cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A003973 Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

Original entry on oeis.org

1, 4, 6, 13, 8, 24, 12, 40, 31, 32, 14, 78, 18, 48, 48, 121, 20, 124, 24, 104, 72, 56, 30, 240, 57, 72, 156, 156, 32, 192, 38, 364, 84, 80, 96, 403, 42, 96, 108, 320, 44, 288, 48, 182, 248, 120, 54, 726, 133, 228, 120, 234, 60, 624, 112, 480, 144, 128, 62, 624, 68
Offset: 1

Views

Author

Marc LeBrun

Keywords

Comments

Sum of the divisors of the prime shifted n, or equally, sum of the prime shifted divisors of n. - Antti Karttunen, Aug 17 2020

Links

Crossrefs

Cf. A000203, A000290 (positions of odd terms), A003961, A007814, A048673, A108228, A151800, A295664, A336840.
Permutation of A008438.
Used in the definitions of the following sequences: A326042, A336838, A336841, A336844, A336846, A336847, A336848, A336849, A336850, A336851, A336852, A336856, A336931, A336932.
Cf. also A003972.

Programs

  • Mathematica
    b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]  (* Jean-François Alcover, Jul 18 2013 *)
  • PARI
    aPrime(p,e)=my(q=nextprime(p+1));(q^(e+1)-1)/(q-1)
a(n)=my(f=factor(n));prod(i=1,#f~,aPrime(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Jul 18 2013
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); }; \\ Antti Karttunen, Aug 06 2020
  • Python
    from math import prod
from sympy import factorint, nextprime
def A003973(n): return prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 05 2022

Formula

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = nextPrime(p). - David W. Wilson, Sep 01 2001
From Antti Karttunen, Aug 06-12 2020: (Start)
a(n) = Sum_{d|n} A003961(d) = Sum_{d|A003961(n)} d.
a(n) = A000203(A003961(n)) = A000593(A003961(n)).
a(n) = 2*A336840(n) - A000005(n) = 2*Sum_{d|n} (A048673(d) - (1/2)).
a(n) = A008438(A108228(n)) = A008438(A048673(n)-1).
a(n) = A336838(n) * A336856(n).
a(n) is odd if and only if n is a square.
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 3.39513795..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022, May 30 2025

Extensions

More terms from David W. Wilson, Aug 29 2001
Secondary name added by Antti Karttunen, Aug 06 2020

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)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • PARI
    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)); };

Formula

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

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • PARI
    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)); };
  • PARI
    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));

Formula

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).

A336699 a(n) = A000265(1+A000265(sigma(A000265(n)))), where A000265(k) gives the odd part of k, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 7, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 7, 5, 3, 1, 1, 11, 1, 3, 1, 5, 1, 1, 1, 29, 1, 5, 1, 7, 3, 5, 1, 3, 1, 1, 1, 1, 1, 7, 1, 11, 1, 9, 5, 1, 1, 5, 7, 19, 5, 1, 3, 1, 1, 3, 1, 61, 11, 11, 1, 7, 3, 1, 1, 23, 5, 1, 1, 1, 1, 1, 1, 25, 29, 5, 1, 13, 5, 7, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2020

Keywords

Comments

See the "lacunae" in the scatter plot. - Antti Karttunen, Mar 27 2022

Links

Crossrefs

Cf. A000203, A000265, A000593, A002131, A082903, A161942, A171435, A322250, A336698, A336700, A336701, A336844 [= a(A003961(n))], A351565.

Programs

Formula

a(n) = A000265(1+A000265(A000593(n))) = A000265(1+A161942(A000265(n))).
a(n) = A336698(A000265(n)).
From Antti Karttunen, Mar 27 2022: (Start)
a(n) = A351565(A000593(n)).
[The following formulas were discovered by Sequence Machine]:
a(n) = A351565(A002131(n)) = A000265(1+A000265(A002131(n))).
a(n) = A336698(1+A322250(n)).
a(n) = A171435(A000593(n)+A082903(n)).
(End)

A337338 Numerator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

2, 14, 32, 122, 58, 404, 134, 1094, 782, 742, 184, 3752, 308, 346, 1768, 9842, 382, 10154, 554, 6898, 4124, 2380, 872, 33884, 2802, 3992, 19532, 1238, 994, 22972, 1408, 88574, 5674, 4954, 7582, 94502, 1724, 7190, 9518, 62302, 1894, 53600, 2258, 22144, 44518, 11324, 2864, 305072, 16106, 36414, 11812, 37148, 3542, 253904
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Links

Crossrefs

Cf. A003961, A003973, A048673, A336697, A336844, A337194, A337337.
Cf. A337339 (denominators).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337338(n) = { my(s=(A003961(n)^2),t=1+sigma(s)); (t/gcd(t, (s+1)/2)); };
\\ Or as:
A337338(n) = { my(s=A003961(n^2)); numerator((1+sigma(s))/((s+1)/2)); };

Formula

a(n) = A337194(A003961(n)^2) / A337337(n).

A336842 Number of trailing 1-bits in the binary representation of A003961(n): a(n) = A007814(1+A003961(n)).

Original entry on oeis.org

1, 2, 1, 1, 3, 4, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 6, 3, 3, 1, 3, 1, 2, 1, 2, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 4, 3, 1, 1, 1, 2, 5, 1, 2, 3, 2, 1, 2, 1, 1, 2, 2, 4, 2, 1, 3, 2, 3, 2, 1, 3, 1, 2, 4, 2, 1, 4, 4, 8, 2, 3, 1, 1, 1, 4, 1, 1, 2, 5, 1, 1, 2, 1, 1, 5, 1, 6, 1, 2, 1, 1, 3, 1, 2, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2020

Keywords

Links

Crossrefs

Cf. A003961, A007814, A246261 (positions of ones), A336843, A336844.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A007814(n) = valuation(n,2);
A336842(n) = A007814(1+A003961(n));
  • Python
    from math import prod
from sympy import factorint, nextprime
def A336842(n): return (~((m:=prod(nextprime(p)**e for p, e in factorint(n).items()))+1)& m).bit_length() # Chai Wah Wu, Jul 01 2022

Formula

a(n) = A007814(1+A003961(n)).
Showing 1-6 of 6 results.

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