, 491, 521, 563, 571, 607, 613, 617, 631, 641, 653, 661, 701, 709, 719, 739, 757, 797

Offset: 1

Emmanuel Vantieghem, Feb 09 2017

nonn

Number of terms < 10^k: 2, 12, 79, 523, 4124, 32678, 267850, etc. Compare these to A006880. - Robert G. Wilson v, Feb 09 2017

Primes p such that A006530(A007504(i)) <= p, where i is the index of p in A000040. - Felix Fröhlich, Feb 12 2017

			5 is in the sequence for the sum of all primes <= 5 is 10, and 10 has no prime divisor > 5.
17 is not in the sequence for the corresponding sum is 58 which has a prime divisor > 17.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A007504, A013916, A013917, A013918, A046731.

p = s = 2; lst = {}; While[p < 1000, If[ FactorInteger[s][[-1, 1]] <= p, AppendTo[lst, p]]; p = NextPrime@ p; s = s + p]; lst (* Robert G. Wilson v, Feb 09 2017 *)

isok(n) = isprime(n) && (vecmax(factor(sum(k=1, primepi(n), prime(k)))[,1]) <= n); \\ Michel Marcus, Feb 12 2017

A280991 Positive integers that can be expressed as the sum of four pairwise coprime squares.

Original entry on oeis.org

3, 4, 7, 12, 15, 19, 27, 28, 31, 36, 39, 43, 51, 52, 55, 60, 63, 67, 75, 76, 79, 84, 87, 91, 99, 103, 108, 111, 115, 123, 124, 127, 132, 135, 139, 147, 148, 151, 156, 159, 163, 171, 172, 175, 180, 183, 187, 195, 196, 199, 204, 207, 211, 219, 220, 223, 228, 231, 235, 243, 244, 247

Offset: 1

Emmanuel Vantieghem, Jan 12 2017

nonn

If n is in the sequence, then n == 0 or 1 mod 3 and n == 3, 4, or 7 mod 8. But the converse is not true: 100 and 268 are not in the sequence (are there other examples?).

Guy [op. cit.] quotes Paul Turan as asking for a characterization of the terms of this sequence. - N. J. A. Sloane, Jan 16 2017

			3 is in the sequence, since 3 is the sum of the squares of 0, 1, 1, 1 and these four numbers are pairwise coprime.
7 is in the sequence, since 7 is the sum of the squares of 1, 1, 1, 2 and these four numbers are pairwise coprime.

R. K. Guy, Unsolved Problems in Theory of Numbers, Section C20

Jean-François Alcover, Table of n, a(n) for n = 1..1000

f[A_]:=Module[{A2, La2},A2=Subsets[A,{2}];La2=Length[A2];Union[Table[GCD@@A2[[i]],{i,1,La2}]]=={1}];
Select[Range[250],MemberQ[Union[f/@PowersRepresentations[#,4,2]],True]&]

A263404 Smallest square containing the first n primes as substrings.

Original entry on oeis.org

25, 324, 3025, 35721, 11730625, 71132356, 1132591716, 17136119025, 1923311317225, 58191311792329, 58191311792329, 231372178511929, 1722376411319529, 1134152319174379129, 472643754131992311729, 17298113195343723473641, 419723711317595314724329, 4319231361106417537475929

Offset: 1

Emmanuel Vantieghem, Oct 17 2015

The sequence is infinite.

			a(6) = 71132356 = 8464^2 contains 2, 3, 5, 7, 11 and 13 as substrings and no smaller square has that property.

Cf. A054261, A069600.

Do[k = 1; While[! AllTrue[Prime@ Range@ n, StringContainsQ[ToString[k^2], ToString@ #] &], k++]; Print[k^2], {n, 9}] (* Michael De Vlieger, Oct 19 2015, Version 10 *)

a(15)-a(16) from Bert Dobbelaere, Oct 28 2018

a(17)-a(18) from Giovanni Resta, Aug 27 2019

cp's OEIS Frontend

User: Emmanuel Vantieghem

A283418 Numbers n such that n and n+1 are primitive abundant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A283658 Numbers d > 1 such that the class number of Q(sqrt(d)) is strictly greater than the class number of Q(sqrt(m)) for all m < d.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A283659 Class numbers of the fields Q(sqrt(A283658(n))).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Extensions

A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A283105 Numbers that are an integer multiple of the mean of their smallest and largest nontrivial divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A280965 Nonsquares whose distances to the two nearest squares are squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A282668 Numbers m whose greatest divisor <= sqrt(m) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A282246 Primes p such that the sum of all primes <= p has no prime divisor > p.

Views