A088164 -id:A088164 - cp's OEIS frontend

A284759 a(n) = (Sum_{i=1..n-1} i^(n-2)) mod n^3.

0, 1, 3, 14, 100, 115, 196, 500, 189, 333, 847, 1022, 1352, 1671, 1920, 3432, 3757, 2937, 1444, 7730, 1092, 427, 4232, 8668, 15000, 13037, 19197, 20902, 1682, 17999, 16337, 27856, 32043, 31873, 16170, 14298, 47915, 5603, 12792, 8260, 16810, 18949, 51772, 64526

Offset: 1

Felix Fröhlich, Apr 02 2017

nonn

Conjecture: For n > 1, a(n) = 0 if and only if n is a term of A088164, i.e., n is a Wolstenholme prime (cf. Mestrovic, 2012, Conjecture 2.10).

Seiichi Manyama, Table of n, a(n) for n = 1..1000
R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.

Cf. A000578, A076015, A088164, A284760.

seq(add(i^(n-2),i=1..n-1) mod n^3, n=1..100);

Table[Mod[Sum[i^(n - 2), {i, n - 1}], n^3], {n, 44}] (* Michael De Vlieger, Apr 02 2017 *)

a(n) = lift(Mod(sum(i=1, n-1, i^(n-2)), n^3))

a(n)=my(m=n^3,e=n-2); lift(sum(i=1,n-1, Mod(i,m)^e)) \\ Charles R Greathouse IV, Apr 07 2017

a(n) = A076015(n-1) modulo A000578(n).

A290059 a(n) = binomial(2*prime(n)-1, prime(n)-1) where prime(n) is the n-th prime.

Original entry on oeis.org

3, 10, 126, 1716, 352716, 5200300, 1166803110, 17672631900, 4116715363800, 15033633249770520, 232714176627630544, 873065282167813104916, 212392290424395860814420, 3318776542511877736535400, 812850570172585125274307760, 3136262529306125724764953838760

Offset: 1

Martin Renner, Jul 19 2017

nonn

Charles Babbage (1791-1871) proved in 1819 that for every prime p > 2 this sequence is congruent to 1 (mod p^2).

Joseph Wolstenholme (1829-1891) proved in 1862 that for every prime p > 3 this sequence is congruent to 1 (mod p^3).

Seiichi Manyama, Table of n, a(n) for n = 1..261
Charles Babbage, Demonstration of a theorem relating to prime numbers, The Edinburgh philosophical journal 1 (1819), p. 46-49.
Joseph Wolstenholme, On certain properties of prime numbers, The quarterly journal of pure and applied mathematics 5 (1862), p. 35-39.

Cf. A088164. Subsequence of A001700.

seq(binomial(2*ithprime(i)-1,ithprime(i)-1),i=1..16);

Array[Function[p, Binomial[2*p - 1, p - 1]]@ Prime@ # &, 16] (* Michael De Vlieger, Jul 19 2017 *)
Binomial[2#-1,#-1]&/@Prime[Range[20]] (* Harvey P. Dale, Mar 29 2024 *)

a(n) = my(p=prime(n)); binomial(2*p-1, p-1); \\ Michel Marcus, Jul 19 2017

from sympy import prime, binomial
def a(n):
    p=prime(n)
    return binomial(2*p - 1, p - 1)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 20 2017

log(a(n)) ~ 2*log(2)*n * (log(n) + log(log(n)) - 1). - Vaclav Kotesovec, May 07 2022

A290347 Numerators of the Harary index for the n-halved cube graph.

Original entry on oeis.org

0, 1, 6, 26, 100, 1096, 3920, 13936, 16544, 296256, 1068672, 11652352, 42658304, 1100471296, 4079876096, 15205967872, 56939270144, 642281037824, 2423854317568, 9177027411968, 34846713511936, 1459319692460032, 5568939824513024, 21297365878571008

Offset: 1

Eric W. Weisstein, Jul 28 2017

For p > 3, if p is prime, then p^2 divides a(p). Conjecture: for n > 3, if n^2 divides a(n), then n is prime. Primes p such that p^3 diviedes a(p) are probably A088164. - Thomas Ordowski, Mar 30 2025

			First few terms are 0, 1, 6, 26, 100, 1096/3, 3920/3, 13936/3, 16544, 296256/5, ....

Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Halved Cube Graph

Cf. A000265, A088164, A290348 (denominators), A330718.

Table[-2^(n - 1) HarmonicNumber[n] - 2^(2 n - 1) Re[LerchPhi[2, 1, n + 1]], {n, 20}] // Numerator

a(n) = -2^(n-1)*HarmonicNumber(n)-2^(2*n-1)*Re(LerchPhi(2,1,n+1)).

For n > 1, A000265(a(n)) = A330718(n). - Thomas Ordowski, Mar 30 2025

A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.

Original entry on oeis.org

1, 39, 78, 155, 310, 465, 546, 793, 798, 930, 1092, 1586, 1638, 1860, 2170, 2379, 2394, 3172, 3276, 3965, 4340, 4758, 4914, 5219, 6045, 6510, 7137, 7182, 7930, 9516, 9828, 10374, 10438, 11102, 11895, 12090, 13020, 14274, 15657, 15860, 16843, 16891, 18135

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

A generalization of Wolstenholme primes (A088164) for composite number.
Leudesdorf proved in 1888 that the numerator of Sum_{k=1..n, gcd(k,n)=1} 1/k is divisible by n^2 for all (but not only) numbers n with gcd(n,6)=1, which is a generalization of Wolstenholme's theorem.
Terms that are coprime to 6: 1, 155, 793, 3965, 5219, 16843, 16891, 51305, ...
a(41) = A088164(1) = 16843.
A general conjecture: if, for some e > 0, m^e | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k), then m^(e-1) | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k^2). Note: in this case, the exponent e = 3. Problem: are there numbers m > 1 such that m^4 | Numerator(Sum_{k=1..m, gcd(k,m)=1} 1/k)? - Thomas Ordowski, Aug 10 2019
This general conjecture was checked up to 10^4. This problem has no solution up to 10^5. - Amiram Eldar, Aug 10 2019
It appears that all odd terms of this sequence are odd numbers m such that the numerator of Sum_{k=1..m, gcd(k,m)=1} 1/k^2 is divisible by m^2. - Thomas Ordowski, Aug 12 2019

			Sum_{k=1..39, gcd(k,39)=1} 1/k = 46855131783993/15222026943200, and 46855131783993 = 39^3 * 789884047, thus 39 is in the sequence.

G. H. Hardy and E. M. Wright, Introduction to the theory of numbers, 5th edition, Oxford, England: Clarendon Press, 1979, pp. 100-102.

David A. Corneth, Table of n, a(n) for n = 1..174 (terms <= 4*10^5; first 100 terms from Amiram Eldar)
C. Leudesdorf, Some results in the elementary theory of numbers, Proceedings of the London Mathematical Society, Vol. 20 (1888), pp. 199-212.
Eric Weisstein's World of Mathematics, Leudesdorf Theorem.
Wikipedia, Wolstenholme's theorem.

Cf. A088164, A093600.

seqQ[n_] :=  Module[{}, g[m_] := GCD[n, m] == 1; Divisible[Numerator[Plus @@ (1/Select[Range[n], g])], n^3]]; Select[Range[10^5], seqQ]

isok(n) = numerator(sum(k=1, n, if (gcd(n, k)==1, 1/k))) % n^3 == 0; \\ Michel Marcus, Aug 11 2017

upto(n) = {my(v = vector(n), d = divisors(n), res = List(), squarefreepart(n) = factorback(factorint(n)[, 1])); v[1] = 1; for(i = 2, n, v[i] = v[i-1] + 1/i; ); for(j = 1, n, fr = v[j]; d = divisors(squarefreepart(j)); for(i = 2, #d, fr += 1/d[i] * v[j/d[i]] * (-1)^omega(d[i]) ); if(numerator(fr) % j^3 == 0, listput(res, j); ) ); res } \\ David A. Corneth, Aug 23 2019

A332786 a(n) = numerator(-1/n + Sum_{k=1..n} 2^(k-1)/k).

