Tomohiro Yamada - cp's OEIS frontend

A369205 Numbers m such that A188999(A034448(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

1, 2, 9, 10, 15, 18, 21, 30, 40, 42, 60, 120, 288, 567, 630, 720, 756, 1023, 1134, 1428, 2046, 2160, 2268, 2520, 3024, 3276, 3570, 4092, 6048, 8184, 8925, 9240, 11424, 11550, 15345, 17850, 18144, 30690, 35700, 46200, 57120, 85680, 147312, 285600, 491040, 556920

Offset: 1

Tomohiro Yamada, Jan 16 2024

nonn

			A034448(18) = 4 * 10 = 40 and A188999(40) = 15 * 6 = 90 = 5 * 18, so 18 is a term with k = 5.

Amiram Eldar, Table of n, a(n) for n = 1..105

Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).

Cf. A369204 (analog for A034448(A188999(m))).

a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
a188999(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=if(e%2,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2));f[i,2]=1);factorback(f)};
isok(n) = (a188999(a034448(n))%n) == 0;

A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 18, 24, 27, 30, 54, 165, 238, 288, 512, 656, 660, 864, 952, 1536, 1968, 2464, 2880, 4608, 4680, 13824, 14448, 14976, 16728, 19008, 19992, 23040, 29376, 60928, 152064, 155520, 172368, 279552, 474936, 746928, 1070592, 1114560, 1524096, 1703520

Offset: 1

Tomohiro Yamada, Jan 16 2024

nonn

			A188999(18) = 4 * 10 = 40 and A034448(40) = 9 * 6 = 54 = 3 * 18, so 18 is a term with k = 3.

Amiram Eldar, Table of n, a(n) for n = 1..80

Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).

Cf. A369205 (analog for A188999(A034448(m))).

a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
a188999(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=if(e%2,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2));f[i,2]=1);factorback(f)};
isok(n) = (a034448(a188999(n))%n) == 0;

A368424 Numbers k such that gcd(A019320(k), A019321(k)) > 1.

Original entry on oeis.org

4, 11, 18, 20, 23, 28, 35, 43, 48, 52, 83, 95, 100, 119, 131, 138, 148, 155, 162, 166, 172, 179, 191, 196, 204, 210, 214, 239, 251, 253, 268, 292, 299, 300, 316, 323, 342, 359, 371, 378, 388, 419, 431, 443, 460, 463, 491, 500, 508, 515, 537, 556, 564, 575

Offset: 1

Tomohiro Yamada, Dec 24 2023

nonn

The corresponding greatest common divisors are given in A368425.

			a(1) = 4 since A019320(4) = 5 and A019321(4) = 10.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A019320, A019321, A191609 (prime factors of such gcds), A368425.

select(k -> igcd(numtheory:-cyclotomic(k,2),
numtheory:-cyclotomic(k,3)) > 1, [$1..1000]); # Robert Israel, Dec 26 2023

Select[Range[600],GCD[Cyclotomic[#,2],Cyclotomic[#,3]]>1&] (* Stefano Spezia, Dec 26 2023 *)

for(n=1,1000,if(gcd(polcyclo(n,2),polcyclo(n,3))>1,print1(n,", ")))

A368425 The corresponding greatest common divisors to A368424(n).

Original entry on oeis.org

5, 23, 19, 5, 47, 29, 71, 431, 97, 53, 167, 191, 505, 239, 263, 139, 149, 311, 163, 499, 173, 359, 383, 197, 409, 211, 643, 479, 503, 23, 269, 293, 599, 1201, 317, 647, 19, 719, 743, 379, 389, 839, 863, 887, 461, 11113, 983, 5, 509, 1031, 4297, 557, 1129

Offset: 1

Tomohiro Yamada, Dec 24 2023

			a(2) = 23 since gcd(A019320(A368424(2)), A019321(A368424(2))) = gcd(2047, 88573) = 23.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A019320, A019321, A191609 (primes dividing some term of this sequence), A368424.

subs(1=NULL, [seq(igcd(numtheory:-cyclotomic(n,2), numtheory:-cyclotomic(n,3)),n=1..1000)]); # Robert Israel, Dec 26 2023

Select[GCD[Cyclotomic[Range[600], 2], Cyclotomic[Range[600], 3]],#>1&] (* Stefano Spezia, Dec 26 2023 *)

for(n=1,1000,m=gcd(polcyclo(n,2),polcyclo(n,3));if(m>1,print1(m,", ")))

A362531 The smallest integer m such that m mod 2k >= k for k = 1, 2, 3, ..., n.

