A268112 -id:A268112 - cp's OEIS frontend

A335189 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 0, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

21, 23, 43, 47, 66, 68, 78, 82, 86, 111, 115, 119, 157, 160, 164, 167, 273, 287, 343, 359, 438, 442, 456, 460, 507, 527, 579, 581, 615, 665, 813, 818, 834, 839, 931, 943, 947, 959, 1082, 1090, 1111, 1119, 1140, 1148, 1248, 1288, 1333, 1340, 1346, 1354, 1360, 1367, 1592, 1640, 1641, 1679, 1807, 1847, 2034, 2067, 2069, 2163, 2190

Offset: 1

Petros Hadjicostas, May 26 2020

nonn

For more numbers in this list (up to 10^6), see one of the links below by Krattenthaler and Rivoal. The first few numbers L for which v_p(H_L-1) = 2 (rather than 1) for some prime p <= L are 43, 2034 and 2069 with corresponding primes 7, 13 and 7.

The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.

David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H_L - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H_L - 1) > 0.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
Carlo Sanna, On the p-adic valuation of harmonic numbers, Journal of Number Theory, 166 (2016), 41-46.

Cf. A007757, A131657, A131658, A268112, A334981.

h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n, if(valuation((numerator(h(n)-1)), p) > 0, return(1))); return(0)};
for(n=1, 1000, if(is(n)==1, print1(n,",")))

A334981 Numbers k such that the numerator of the k-th alternating harmonic number H'(k) is divisible by the square of a prime less than k.

Original entry on oeis.org

30, 241, 1057, 1499, 10499, 318425

Offset: 1

Petros Hadjicostas, May 25 2020

This sequence was inspired by unsolved conjectures related to the papers by Boyd (1994) and Krattenhaler and Rivoal (2007-2009, 2009) about the harmonic numbers H(k) = Sum_{i=1..k} 1/i. See also the comments for sequences A007757, A131657, A131658, and A268112. Here we are dealing with the alternating harmonic numbers H'(k) = Sum_{i=1..k} (-1)^(i+1)/i.
For the harmonic numbers H(k), it is not known whether there is k >= 1 and a prime p such that v_p(H(k)) >= 4, where v_p(x) is the p-adic valuation of x. Since p cannot be present in both the numerator and the denominator of H(k), this is equivalent to saying that the numerator of H(k) cannot be divisible by the fourth power of a prime p.
If variations of the above conjecture are true, then some conditional results in Krattenhaler and Rivoal (2007-2009, 2009) would hold. Boyd (1994) found only 5 integers k such that there is a prime p < k with v_p(H(k)) >= 3. Since 1994 no other k's have been found that satisfy the latter inequality.
We claim that a similar conjecture holds for the alternating harmonic numbers H'(k): there is no pair of an integer k and a prime p such that v_p(H'(k)) >= 4; i.e., there is no k for which the numerator of H'(k) is divisible by the fourth power of a prime.
This sequence contains those k's for which there is a prime p < k with v_p(H'(k)) >= 2. Up to 2000, we have only been able to find four such k's. The corresponding primes for 30, 241, 1057, and 1499 are 7, 19, 37, and 7. We have v_7(H'(30)) = v_19(H'(241)) = v_37(H'(1057)) = 2, while v_7(H'(1499)) = 3.
It holds v_7(H'(10499)) = 2 and  v_691(H'(318425)) = 2. a(7) > 5*10^5. - Giovanni Resta, May 26 2020

David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 73-94.