Original entry on oeis.org

0, 3, 3, 61, 25, 137, 343, 32663, 2357, 74689, 66671, 5299069, 2416531, 115545821, 106974277, 637525199, 74575583, 1588674349, 4496071973, 3234136824109, 1535024393629, 5843920343363, 5575228585159, 1961561381531581, 93953561866435, 9016382638527647, 2888981280567587, 200248741591132607, 96525489421136333

Offset: 1

Thomas Ordowski, Feb 24 2020

If p > 3 is a prime, then p^2 | a(p).
Does the above statement follow from Wolstenholme's theorem?
If p is a Wolstenholme prime (A088164), then p^3 | a(p).
However, it should be noted that also 7^3 | a(7).
Conjecture: there are no pseudoprimes m such that m^2 | a(m).
Is 7^2 the only weak pseudoprime (i.e., a composite m such that m | a(m))?

			a(5) = numerator(-1/5 + 1/1+2/2+4/3+8/4+16/5) = numerator(128/15 - 1/5) = numerator(25/3) = 25.

Robert Israel, Table of n, a(n) for n = 1..1367

Cf. A001008, A088164, A108866, A330718.

f:= proc(n) local k; numer(-1/n + add(2^(k-1)/k,k=1..n)) end proc:
map(f, [$1..30]); # Robert Israel, Sep 15 2024

n = 30; Numerator[Accumulate @ Table[(2^(k-1))/k, {k, 1, n}] - 1/Range[n]] (* Amiram Eldar, Feb 24 2020 *)

a(n) = numerator(-1/n + sum(k=1, n, 2^(k-1)/k)); \\ Michel Marcus, Feb 24 2020

a(n) = numerator(-2/n + S(n))/2 for odd n and a(n) = numerator(-2/n + S(n)) for even n, where S(n) = Sum_{k=1..n} 2^k/k, see A108866 / A229726.

a(n) = numerator(Sum_{k=1..n} (2^(k-1)-1)/k + Sum_{k=1..n-1} 1/k), see A330718 / A330719 and A001008 / A002805.

More terms from Amiram Eldar, Feb 24 2020

A212557 Primes p such that (p, p-9) is an irregular pair.

Original entry on oeis.org

67, 877

Offset: 1

Felix Fröhlich, May 21 2012

No further terms found up to 163577833 in extended tables (see Buhler link). - Michel Marcus, Apr 17 2014

Joe Buhler and David Harvey, Irregular primes to 163 million, 2009.
W. Johnson, Irregular primes and Cyclotomic Invariants, Math. Comp., 1975, Vol. 29, No. 129, pp. 113-120.

Special instances of A000928. Variant of A088164 (for the terms of A088164, (p, p-3) is an irregular pair).

is(n)=lift(Mod(numerator(bernfrac(n-9)), n)==0)
forprime(p=8, , if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Jan 19 2015

A263429 Smallest prime p such that binomial(2*p-1, p-1) == 1 (mod p^n), or 0 if no such p exists.

Original entry on oeis.org

2, 3, 5, 16843

Offset: 1

Felix Fröhlich, Oct 18 2015

For n > 1, smallest p = prime(i) such that A244919(i) = n.
For n > 3, p is a term of A088164.
Conjecture: a(n) = 0 for n > 4 (McIntosh, 1995, p. 387).

R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), 381-389.

Cf. A088164, A244919.

a(n) = my(p=2); while(Mod(binomial(2*p-1, p-1), p^n)!=1, p=nextprime(p+1)); p

A282410 a(n) = binomial(2*p-1, p-1) modulo p^5, where p = prime(n).

