A337338 -id:A337338 - cp's OEIS frontend

A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 109, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 377, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 8821, 6613, 865

Offset: 1

Antti Karttunen, Aug 24 2020

All terms are members of A007310, because all terms of A337336 and A337337 are.
No 1's after the initial one at a(1) => No quasiperfect numbers. See comments in A336700 & A337342.
If any quasiperfect numbers qp exist, they must occur also in A325311.
Question: Is there any reliable lower bound for this sequence? See A337340, A337341.
Duplicate values are rare, but at least two cases exist: a(21) = a(74) = 1513 and a(253) = a(554) = 71065. - Antti Karttunen, Jan 03 2024

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003961, A003973, A007310, A048673, A074627, A074630, A209922, A324899, A325311, A336697, A336700, A336844, A337194, A337336, A337337, A337340, A337341, A337342.
Cf. A337338 (numerators).
Cf. also A336848, A336849.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337339(n) = { my(s=(A003961(n)^2),u=(s+1)/2); (u/gcd(1+sigma(s), u)); };
\\ Or alternatively as:
A337339(n) = { my(s=A003961(n^2)); denominator((1+sigma(s))/((s+1)/2)); };

a(n) = A337336(n) / A337337(n) = A048673(n^2) / gcd(A048673(n^2), A336844(n^2)).

a(n) = A337336(n) / gcd(A337336(n), 1+A003973(n^2)).

A337337 a(n) = gcd(1+sigma(s), (s+1)/2), where s is the square of n once prime-shifted (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

All terms are in A007310, because all terms of A337336 are.

Cf. A003961, A003973, A007310, A048673, A336697, A336844, A337194, A337335, A337336, A337338, A337339.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337337(n) = { my(s=(A003961(n)^2)); gcd((s+1)/2, 1+sigma(s)); };

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336697(n) = { my(s=((n+n-1)^2)); gcd((s+1)/2, 1+sigma(s)); };
A337337(n) = A336697(A048673(n));

a(n) = gcd((s+1)/2, 1+sigma(s)), where s = A003961(n)^2 = A003961(n^2).
a(n) = gcd(A048673(n^2), 1+A003973(n^2)).
a(n) = gcd(A048673(n^2), A337194(A003961(n)^2)) = gcd(A337336(n), A336844(n^2)).
a(n) = A336697(A048673(n)).
a(n) = A337335(n^2).

A337336 a(n) = A048673(n^2).

Original entry on oeis.org

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 545, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 4901, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 44105

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

All terms are odd and neither there are multiples of 3, thus only terms of A007310 occur here.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000290, A003961, A007310, A040001, A048673, A337337, A337338, A337339.

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A337336(n) = A048673(n^2);

a(n) = A048673(A000290(n)) = (1+(A003961(n)^2))/2.

For all n>= 1, A010872(a(n)) = A040001(n).

A367600 Numbers that are not the comma-successor of any number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 49, 50, 51, 52, 53, 54, 60, 62, 63, 64, 65, 70, 74, 75, 76, 80, 86, 87, 90, 98, 200, 300, 400, 500, 600, 700, 800, 900, 2000, 3000, 4000, 5000, 6000, 7000

Offset: 1

Giovanni Resta, Nov 23 2023

These are the positive integers that do not appear in A367338.

All terms > 98 are of the form c*10^i for i >= 2 and 2 <= c <= 9; see proof in links. - Michael S. Branicky, Nov 28 2023

Michael S. Branicky, Table of n, a(n) for n = 1..110 (all terms < 10^9)
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Michael S. Branicky, Comma-Predecessor Theorem

Cf. A121805, A337338, A367614.

from itertools import count, islice
def A367338(n):
    nn = n + 10*(n%10)
    return next((nn+y for y in range(1, 10) if str(nn+y)[0] == str(y)), -1)
def agen():
    A367338_set = set()
    for n in count(1):
        A367338_set.add(A367338(n))
        if n not in A367338_set:
            yield n
        # A367338_set.discard(n-100) # uncomment if memory is an issue
print(list(islice(agen(), 86))) # Michael S. Branicky, Nov 28 2023

cp's OEIS Frontend

A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

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A337337 a(n) = gcd(1+sigma(s), (s+1)/2), where s is the square of n once prime-shifted (s = A003961(n)^2 = A003961(n^2)).

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PARI

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A337336 a(n) = A048673(n^2).

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PARI

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A367600 Numbers that are not the comma-successor of any number.

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