Richard Peterson - cp's OEIS frontend

A377266 Primes p with the property that there exist nonnegative integers m,n such that m!*n! is congruent to either +1 or -1 mod p^2, with m + n = p - 1.

2, 3, 5, 7, 11, 13, 17, 31, 47, 53, 59, 61, 71, 107, 137, 149, 173, 227, 251, 277, 313, 347, 349, 359, 367, 373, 409, 419, 443, 463, 467, 479, 491, 499, 521, 523, 541, 563, 577, 599, 607, 613, 617, 631, 643, 647, 677, 683, 739, 751, 757, 809, 811, 821, 823, 827, 829

Offset: 1

Richard Peterson, Oct 22 2024

nonn

The generalization of Wilson's Theorem that x!*(p-1-x)! is either +1 or -1 mod p when p is prime is known. This is investigated here for cases of p such that there is an x with x!*(p-1-x)! congruent to either +1 or -1 mod p^2.

			0!*4! + 1 = 5^2 and 4+0 = 5-1, so 5 is in the sequence.
1!*9! - 1 = 11^2*2999 and 1+9 = 10-1, so 11 is in the sequence.
0!*12! + 1 = 13^2*2834329 and 0+12 = 13-1, so 13 is in the sequence.
10!*6! + 1 = 17^2*83*108923 and 10+6=17-1, so 17 is in the sequence.
19!*39!-1 is divisible by 59^2 and 19+39=59-1, so 59 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..4011

A007540 is a subsequence.

Cf. A157249, A157250.

filter:= proc(p) local m,n,A,v;
  A:= Array(0..p);
  A[0]:= 1:
  for n from 1 to p do A[n]:= n*A[n-1] mod p^2 od:
  for m from 0 to (p-1)/2 do
    v:= A[m] * A[p-1-m] mod p^2;
    if v = 1 or v = p^2-1 then return true fi;
  od;
  false
end proc:
select(filter, [seq(ithprime(i),i=1..150)]); # Robert Israel, Dec 30 2024

isok(p)={if(isprime(p), for(m=0, p\2, my(t=(m!*(p-1-m)!%p^2)); if(t==1||t==p^2-1, return(1)))); 0} \\ Andrew Howroyd, Oct 22 2024

a(14) onwards from Andrew Howroyd, Oct 22 2024

A362829 Positions in lexicographic order of odd partitions of sufficiently large numbers.

Original entry on oeis.org

1, 3, 7, 10, 15, 20, 27, 30, 39, 41, 51, 56, 69, 72, 75, 93, 95, 101, 123, 128, 132, 134, 160, 163, 166, 172, 176, 212, 214, 220, 227, 229, 273, 278, 282, 284, 291, 297, 353, 356, 359, 365, 369, 379, 382, 384, 453, 455, 461, 468, 470, 481, 483, 490, 579, 584

Offset: 1

Richard Peterson, Aug 01 2023

nonn

a(n) is the position in lexicographic order of the n-th odd partition of a sufficiently large number k. As long as the number k whose partitions we are examining is large enough, a(n) will exist and won't change for different k. The number of partitions of an odd number, for example, 101 for k=13, will always appear in the sequence, since 13 is the 101st partition in lexicographic order.

Equivalently, positions of partitions with all parts odd among all partitions with no parts of size 1, ordered first by sum, then lexicographically (with the parts in nondecreasing order); or positions of partitions with all parts even among all partitions ordered first by the number of parts plus the sum of the parts, then lexicographically. - Pontus von Brömssen, Sep 14 2023

			a(1)=1 because 1+1+...+1 (k times) is the first partition in lexicographic order of any positive integer k, and it is odd.
a(2)=3 because 1+1+...+1(k-3 times)+3=k is the third partition of k lexicographically and it is odd.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000009, A087897, A000041.

More terms from Pontus von Brömssen, Sep 14 2023

A350813 a(n) is the least positive number k such that the product of the first n primes that are congruent to 1 (mod 4) is equal to y^2 - k^2 for some integer y.

Original entry on oeis.org

2, 4, 24, 38, 16, 588, 5782, 5528, 80872, 319296, 3217476, 32301914, 20085008, 166518276, 2049477188, 17443412442, 27905362944, 233647747282, 886295348972, 134684992249108, 98002282636962, 392994156083892, 5283713761100536, 76642755213473624, 923250078609721236

Offset: 1

Richard Peterson, Jan 17 2022

nonn

Because y^2-k^2=(y-k)(y+k), a method to make k as small as possible is to try to make y-k and y+k as nearly equal as possible.

Because each of y-k and y+k are made up of primes of form 1 mod 4, algebra shows that k=a(n) is always even.

			For n=3, m = 5*13*17. The "middle" most nearly equal divisor and codivisor of m are y-k=17 and y+k=65, whence a(n) = (65 - 17)/2 = 24.

Chai Wah Wu, Table of n, a(n) for n = 1..36

Cf. A006278, A236381, A061060.

from math import prod, isqrt
from itertools import islice
from sympy import sieve, divisors
def A350813(n):
    m = prod(islice(filter(lambda p: p % 4 == 1, sieve),n))
    a = isqrt(m)
    d = max(filter(lambda d: d <= a, divisors(m,generator=True)))
    return (m//d-d)//2 # Chai Wah Wu, Mar 29 2022

Terms corrected by and more terms from Jinyuan Wang, Mar 17 2022

A348635 a(n) is the smallest positive number k coprime to (2n+1)!! such that (2n+1)!! + k^2 is a square.

Original entry on oeis.org

1, 1, 4, 4, 29, 17, 436, 356, 569, 1847, 27704, 72944, 1283333, 726079, 23833532, 45232276, 302068799, 616565857, 26369361188, 23157514888, 70991664061, 505527042479, 1150735735948, 13238389944712, 58668785675111, 209280259070287, 7809609503808088, 530566746979816

Offset: 1

Richard Peterson, Dec 13 2021

nonn

a(n) always exists since the set of k coprime to (2n+1)!! and with (2n+1)!! + k^2 equal to a square is nonempty, because k = ((2n+1)!!-1)/2 is in the set.

			a(5)=29 since 106^2 - 29^2 = 10395 = 3*5*7*9*11 and 29 is relatively prime to 10395 and is as small as possible.

Cf. A001147, A230710, A236381.

df(n) = (2*n)! / n! / 2^n; \\ A001147
a(n) = my(d=df(n+1), k=1); while (!((gcd(d,k)==1) && issquare(d+k^2)), k++); k; \\ Michel Marcus, Jan 06 2022

df(n) = (2*n)! / n! / 2^n; \\ A001147
a(n) = my(d=df(n+1), m=sqrtint(d), k); while (!(issquare(m^2-d, &k) && gcd(d,k)==1), m++); k; \\ Michel Marcus, Jan 06 2022

a(21)-a(24) and a(28) from Jon E. Schoenfield, Jan 06 2022

a(25)-a(27) from Jinyuan Wang, Jan 07 2022

A349708 a(n) is the smallest positive number k such that (product of the first n odd primes) + k^2 is a square.

Original entry on oeis.org

1, 1, 4, 1, 19, 53, 58, 97, 181, 4244, 2122, 31126, 16451, 297392, 2444006, 622249, 2909047, 216182072, 62801719, 769709491, 32522441312, 37859955467, 129549407177, 286721160343, 101419856449, 107709289064864, 72441253480727, 56099073382147, 5249126879235893

Offset: 1

Richard Peterson, Dec 31 2021

nonn

a(n) is half the difference between the middle two divisors of A070826(n + 1). - David A. Corneth, Jan 17 2022

			a(4)=1 because the product of the first 4 odd primes, 3*5*7*11 = 1155, is 34^2 - 1. a(5)=19 because 15015=3*5*7*11*13=124^2-19^2, and no positive integer less than 19 will work in this situation.

Jon E. Schoenfield, Table of n, a(n) for n = 1..42

Cf. A056737, A061060, A070826, A236381.

a(n) = my(k=1, p=prod(k=2, n+1, prime(k))); while (!issquare(k^2+p), k++); k; \\ Michel Marcus, Jan 10 2022

from math import isqrt
from sympy import primorial, divisors
def A349708(n):
    m = primorial(n+1)//2
    a = isqrt(m)
    d = max(filter(lambda d: d <= a, divisors(m,generator=True)))
    return (m//d-d)//2 # Chai Wah Wu, Mar 29 2022

a(15)-a(26) and corrections to a(9) and a(11) from Jinyuan Wang, Jan 07 2022

a(27)-a(30) from Jon E. Schoenfield, Jan 16 2022

A341678 Irregular triangle read by rows: row n consists of all numbers x such that x^2 + y^2 = A006278(n), with 0 < x < y.

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 23, 2, 19, 46, 67, 74, 86, 109, 122, 64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743, 59, 132, 531, 581, 627, 876, 1008, 1284, 1588, 1659, 1723, 2092, 2136, 2317, 2373, 2736, 2757, 2803, 3072, 3164, 3333, 3469, 3704, 3821, 4028, 4077, 4136, 4371, 4596, 4668, 4712, 4851

Offset: 1

Richard Peterson, Feb 17 2021

The n-th row of the triangle is of length 2^(n-1), since a product of n distinct primes congruent to 1 (mod 4) has 2^(n-1) solutions to being the sum of two squares.

			Triangle starts:
1,
1, 4,
4, 9, 12, 23,
2, 19, 46, 67, 74, 86, 109, 122,
64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743,
...
In the second row, calculations are as follows. 5*13 is the product of the first two primes congruent to 1 (mod 4), and 65 = 1^2 + 8^2 = 4^2 + 7^2, so the second row is 1, 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1023 (rows 1..10)
Eric Weisstein's World of Mathematics, Sum of Squares Function

Cf. A006278, A230710.

Cf. A236381 (1st column).

row(n) = {my(t=1, q=3, v=vector(2^n/2)); for(k=1, n, until(q%4==1, q=nextprime(q+1)); t*=q); q=0; for(k=1, #v, until(issquare(t-q^2), q++); v[k]=q); v; } \\ Jinyuan Wang, Mar 03 2021

A338085 a(n) is the cardinality of S(n), the subset of partitions of n such that there are enough smaller parts to add together to be greater than a larger part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 5, 12, 18, 30, 36, 65, 83, 120, 159, 225, 284, 395, 495, 665, 848, 1094, 1348, 1757, 2184, 2746, 3399, 4250, 5199, 6469, 7867, 9667, 11756, 14310, 17266, 20988, 25216, 30372, 36371, 43648, 52041, 62187, 73866, 87837, 104105, 123279, 145453

Offset: 1

Richard Peterson, Oct 08 2020

nonn

In George Andrews’s partition notation, exponents mean repeated addition, not repeated multiplication. So (p^K)(q^L) with p0 and the parts pk arranged in increasing order, suppose E1p1+E2p2+..Ekpk>p(k+1) for some 1

For any partition in S, the number of parts must be at least 3, with at least 2 distinct parts.

The sequence a(n) is nondecreasing since if a(n-1)=t, then t distinct elements of S(n) can be formed by putting a dot in the lower left corner of the Ferrers diagram for each element of S(n-1).

Closure: Given a partition X in S(x) and partition Y in S(y): The partition X+Y given by concatenation is in S(x+y). So a(x)+a(y) might be provably less than or equal to S(x+y). X and Y can be multiplied to give a partition XY in S(xy). The two operations obey distributivity of multiplication over addition.

			(4^2)(7^3), a partition of 29, is in S(29) since 2*4=8>7.
Also, (1^3)(3^2)(7^1)(20^4), a partition of 96, is in S(96) since 3*1+2*3=9>7.
But (1^3)(4^5) is not in S(23) because 3*1 is not greater than 4.

ispart[p_] := Module[{s = 0}, For[i = 1, i <= Length[p], i++, If[s > p[[i]] && p[[i]] > p[[i-1]], Return[1]]; s += p[[i]]]; 0];
a[n_] := a[n] = Module[{c = 0}, Do[ c += ispart[p], {p, Reverse /@ IntegerPartitions[n]}]; c];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 13 2020, after Andrew Howroyd *)

ispart(p)={my(s=0);for(i=1, #p, if(s>p[i]&&p[i]>p[i-1], return(1)); s+=p[i]);0}
a(n)={my(c=0); forpart(p=n, c+=ispart(p)); c} \\ Andrew Howroyd, Oct 25 2020

a(n)={local(Cache=Map()); my(F(r,k,b)=my(hk=[r,k,b],z); if(!mapisdefined(Cache, hk, &z), z = if(k<=1, b, sum(m=0, r\k, self()(r-m*k, k-1, b||(m&&r-m*k>k)))); mapput(Cache, hk, z)); z); F(n,n,0)} \\ Andrew Howroyd, Nov 03 2020

Terms a(15) and beyond from Andrew Howroyd, Nov 03 2020

A337674 Numbers k whose prime divisors are all less than or equal to the number of divisors of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 64, 70, 72, 75, 80, 81, 84, 90, 96, 100, 105, 108, 112, 120, 126, 128, 132, 135, 140, 144, 150, 160, 162, 168, 180, 189, 192, 196, 198, 200, 210, 216, 220, 224, 225, 240, 243, 250

Offset: 1

Richard Peterson, Sep 15 2020

nonn

Density: 33 terms between 1 and 100, 17 between 201 and 300, 11 between 1001 and 1100, and 2 between 1000001 and 1000100.