Original entry on oeis.org

1, 3, 3, 15, 15, 47, 95, 95, 287, 335, 1199, 1199, 1295, 2015, 2879, 2879, 2879, 2879, 2879, 2879, 2879, 43199, 211679, 211679, 211679, 211679, 211679, 211679, 211679, 211679, 3084479, 3084479, 3084479, 3084479, 3084479, 3084479, 302702399, 469909439

Offset: 1

Tomohiro Yamada, Apr 24 2023

nonn

Take the square array A(k, l) with k= 1, 2, ... and l = 0, 1, ... such that for each k, A(k, l) takes k zeros and then k ones alternately:
0, 1, 0, 1, 0, 1, 0, 1, ...
0, 0, 1, 1, 0, 0, 1, 1, ...
0, 0, 0, 1, 1, 1, 0, 0, ...
...
Then the a(n)-th column is the first column which begins with n ones.

			a(3) = 3 since 3 mod 2 = 1, 3 mod 4 = 3 >= 2, 3 mod 6 = 3 (but 3 mod 8 = 3 < 4) while 1 mod 4 = 1 < 2, 2 mod 2 = 0 < 1.

Tomohiro Yamada, Fast Python program, the original program written by nazratt2.

Cf. A053664.

a(n)={my(m);m=1;while(vecmin(vector(n,j,(m%(2*j))/j))<1,m=m+1);m}

n=1;for(m=1,10^9,while(vecmin(vector(n,j,(m%(2*j))/j))>=1,print(n," ",m);n=n+1))

def A362531(n):
    m = 1
    while True:
        for k in range(n,0,-1):
            if (l:=k-m%(k<<1))>0:
                break
        else:
            return m
        m += l # Chai Wah Wu, Jun 21 2023

a(37)-a(38) from Yifan Xie, Apr 24 2023

A362532 The smallest positive integer m such that m mod 2k < k for k = 1, 2, 3, ..., n.

Original entry on oeis.org

2, 4, 8, 8, 24, 24, 72, 144, 144, 144, 384, 384, 2160, 2160, 2160, 6720, 54240, 57600, 131040, 131040, 131040, 131040, 612000, 612000, 612000, 612000, 612000, 776160, 776160, 776160, 6219360, 23184000, 28627200, 28627200, 28627200, 28627200, 28627200

Offset: 1

Tomohiro Yamada, Apr 24 2023

nonn

Take the square array A(k, l) with k = 1, 2, ... and l = 0, 1, ... such that for each k, A(k, l) takes k zeros and then k ones alternately:
   0, 1, 0, 1, 0, 1, 0, 1, ...
   0, 0, 1, 1, 0, 0, 1, 1, ...
   0, 0, 0, 1, 1, 1, 0, 0, ...
   ...
Then the a(n)-th column is the first column which begins with n zeros except the 0th column.

			a(2) = 4 since 4 mod 2 = 0, 4 mod 4 = 0 (but 4 mod 6 = 4 >= 3) while 1 mod 2 = 1, 2 mod 4 = 2, 3 mod 2 = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..136
Tomohiro Yamada, Fast python program, the original program written by nazratt2.