Cf. A007757, A131657, A131658, A268112 (similar sequence for harmonic numbers).

ah(n) = sum(i=1, n, (-1)^(i+1)/i);
is(n) = {forprime(p=1, n-1, if(valuation((numerator(ah(n))), p) > 1, return(1))); return(0)}

listaa(nn) = {my(h=0,s=1,nh); for (n=1, nn, h += s/n; nh = numerator(h); forprime(p=1, n-1, if(valuation(nh, p) > 1, print1(n, ", "); break)); s = -s;);} \\ Michel Marcus, May 26 2020

a(5) from Michel Marcus, May 26 2020

a(6) from Giovanni Resta, May 26 2020

A335207 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

Original entry on oeis.org

43, 2034, 2069, 9702, 9712, 67258, 102691, 102727, 147253, 904332

Offset: 1

Petros Hadjicostas, May 26 2020

This is a subset of A335189. All numbers in this list were copied from one of the links below by Krattenthaler and Rivoal.
For all L in this list (up to 904332), we have v_p(H_L - 1) = 2 with corresponding primes as follows: p(1) = 7, p(2) = 13, p(3) = 7, p(4) = p(5) = 11, p(6) = 41, p(7) = p(8) = 11, p(9) = 53, and p(10) = 97.
The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.

Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H_L - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H_L - 1) > 0.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.

Cf. A007757, A131657, A131658, A268112, A335189.

A335207_list := proc(bound) local p, h, H, L, n;
L := NULL; h := 0;
for n from 1 to bound do
    h := h + 1/n; H := h - 1; p:= 2;
    while p <= n do
        if padic:-ordp(H, p) <= 1
           then p := nextprime(p);
           else L := L, n; break;
        fi
    od;
od; L end:
A335207_list(2222); # Peter Luschny, May 29 2020

list(nn) = {my(h=-1); for (n=1, nn, h += 1/n; forprime(p=1, n-1, if(valuation(h, p) > 1, print1(n, ", "); break)););} \\ Petros Hadjicostas, May 26 2020, courtesy of Michel Marcus

A335210 Numbers L such that there is a prime p <= L for which v_p(H'(L) - 1) > 0, where v_p(x) is the p-adic valuation of x and H'(L) is the L-th alternating harmonic number.

Original entry on oeis.org

16, 19, 81, 211, 231, 232, 242, 243, 267, 274, 340, 357, 559, 637, 644, 898, 1121, 1391, 1399, 1412, 1433, 1436, 1439, 1470, 1474, 1501, 1892, 2304, 2336, 2477, 2496, 2520, 2768, 2948, 2992, 3351, 3367, 3563, 3953, 3966, 4431, 4505, 4587, 4596, 4626, 5061, 6058, 6781, 6847, 6861

Offset: 1

Petros Hadjicostas, May 26 2020

nonn

This sequence was inspired by the database of Krattenthaler and Rivoal (see the link below) about all triplets of numbers (L, p, v_p(H(L) - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H(L) - 1) > 0. Here v_p(x) is the p-adic valuation of x and H(L) is the L-th harmonic number. See also the sequences A268112, A335189, and A335207.

Here we tabulate the numbers L >= 1 for which there is a prime p <= L such that v_p(H'(L) - 1) >= 1, where H'(L) = Sum_{k=1..L} (-1)^(k+1)/k. The first few numbers L for which v_p(H'(L) - 1) = 2 (rather than 1) for some p <= L are 1501, 4596, and 9367 with corresponding p equal to 7, 19, and 37, respectively.

Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H(L) - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H(L) - 1) > 0.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.

Cf. A268112, A335189, A335207.

listaa(nn) = {my(h=0, s=1, nh); for (n=1, nn, h += s/n; nh = numerator(h-1); forprime(p=1, n-1, if(valuation(nh, p) > 0, print1(n, ", "); break)); s = -s; ); }

cp's OEIS Frontend

A335189 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 0, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

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A334981 Numbers k such that the numerator of the k-th alternating harmonic number H'(k) is divisible by the square of a prime less than k.

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A335207 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

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A335210 Numbers L such that there is a prime p <= L for which v_p(H'(L) - 1) > 0, where v_p(x) is the p-adic valuation of x and H'(L) is the L-th alternating harmonic number.

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PARI