A007520 -id:A007520 - cp's OEIS frontend

A383075 Smallest number m such that m(m + 1)(2*m + 1)/6 is divisible by n.

1, 3, 4, 7, 2, 4, 3, 15, 13, 4, 5, 8, 6, 3, 4, 31, 8, 27, 9, 7, 13, 11, 11, 31, 12, 12, 40, 7, 14, 4, 15, 63, 22, 8, 7, 40, 18, 19, 13, 15, 20, 27, 21, 16, 27, 11, 23, 31, 24, 12, 8, 32, 26, 40, 5, 31, 9, 28, 29, 40, 30, 15, 13, 127, 12, 27, 33, 8, 22, 7, 35

Offset: 1

Ctibor O. Zizka, Apr 15 2025

nonn

m*(m + 1)*(2*m + 1)/6 is divisible by n if and only if m*(m + 1)*(2*m + 1) is divisible by 6*n. - David A. Corneth, Apr 21 2025

If n > 3 is a prime of the form (8*k + 3), then a(2*n) = n. - Ctibor O. Zizka, May 21 2025

			n = 2: smallest m such that m*(m + 1)*(2*m + 1) is divisible by 2*6 is m = 3.
a(4) = 7. The first few numbers of the form m*(m + 1)*(2*m + 1)/6, m >= 1 are 1, 5, 14, 30, 55, 91, 140,... The first 6 are not divisible by 4 but the seventh is. - _David A. Corneth_, Apr 21 2025

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Cf. A000330, A007520, A011772.

a[n_]:=Module[{m=1},While[!Divisible[m(m+1)(2m+1)/6,n], m++]; m]; Array[a,71] (* Stefano Spezia, Apr 15 2025 *)

a(n) = my(m=1); while (m*(m+1)*(2*m+1)/6 % n, m++); m; \\ Michel Marcus, Apr 20 2025

PARI
```
\\ See Corneth link
```

from itertools import count
from sympy import integer_nthroot
def A383075(n): return next(m for m in count(integer_nthroot(3*n,3)[0]) if not m*(m+1)*((m<<1)+1)%(6*n)) # Chai Wah Wu, Apr 21 2025

A022763 n-th 8k+3 prime plus n-th 8k+5 prime.

Original entry on oeis.org

8, 24, 48, 80, 112, 128, 184, 216, 280, 296, 336, 360, 408, 456, 520, 560, 600, 648, 696, 752, 808, 840, 888, 952, 1008, 1064, 1104, 1176, 1224, 1248, 1296, 1344, 1368, 1416, 1448, 1512, 1584, 1632, 1656, 1712, 1760, 1848, 1944, 1984, 2040

Offset: 1

Clark Kimberling

nonn

lista(nn) = {prm = primes(nn); tp = select(p->(Mod(p,8)==3), prm); fp = select(p->(Mod(p, 8)==5), prm); for (i = 1, min(#tp, #fp), print1(tp[i] + fp[i], ", "););} \\ Michel Marcus, Sep 30 2013

a(n) = A007520(n) + A007521(n). - Michel Marcus, Sep 30 2013

A153236 Numbers n such that 8*n + 3 is not prime.

Original entry on oeis.org

3, 4, 6, 9, 11, 12, 14, 15, 18, 19, 21, 23, 24, 25, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 60, 63, 64, 66, 67, 69, 72, 74, 75, 76, 78, 79, 81, 83, 84, 87, 88, 89, 90, 91, 93, 94, 95, 96, 97

Offset: 1

Vincenzo Librandi, Dec 21 2008

			Distribution of the terms in the following triangular array:
*;
*,*;
*,4,*;
3,*,*,*;
*,*,*,12,*;
*,*,11,*,*,*;
*,9,*,*,*,24,*;
6,*,*,*,23,*,*,*;
*,*,*,21,*,*,*,40,*;
*,*,18,*,*,*,39,*,*,*;
*,14,*,*,*,37,*,*,*,60,*;
9,*,*,*,34,*,*,*,59,*,*,*; etc.
where * marks the non-integer values of (2*h*k + k + h - 1)/4 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007520, A005124, A139487.

[n: n in [0..110] | not IsPrime(8*n+3)]; // Vincenzo Librandi, Jan 12 2013

Select[Range[0, 200], !PrimeQ[8 # + 3] &] (* Vincenzo Librandi, Jan 12 2013 *)

A163184 Primes of the form 8k + 1 dividing 2^j + 1 for some odd j.

Original entry on oeis.org

281, 617, 1033, 1049, 1097, 1193, 1481, 1553, 1753, 1777, 2281, 2393, 2473, 2657, 2833, 2857, 3049, 3529, 3673, 3833, 4049, 4153, 4217, 4273, 4457, 4937, 5113, 5297, 5881, 6121, 6449, 6481, 6521, 6529, 6569, 6761, 6793, 6841, 7121, 7129, 7481, 7577, 7817, 8081, 8233, 8537, 9001, 9137, 9209, 9241

Offset: 1

Christopher J. Smyth, Jul 22 2009

Each term p has the form 2^r*j + 1, where r >= 3, j is odd, and ord_p(-2) divides j.

