A378983 -id:A378983 - cp's OEIS frontend

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

1, 2, 3, 4, 6, 7, 10, 25, 26, 28, 33, 46, 55, 57, 69, 91, 93, 496, 1034, 1054, 1558, 2211, 2626, 4825, 8128, 11222, 12046, 12639, 28225, 32043, 68727, 89575, 970225, 1392386, 2245557, 8550146, 12371554, 16322559, 22799825, 33550336, 48980427, 51326726, 55037217, 60406599, 68258725, 142901438, 325422273, 342534446

Offset: 1

Antti Karttunen, Dec 12 2024

nonn

Numbers k such that A252748(k) divides A286385(k).

Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.

Amiram Eldar, Table of n, a(n) for n = 1..69 (terms below 10^11; terms 1..60 from Antti Karttunen)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.
Positions of 0's in A378981.
Subsequence of A263837.
Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).
Cf. also A378983.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* Amiram Eldar, Dec 19 2024 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
isA378980(n) = !A378981(n);

A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

Original entry on oeis.org

1, 2, 4, 8, 16, 25, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Original definition, now conjectural, is "Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 4m-1". This is certainly equal to the new definition if 25 is the only term that is not a power of 2, as there is no x such that sigma(x) = 99 = 4*25-1. - Antti Karttunen, Dec 13 2024
If 31 is only number h of form 2^k-1 for any k>=1 such that sigma(x) = h has solution for more than one value of x then a(n) is union number 25 with A000079 (powers of 2).
Numbers k such that A000203(k) is in A000225. If Goormaghtigh conjecture is valid, then it is certain that 25 is the only odd prime power (after 1) in this sequence. - Antti Karttunen, Dec 13 2024

			These examples relate to the original definition:
m=1, 4m-1=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 4m-1=7, sigma(h)=7 for h=4; number 4 is in sequence.
m=4, 4m-1=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 4m-1=31, sigma(h)=31 for h=16 and 25; numbers 16 and 25 are in sequence.

Wikipedia, Goormaghtigh conjecture

Cf. A000079 (subsequence), A000203, A000225, A002191, A292369 (conjectured subsequence).
Subsequence of A028982. Conjectured intersection of A028982 and A378983.
Positions of 0's in A336694, A336695.
Positions of 1's in A324294, A332459, A336692, A336693, A336696.
Cf. also A202273.

is_A202274(n) = ((x->!bitand(x,x+1))(sigma(n)));
for(n=1,2^20,if(is_A202274(n^2), print1(n^2,", ")); if(n>1 && is_A202274(2*((n-1)^2)), print1(2*((n-1)^2),", "))); \\ Remember to sort! - Antti Karttunen, Dec 13 2024

Data section corrected (terms 1024, 2048 were duplicated), more terms added, and the name replaced with a new definition, with the original definition moved to the comments - Antti Karttunen, Dec 13 2024

A378982 a(n) = (A003961(n)-(1+sigma(n))) mod (A003961(n)-2*n), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 0, 4, 0, 0, 16, 2, 3, 0, 0, 0, 35, 2, 20, 9, 2, 4, 74, 0, 0, 13, 42, 0, 32, 4, 0, 0, 2, 0, 133, 2, 1, 0, 98, 0, 68, 2, 3, 11, 4, 4, 280, 17, 6, 1, 5, 4, 254, 18, 176, 0, 2, 0, 146, 4, 1, 21, 0, 1, 50, 2, 9, 6, 86, 0, 479, 4, 8, 25, 11, 2, 86, 2, 380, 40, 2, 4, 270, 24, 8, 15, 170, 6, 290, 4, 15

Offset: 1

Antti Karttunen, Dec 13 2024

nonn

Cf. A000203, A003961, A252748, A286385, A378983 (positions of 0's).

Cf. also A378981.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378982(n) = ((A003961(n)-(sigma(n)+1))%((2*n)-A003961(n)));

a(n) = (A286385(n)-1) mod A252748(n).

cp's OEIS Frontend

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A378982 a(n) = (A003961(n)-(1+sigma(n))) mod (A003961(n)-2*n), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula