A008329 -id:A008329 - cp's OEIS frontend

A145339 a(n) = the minimum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

1, 2, 3, 4, 4, 4, 5, 6, 4, 6, 6, 4, 8, 6, 4, 6, 4, 4, 6, 8, 4, 8, 4, 8, 6, 8, 8, 4, 8, 8, 8, 8, 8, 8, 6, 8, 4, 6, 4, 6, 4, 8, 8, 4, 9, 12, 6, 8, 4, 8, 8, 8, 6, 8, 8, 4, 6, 10, 4, 8, 6, 6, 12, 8, 4, 6, 6, 6, 4, 12, 8, 4, 8, 8, 12, 4, 6, 4, 8, 8, 8, 4, 8, 8, 8, 8, 14, 4, 12, 10, 4, 4, 8, 12, 8, 4, 6, 12, 6, 4, 6

Offset: 1

Leroy Quet, Oct 08 2008

nonn

Cf. A145340, A008328, A008329, A067889, A103664, A103665.

Table[Min[DivisorSigma[0, Prime[n]-1], DivisorSigma[0, Prime[n]+1]], {n, 1, 100}] (* Stefan Steinerberger, Oct 11 2008 *)

a(n) = my(p = prime(n)); min(numdiv(p-1), numdiv(p+1)); \\ Michel Marcus, Sep 28 2018

More terms from Stefan Steinerberger and Ray Chandler, Oct 11 2008

A189977 Primes p such that d(p+1) = 2*d(p-1), where d(k) counts the divisors of k.

Original entry on oeis.org

2, 23, 149, 293, 311, 439, 557, 569, 743, 773, 857, 1031, 1151, 1493, 1607, 1663, 1709, 1733, 1879, 1913, 2069, 2141, 2423, 2711, 2719, 2729, 2789, 2969, 3191, 3209, 3559, 3607, 3767, 3821, 3833, 3847, 3929, 3967, 4019, 4073, 4229, 4339, 4451, 4517, 4549

Offset: 1

Ctibor O. Zizka, May 03 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000005, A008328, A008329.

Select[Prime[Range[PrimePi[5059]]], 2*DivisorSigma[0, # - 1] == DivisorSigma[0, # + 1] &] (* T. D. Noe, May 03 2011 *)

isok(p) = isprime(p) && (numdiv(p+1) == 2*numdiv(p-1)); \\ Michel Marcus, Jan 13 2018

A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

Original entry on oeis.org

1, 5, 13, 4, 3, 7, 23, 25, 53, 7, 5, 11, 23, 73, 377, 49, 67, 18, 113, 77, 41, 21, 257, 131, 68, 559, 287, 73, 599, 307, 629, 213, 109, 83, 337, 689, 719, 739, 1493, 1523, 4609, 4699, 33253, 34513, 34933, 35353, 36193, 36613, 37033, 37663, 38083, 7667, 7835, 7891

Offset: 1

Amiram Eldar, Aug 08 2025

			Fractions begin with 1/2, 5/6, 13/12, 4/3, 3/2, 7/4, 23/12, 25/12, 53/24, 7/3, 5/2, 11/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mikhail R. Gabdullin, Vitalii V. Iudelevich, and Sergei V. Konyagin, Karatsuba's divisor problem and related questions, arXiv:2304.04805 [math.NT], 2023.
Vitalii V. Iudelevich, On the Karatsuba divisor problem, Izvestiya: Mathematics, Vol. 86, No. 5 (2022), pp. 992-1019; arXiv preprint, arXiv:2304.03049 [math.NT], 2023.

Cf. A000005, A008329, A008864, A104528, A386922 (denominators).

Numerator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]

list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(numerator(s), ", "));}

a(n) = numerator(Sum_{k=1..n} 1/A008329(k)).

a(n)/A386922(n) <= 4 * K * n/log(n)^(3/2) + O(n*log(log(n))/log(n)^(5/2)), where K = (1/sqrt(Pi)) * Product_{p prime} sqrt(p/(p-1)) * (p * log(p/(p-1)) - 1/(p-1)) = 0.25320111501639846923... (Iudelevich, 2022). Gabdullin et al. (2023) conjectured that a(n)/A386922(n) ~ K * n/log(n)^(3/2).

A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

Original entry on oeis.org

2, 6, 12, 3, 2, 4, 12, 12, 24, 3, 2, 4, 8, 24, 120, 15, 20, 5, 30, 20, 10, 5, 60, 30, 15, 120, 60, 15, 120, 60, 120, 40, 20, 15, 60, 120, 120, 120, 240, 240, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 1008, 1008, 1008, 1008, 63, 1008, 5040, 5040

Offset: 1

Amiram Eldar, Aug 08 2025

			Fractions begin with 1/2, 5/6, 13/12, 4/3, 3/2, 7/4, 23/12, 25/12, 53/24, 7/3, 5/2, 11/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mikhail R. Gabdullin, Vitalii V. Iudelevich, and Sergei V. Konyagin, Karatsuba's divisor problem and related questions, arXiv:2304.04805 [math.NT], 2023.
Vitalii V. Iudelevich, On the Karatsuba divisor problem, Izvestiya: Mathematics, Vol. 86, No. 5 (2022), pp. 992-1019; arXiv preprint, arXiv:2304.03049 [math.NT], 2023.

Cf. A000005, A008329, A008864, A104529, A386921 (numerators).

Denominator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]

list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(denominator(s), ", "));}

a(n) = denominator(Sum_{k=1..n} 1/A008329(k)).

A068356 Numbers k such that bigomega(k) = tau(prime(k)+1) where tau(k) = A000005(k).

Original entry on oeis.org

88, 100, 198, 260, 306, 375, 384, 459, 540, 585, 714, 798, 825, 856, 864, 960, 1012, 1062, 1148, 1278, 1312, 1484, 1725, 1784, 1924, 1988, 2125, 2130, 2160, 2204, 2210, 2240, 2241, 2250, 2346, 2480, 2516, 2619, 2662, 2776, 2814, 2826, 2889, 2892, 2910

Offset: 1

Benoit Cloitre, Feb 28 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A008329.

Select[Range[3000],PrimeOmega[#]==DivisorSigma[0,Prime[#]+1]&] (* Harvey P. Dale, Aug 18 2013 *)

isok(k) = bigomega(k) == numdiv(prime(k)+1); \\ Michel Marcus, May 10 2022

A145340 a(n) = the maximum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

Original entry on oeis.org

2, 3, 4, 4, 6, 6, 6, 6, 8, 8, 8, 9, 8, 8, 10, 8, 12, 12, 8, 12, 12, 10, 12, 12, 12, 9, 8, 12, 12, 10, 12, 12, 8, 12, 12, 12, 12, 10, 16, 8, 18, 18, 14, 14, 12, 12, 16, 12, 12, 12, 12, 20, 20, 18, 9, 16, 16, 16, 12, 16, 8, 12, 12, 16, 16, 8, 16, 20, 12, 12, 12, 24, 10, 12, 16, 16, 16

Offset: 1

Leroy Quet, Oct 08 2008

nonn

Cf. A145339, A008328, A008329, A067889, A103664, A103665.

Table[Max[DivisorSigma[0, Prime[n]-1], DivisorSigma[0, Prime[n]+1]], {n, 1, 100}] (* Stefan Steinerberger, Oct 11 2008 *)

More terms from Stefan Steinerberger and Ray Chandler, Oct 11 2008

cp's OEIS Frontend

A145339 a(n) = the minimum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

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A189977 Primes p such that d(p+1) = 2*d(p-1), where d(k) counts the divisors of k.

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A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

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A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

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A068356 Numbers k such that bigomega(k) = tau(prime(k)+1) where tau(k) = A000005(k).

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A145340 a(n) = the maximum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

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