Original entry on oeis.org

3, 10, 126, 1716, 30614, 2198, 1100513, 713337, 4635628, 4511966, 15729649, 49285370, 10820598, 115444165, 110571496, 84562137, 145202954, 386548644, 208729523, 1232287574, 790871562, 2277840181, 3525066856, 4912928962, 7258488370, 8723558568, 9006255935

Offset: 1

Felix Fröhlich, Feb 14 2017

nonn

Conjecture: a(n) != 1 for all n (cf. McIntosh, 1995, p. 387).
See arXiv:1502.05750, Theorem 2 for several conditions equivalent to p having a(n) = 1.
Clearly, a prime p such that a(n) = 1 must be a Wolstenholme prime, i.e., a term of A088164.
a(n) is prime for n: 1, 7, 19, 59, 76, 92, 109, 112, 165, 196, 221, 249, 263, 326, etc. Robert G. Wilson v, Feb 14 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. Aebi and G. Cairns, Wolstenholme again, arXiv:1502.05750 [math.NT], 2015.
R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), 381-389.

Cf. A088164, A242473.

f[n_] := Block[{p = Prime@n}, Mod[ Binomial[ 2p -1, p -1], p^5]]; Array[f, 27] (* Robert G. Wilson v, Feb 14 2017 *)
Table[Mod[Binomial[2p-1,p-1],p^5],{p,Prime[Range[30]]}] (* Harvey P. Dale, Jul 07 2022 *)

a(n) = my(p=prime(n)); lift(Mod(binomial(2*p-1, p-1), p^5))

from sympy import Mod, binomial, prime
def A282410(n): return int(Mod(binomial(2*(p:=prime(n))-1,p-1,evaluate=False),p**5)) # Chai Wah Wu, Apr 24 2025

A276808 Odd prime numbers p such that pBernoulli(p-1) + (p-1)!(p-1) == 0 (mod p^3).

Original entry on oeis.org

17, 1733, 18433

Offset: 1

René Gy, Sep 18 2016

For all other odd primes, the congruence holds mod p^2 only.

Cf. A088164, A197632.

lista(nn) = {forprime(p=3, nn, if (!((p*bernfrac(p-1) + (p-1)!*(p-1)) % p^3) , print1(p, ", ")););} \\ Michel Marcus, Sep 18 2016

A309398 a(n) is the nearest integer to log(log(10^n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Felix Fröhlich, Sep 27 2019

The sequence grows relatively slowly. For example, for n < 10^7, a(n) <= 17.

a(n) is roughly the expected number of Wieferich primes (cf. A001220 and Knauer, Richstein, 2005, p. 1560) as well as the expected number of Fibonacci-Wieferich primes (Wall-Sun-Sun primes) (cf. McIntosh, Roettger, 2007, p. 2091) and Wolstenholme primes (cf. A088164 and McIntosh, 1995, p. 387) with at most n digits. It is also roughly the expected number of Wilson primes with at most n digits (cf. A007540 and Costa, Gerbicz, Harvey, 2014).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, arXiv:1209.3436 [math.NT], 2012; Mathematics of Computation, Vol. 83, No. 290 (2014), 3071-3091, DOI:10.1090/S0025-5718-2014-02800-7.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation, Vol. 74, No. 251 (2005), 1559-1563.
R. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica 71 (1995), 381-389.
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Mathematics of Computation, Vol. 76, No. 260 (2007), 2087-2094.

Cf. A000193, A001220, A007540, A088164.

Round[Log[Log[10^Range[90]]]] (* Harvey P. Dale, Jan 16 2024 *)

PARI
```
a(n) = round(log(log(10^n)))
```

a(n) = round(log(log(10^n))) = log n + O(1).

cp's OEIS Frontend

A284759 a(n) = (Sum_{i=1..n-1} i^(n-2)) mod n^3.

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A290059 a(n) = binomial(2*prime(n)-1, prime(n)-1) where prime(n) is the n-th prime.

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A290347 Numerators of the Harary index for the n-halved cube graph.

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A290815 Numbers m such that the numerator of Sum_{k=1..m, gcd(k,m) = 1} 1/k is divisible by m^3.

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A332786 a(n) = numerator(-1/n + Sum_{k=1..n} 2^(k-1)/k).

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A212557 Primes p such that (p, p-9) is an irregular pair.

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A263429 Smallest prime p such that binomial(2*p-1, p-1) == 1 (mod p^n), or 0 if no such p exists.

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A276808 Odd prime numbers p such that pBernoulli(p-1) + (p-1)!(p-1) == 0 (mod p^3).