			42=a(17) is a term, since 2,3 and 7 are the prime divisors of 42, which has 8 divisors. 156=2^2*3*13 is not a term, since 13 is greater than 12, the number of divisors of 156.

A199768 has "strictly less", while this sequence has "less than or equal to".
The union of A199768 and A036878.
A146982 does not include terms 42, 56, 132, 198, 220, 264, 308, 312, 330, ...
Cf. A000005.

Select[Range[250], FactorInteger[#][[-1, 1]] <= DivisorSigma[0, #] &] (* Amiram Eldar, Sep 22 2020 *)

isok(m) = #select(x->(x>numdiv(m)), factor(m)[,1]) == 0; \\ Michel Marcus, Sep 22 2020

A333986 Greatest prime factor of A007497(n).

Original entry on oeis.org

2, 3, 2, 7, 2, 5, 3, 5, 7, 5, 7, 5, 127, 7, 8191, 5, 127, 13, 31, 127, 127, 89, 31, 17, 127, 31, 3221, 179, 151, 127, 8191, 13, 262657, 11939, 199, 257, 127, 127, 337, 257, 524287, 73, 1093, 547, 137, 241, 1093, 547, 331, 131071, 1093, 599479, 8191, 137, 127, 8191

Offset: 1

Richard Peterson, Sep 04 2020

nonn

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

Cf. A006530, A007497.

f[1] = 2; f[n_] := f[n] = DivisorSigma[1, f[n - 1]]; Table[FactorInteger[f[n]][[-1, 1]], {n, 50}] (* Amiram Eldar, Sep 23 2020 *)

a(n) = A006530(A007497(n)). - Michel Marcus, Sep 09 2020

More terms from Michel Marcus, Sep 16 2020

A335534 a(n) = tribonacci(n) modulo Fibonacci(n).

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 0, 3, 10, 26, 60, 130, 38, 173, 485, 175, 977, 273, 2789, 2065, 336, 15149, 22718, 39800, 5226, 54214, 2323, 251416, 418400, 93831, 977776, 1518664, 261912, 5208104, 2557037, 3549042, 21177270, 11203146, 36247269, 87596844, 44950918, 261069681

Offset: 1

Richard Peterson, Jun 12 2020

nonn

a(n) is congruent to tribonacci(n) modulo k if Fibonacci(n) is divisible by k, although the converse does not hold.

			For n=10, since tribonacci(10)=81 and Fibonacci(10)=55, a(10)=81 modulo 55 = 26.

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000073, A000045, A001177, A133021, A046738.

a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3] mod (<<0|1>, <1|1>>^n)[1, 2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Aug 19 2020

m = 42; Mod[LinearRecurrence[{1, 1, 1}, {0, 1, 1}, m], Array[Fibonacci, m]] (* Amiram Eldar, Aug 19 2020 *)

t(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
a(n) = t(n) % fibonacci(n); \\ Michel Marcus, Aug 19 2020

cp's OEIS Frontend

User: Richard Peterson

A377266 Primes p with the property that there exist nonnegative integers m,n such that m!*n! is congruent to either +1 or -1 mod p^2, with m + n = p - 1.

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A362829 Positions in lexicographic order of odd partitions of sufficiently large numbers.

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A350813 a(n) is the least positive number k such that the product of the first n primes that are congruent to 1 (mod 4) is equal to y^2 - k^2 for some integer y.

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A348635 a(n) is the smallest positive number k coprime to (2n+1)!! such that (2n+1)!! + k^2 is a square.

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A349708 a(n) is the smallest positive number k such that (product of the first n odd primes) + k^2 is a square.

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A341678 Irregular triangle read by rows: row n consists of all numbers x such that x^2 + y^2 = A006278(n), with 0 < x < y.

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A338085 a(n) is the cardinality of S(n), the subset of partitions of n such that there are enough smaller parts to add together to be greater than a larger part.

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A337674 Numbers k whose prime divisors are all less than or equal to the number of divisors of k.

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