Cf. A053664.

a(n)=my(m);m=1;while(vecmax(vector(n,j,(m%(2*j))/j))>=1,m=m+1);m

n=1;for(m=1,10^9,while(vecmax(vector(n,j,(m%(2*j))/j))<1,print(n," ",m);n=n+1))

a(n,startAt=1)=if(n<5, return(2^min(n,3))); my(s=if(n>146,70274254050, n>108,10039179150, n>94,436486050, n>65,22972950, n>50,417690, n>46,8190, n>27,630, n>17,90, n>12,30, 6)<=k, next(2))); return(m)) \\ Charles R Greathouse IV, Apr 28 2023

from itertools import count, islice
def agen(): # generator of terms
    m = 1
    for n in count(1):
        while not all(m%(2*k) < k for k in range(1, n+1)): m += 1
        yield m
print(list(islice(agen(), 37))) # Michael S. Branicky, Apr 24 2023

Trivial bounds: a(n-1) <= a(n) <= 2*A003418(n). - Charles R Greathouse IV, May 08 2023

A355298 Primes p such that q divides p + 1, r divides q^2 + q + 1, s divides r^2 + r + 1, and p divides s^2 + s + 1 for some primes q, r, and s.

Original entry on oeis.org

3, 13, 61, 127, 399403

Offset: 1

Tomohiro Yamada, Jun 28 2022

There are no other terms below 2^24.

If rad(n)^2 = sigma(n), where rad(n) = A007927(n) is the largest squarefree number dividing n and sigma(n) = A000203(n) is the sum of divisors of n, and there exists just one odd prime factor p dividing n exactly once, then p must belong to A354427 or this sequence.

			61 is a term since 61 + 1 = 2 * 31, 31^2 + 31 + 1 = 3 * 331, 3^2 + 3 + 1 = 13, and 13^2 + 13 + 1 = 3 * 61.

Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, 10:3 (2021), 249-260, correction, 10:4 (2021), 339.

Cf. A101368, A347988, A354427.

is(p)={my(W, V1, V2, V3, V4, q1, q2, q3, q4, i1, i2, i3, i4, l1, l2, l3, l4); W=0; V1=factor(p+1); l1=length(V1[, 1]); for(i1=1, l1, q1=V1[i1, 1]; V2=factor(q1^2+q1+1); l2=length(V2[, 1]); for(i2=1, l2, q2=V2[i2, 1]; V3=factor(q2^2+q2+1); l3=length(V3[, 1]); for(i3=1, l3, q3=V3[i3, 1]; V4=factor(q3^2+q3+1); l4=length(V4[, 1]); for(i4=1, l4, q4=V4[i4, 1];if(q4==p, W=[p, q1, q2, q3]))))); W}

A354428 Primes p such that q divides p^2 + p + 1, r divides q + 1 and p divides r^2 + r + 1 for some primes q and r.

Original entry on oeis.org

3, 7, 43, 73363, 1477111

Offset: 1

Tomohiro Yamada, May 27 2022

There are no other terms below 2^24.
The first five terms correspond to 7, 2, 79, 9829, and 5569 in A354426 respectively.
Similarly, these correspond to 13, 3 (or 19), 631, 1794067711, and 10855016833 in A354427 respectively.

			43 is a term since 43^2 + 43 + 1 = 3 * 631, 631 + 1 = 2^3 * 79, and 79^2 + 79 + 1 = 3 * 7^2 * 43.

Cf. A101368, A347988.

Cf. A354426 (r corresponding to primes p in this sequence), A354427 (q corresponding to primes p in this sequence).

is(p)={my(W, V1, V2, V3, q1, q2, q3, i1, i2, i3, l1, l2, l3); W=0; V1=factor(p^2+p+1); l1=length(V1[, 1]); for(i1=1, l1, q1=V1[i1, 1]; V2=factor(q1+1); l2=length(V2[, 1]); for(i2=1, l2, q2=V2[i2, 1]; V3=factor(q2^2+q2+1); l3=length(V3[, 1]); for(i3=1, l3, q3=V3[i3, 1]; if(q3==p, W=[p, q1, q2])))); W}

A354427 Primes p such that q divides p + 1, r divides q^2 + q + 1 and p divides r^2 + r + 1 for some primes q and r.

