A336317 -id:A336317 - cp's OEIS frontend

A336353 Numbers k such that sigma(k) does not have any prime factor larger than the largest prime factor of k.

1, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Note that the terms after 1 do not form a subsequence of A102750: the first counterexample is 343 = 7^3. See A336354.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Positions of zeros in A336352.
Cf. A102750, A336317.
Subsequences: A000396, A001599, A065091, A105402, A333646 (see comment there), A336354.

PARI
```
isA336353(n) = !A336352(n);
```

A336314 a(n) = A324121(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 2, 1, 4, 3, 2, 1, 12, 3, 2, 1, 12, 1, 6, 1, 4, 1, 8, 4, 36, 1, 10, 1, 24, 3, 2, 3, 4, 24, 4, 1, 12, 1, 56, 1, 24, 1, 2, 3, 4, 1, 4, 1, 6, 9, 6, 1, 4, 8, 8, 1, 12, 9, 48, 1, 4, 1, 2, 24, 120, 5, 2, 3, 18, 7, 12, 1, 36, 2, 10, 3, 24, 1, 12, 3, 4, 3, 112

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A324121, A336315.
Cf. A336317 (gives the positions where this coincides with A323173).
Cf. also A335914.

A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A324121(n) = gcd(sigma(n),n*numdiv(n));
A336314(n) = A324121(A122111(n));

\\ Or as a standalone program:
A336314(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); gcd(s,x*d));

a(n) = A324121(A122111(n)) = gcd(A323173(n), A122111(n)*A336315(n)).

A336397 Numbers k such that both k and A122111(k) [its conjugated prime factorization] are Ore's Harmonic numbers (A001599).

Original entry on oeis.org

1, 6, 753480, 429520946400

Offset: 1

Antti Karttunen, Jul 20 2020

The initial four terms were found from the b-file of 937 terms provided in A001599.
There also at least these additional terms present: 439151400324000, 889013810412000, 106713790278732000, that were found from the file of conjectured initial 10000 terms of A001599 provided by David A. Corneth under that entry. However, there might be further terms in the range between.
Note that A122111(889013810412000) = 106713790278732000, and vice versa, which proves that the intersection of A001599 and A088902 (equal to the intersection of A088902 with A336317) is not equal to this sequence, but just a proper subsequence.

Intersection of A001599 and A336317.

Cf. A088902, A122111.

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336397(n) = (isA001599(n) && isA001599(A122111(n)));
\\ Or alternatively, by using standalone program given in A336317:
isA336397(n) = (isA001599(n) && isA336317(n));

cp's OEIS Frontend

A336353 Numbers k such that sigma(k) does not have any prime factor larger than the largest prime factor of k.

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PARI

A336314 a(n) = A324121(A122111(n)).

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Formula

A336397 Numbers k such that both k and A122111(k) [its conjugated prime factorization] are Ore's Harmonic numbers (A001599).

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Author

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Programs

PARI