			281 is in the sequence as 281 = 2^3*35 + 1 and 281 | 2^35 + 1.

Set difference of A163183 and A007520.

Subsequence of A033203, A051071, A051073, A051077, A051085, A051101.

with(numtheory):A:=NULL:p:=2: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 and p mod 8 = 1 then A:=A,p;;fi;od:A;

More terms from Max Alekseyev, Sep 29 2016

A201719 Primes of the form x^2 + 2y^2 such that y^2 + 2x^2 is also prime.

Original entry on oeis.org

11, 19, 43, 59, 67, 83, 107, 139, 163, 179, 211, 251, 307, 331, 419, 443, 467, 491, 563, 571, 587, 619, 643, 811, 883, 907, 947, 971, 1019, 1091, 1123, 1171, 1259, 1291, 1307, 1427, 1531, 1571, 1579, 1667, 1699, 1747, 1787, 1811, 1907, 1979, 1987, 2003, 2011

Offset: 1

Zak Seidov, Dec 04 2011

nonn

All terms == 3 mod 8 (cf. A007520).

			Corresponding pairs of primes:
(a(1),a(2))=(11,19): 11=3^2+2*1^2, 19=1^2+2*3^2
(a(3),a(4))=(43,59): 43=5^2+2*3^2, 59=3^2+2*5^2
(a(5),a(7))=(67,107): 67=7^2+2*3^2, 107=3^2+2*7^2.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A154777.

With[{nn=50},Take[Union[Flatten[Select[{#[[1]]^2+2#[[2]]^2,2#[[1]]^2+ #[[2]]^2}&/@Subsets[Range[nn],{2}],And@@PrimeQ[#]&]]],nn]] (* Harvey P. Dale, Sep 15 2013 *)

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2n-1)(2^((n-1)/2)), 4ceiling((3/4)n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).

Original entry on oeis.org

11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283

Offset: 1

Jonas Kaiser, Nov 12 2017

nonn

It appears that A007520 is a subsequence. Up to 10^7 there are no composites in this sequence.

The first composite is a(17465859) = 1397357851; there are probably infinitely many. - Charles R Greathouse IV, Nov 12 2017

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A244626, A294717, A293394, A070179, A007520.

Select[Range[2, 1300], Function[n, AllTrue[Join[Prime[Range@3]^(n - 1), {(2 n - 1) (2^((n - 1)/2)), 4 Ceiling[3 n/4] - 2, (2^((n + 1)/2) + Floor[n/4]*2^((n + 3)/2))}], Mod[#, n] == 1 &]]] (* Michael De Vlieger, Nov 15 2017 *)

is(n) = n%2 && Mod(2, n)^(n-1)==1 && Mod(3, n)^(n-1)==1 && Mod(5, n)^(n-1)==1 && (2*n-1)*Mod(2, n)^((n-1)/2)== 1 && Mod(4*ceil((3/4)*n)-2, n)==1 && Mod(2, n)^((n+1)/2)+floor(n/4)*Mod(2, n)^((n+3)/2)==1

A346400 Composite numbers k such that the numerator of the harmonic mean of the divisors of k is equal to k.

Original entry on oeis.org

20, 21, 22, 27, 35, 38, 39, 45, 49, 55, 56, 57, 65, 68, 77, 85, 86, 93, 99, 110, 111, 115, 116, 118, 119, 125, 129, 133, 134, 143, 147, 150, 155, 161, 164, 166, 169, 183, 184, 185, 187, 189, 201, 203, 205, 207, 209, 212, 214, 215, 217, 219, 221, 235, 237, 245

Offset: 1

Amiram Eldar, Nov 01 2021

nonn

Composite numbers k such that A099377(k) = k.
Since the harmonic mean of the divisors of an odd prime p is p/((p+1)/2), its numerator is equal to p. Therefore, this sequence is restricted to composite numbers.
This sequence is infinite. For example, if p is a prime of the form 8*k+3 (A007520) with k>1, then 2*p is a term.

			20 is a term since the harmonic mean of the divisors of 20 is 20/7.

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

Intersection of A002808 and A250094.

Cf. A099377, A099378.

q[n_] := CompositeQ[n] && Numerator[DivisorSigma[0, n]/DivisorSigma[-1, n]] == n; Select[Range[250], q]

isok(k) = my(d=divisors(k)); (#d>2) && (numerator(#d/sum(i=1, #d, 1/d[i])) == k); \\ Michel Marcus, Nov 01 2021

list(lim)=my(v=List()); forfactored(n=20,lim\1, if(vecsum(n[2][,2])>1 && numerator(sigma(n,0)/sigma(n,-1))==n[1], listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 01 2021

A022764 (n-th 8k+3 prime plus n-th 8k+5 prime)/8.

Original entry on oeis.org

1, 3, 6, 10, 14, 16, 23, 27, 35, 37, 42, 45, 51, 57, 65, 70, 75, 81, 87, 94, 101, 105, 111, 119, 126, 133, 138, 147, 153, 156, 162, 168, 171, 177, 181, 189, 198, 204, 207, 214, 220, 231, 243, 248, 255, 264, 270, 277, 284, 286, 296, 309, 314, 316

Offset: 1

Clark Kimberling

nonn

lista(nn) = {prm = primes(nn); tp = select(p->(Mod(p,8)==3), prm); fp = select(p->(Mod(p, 8)==5), prm); for (i = 1, min(#tp, #fp), print1((tp[i] + fp[i])/8, ", "););} \\ Michel Marcus, Sep 30 2013

a(n) = (A007520(n) + A007521(n))/8 = A022763(n)/8. - Michel Marcus, Sep 30 2013

A035198 From a Dirichlet series.

Original entry on oeis.org

1, 9, 17, 25, 41, 73, 81, 89, 97, 113, 121, 137, 153, 169, 193, 225, 233, 241, 257, 281, 289, 313, 337, 353, 361, 369, 401, 409, 425, 433, 449, 457, 521, 569, 577, 593, 601, 617, 625, 641, 657, 673, 697, 729, 761, 769, 801, 809, 841, 857, 873, 881, 929, 937

Offset: 0

N. J. A. Sloane

Contribution from R. J. Mathar, Jul 16 2010: (Start)
The Dirichlet function is (z_1(s))^2*z_3(2*s)*z_5(2*s) = 1+ 2/9^s+4/17^s+2/25^s+4/41^s+..,
where z_1(s) = prod_{p in A007519} Zeta(s,p) = 1+2/17^s+2/41^s+2/73^s+ ...(see A004625),
z_3(s) = prod_{p in A007520} Zeta(s,p) = 1+2/3^s+2/9^s+2/11^s+2/19^s+2/27^s+4/33^s+..,
z_5(s) = prod_{p in A007521} Zeta(s,p) = 1+2/5^s+2/13^s+...+4/65^s+2/101^s+..., Zeta(s,p)=(1+p^(-s))/(1-p^(-s)). (End)

P. A. B. Pleasants, M. Baake, J. Roth, Planar coincidences for N-fold symmetry J. Math. Phys. 37 (1996) 1029.

More terms from R. J. Mathar, Jul 16 2010

More terms from Sean A. Irvine, Sep 29 2020

A100876 Least number of squares that sum to prime(n).

Original entry on oeis.org

2, 3, 2, 4, 3, 2, 2, 3, 4, 2, 4, 2, 2, 3, 4, 2, 3, 2, 3, 4, 2, 4, 3, 2, 2, 2, 4, 3, 2, 2, 4, 3, 2, 3, 2, 4, 2, 3, 4, 2, 3, 2, 4, 2, 2, 4, 3, 4, 3, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 2, 2, 4, 4, 2, 3, 4, 2, 2, 2, 2, 3, 2, 4, 2, 4, 3, 2, 2, 2, 4, 3, 4, 4, 3, 3, 4, 2, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Giovanni Teofilatto, Jan 09 2005

nonn

Note that a(n) <= 4 by Lagrange's four-square theorem. - T. D. Noe, Jan 10 2005

Primes 2 and 4k+1 (A002313) require only 2 positive squares; primes 8k+3 (A007520) require 3 positive squares; primes 8k+7 (A007522) require 4 positive squares.

			a(2)=3 because 3=1^2+1^2+1^2;
a(3)=2 because 5=1^2+2^2;
a(4)=4 because 7=2^2+1^2+1^2+1^2.

Cf. A002828 (least number of squares needed to represent n).

SquareCnt[n_] := If[SquaresR[1, n] > 0, 1, If[SquaresR[2, n] > 0, 2, If[SquaresR[3, n] > 0, 3, 4]]]; Table[p = Prime[n]; SquareCnt[p], {n, 150}] (* T. D. Noe, Jan 10 2005, revised Sep 27 2011 *)

a(n) = A002828(prime(n)) - T. D. Noe, Jan 10 2005

More terms from T. D. Noe, Jan 10 2005

cp's OEIS Frontend

A383075 Smallest number m such that m*(m + 1)*(2*m + 1)/6 is divisible by n.

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A022763 n-th 8k+3 prime plus n-th 8k+5 prime.

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PARI

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A153236 Numbers n such that 8*n + 3 is not prime.

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Magma

Mathematica

A163184 Primes of the form 8k + 1 dividing 2^j + 1 for some odd j.

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Maple

Extensions

A201719 Primes of the form x^2 + 2y^2 such that y^2 + 2x^2 is also prime.

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Mathematica

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2*n-1)*(2^((n-1)/2)), 4*ceiling((3/4)*n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).

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PARI

A346400 Composite numbers k such that the numerator of the harmonic mean of the divisors of k is equal to k.

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Mathematica

PARI

PARI

A022764 (n-th 8k+3 prime plus n-th 8k+5 prime)/8.

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Author

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Programs

PARI

Formula

A383075 Smallest number m such that m(m + 1)(2*m + 1)/6 is divisible by n.

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2n-1)(2^((n-1)/2)), 4ceiling((3/4)n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).