Original entry on oeis.org

3, 13, 19, 631

Offset: 1

Tomohiro Yamada, May 27 2022

There are no other terms below 2^24.
1794067711 and 10855016833 are also terms, which arise from q = 9829 and q = 5569 in A354426 respectively, since
1794067711 + 1 = 2^8 * 23 * 31 * 9829, 9829^2 + 9829 + 1 = 3 * 439 * 73363, 73363^2 + 73363 + 1 = 3 * 1794067711 and
10855016833 + 1 = 2 * 17 * 5569 * 57329, 5569^2 + 5569 + 1 = 3 * 7 * 1477111, 1477111^2 + 1477111 + 1 = 3 * 67 * 10855016833.
If rad(n)^2 = sigma(n), where rad(n) = A007927(n) is the largest squarefree number dividing n and sigma(n) = A000203(n) is the sum of divisors of n, and there exists just one odd prime factor p dividing n exactly once, then p must belong to this sequence, or q divides p + 1, r divides q^2 + q + 1, s divides r^2 + r + 1, and p divides s^2 + s + 1 for some primes q, r, and s (Yamada, 2021). [Corrected by Tomohiro Yamada, Jun 05 2022]

			631 is a term since 631 + 1 = 2^3 * 79, 79^2 + 79 + 1 = 3 * 7^2 * 43, and 43^2 + 43 + 1 = 3 * 631.

Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, 10:3 (2021), 249-260, correction, 10:4 (2021), 339.

Cf. A101368, A347988.

Cf. A354426 (q corresponding to primes p in this sequence), A354428 (r corresponding to primes p in this sequence).

is(p)={my(W, V1, V2, V3, q1, q2, q3, i1, i2, i3, l1, l2, l3); W=0; V1=factor(p+1); l1=length(V1[, 1]); for(i1=1, l1, q1=V1[i1, 1]; V2=factor(q1^2+q1+1); l2=length(V2[, 1]); for(i2=1, l2, q2=V2[i2, 1]; V3=factor(q2^2+q2+1); l3=length(V3[, 1]); for(i3=1, l3, q3=V3[i3, 1]; if(q3==p, W=[p, q1, q2])))); W}

A354426 Primes p such that q divides p^2 + p + 1, r divides q^2 + q + 1 and p divides r + 1 for some primes q and r.

Original entry on oeis.org

2, 7, 79, 5569, 9829

Offset: 1

Tomohiro Yamada, May 27 2022

There are no other terms below 2^24.

There are no other terms below 10^8. - Lucas A. Brown, Aug 11 2024

			7 is a term since 7^2 + 7 + 1 = 3 * 19, 3^2 + 3 + 1 = 13 and 13 + 1 = 2 * 7.

Lucas A. Brown, Python program.
Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, 10:3 (2021), 249-260, correction, 10:4 (2021), 339.

Cf. A101368, A347988.

is(p)={my(W, V1, V2, V3, q1, q2, q3, i1, i2, i3, l1, l2, l3); W=0; V1=factor(p^2+p+1); l1=length(V1[, 1]); for(i1=1, l1, q1=V1[i1, 1]; V2=factor(q1+1); l2=length(V2[, 1]); for(i2=1, l2, q2=V2[i2, 1]; V3=factor(q2^2+q2+1); l3=length(V3[, 1]); for(i3=1, l3, q3=V3[i3, 1]; if(q3==p, W=[p, q1, q2])))); W}

cp's OEIS Frontend

User: Tomohiro Yamada

A369205 Numbers m such that A188999(A034448(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

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A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

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A368424 Numbers k such that gcd(A019320(k), A019321(k)) > 1.

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A368425 The corresponding greatest common divisors to A368424(n).

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A362531 The smallest integer m such that m mod 2k >= k for k = 1, 2, 3, ..., n.

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A362532 The smallest positive integer m such that m mod 2k < k for k = 1, 2, 3, ..., n.

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A355298 Primes p such that q divides p + 1, r divides q^2 + q + 1, s divides r^2 + r + 1, and p divides s^2 + s + 1 for some primes q, r, and s.

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