A000688 -id:A000688 - cp's OEIS frontend

A369165 a(n) = A001222(A000688(n)).

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A369164 at n = 36.
The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 42, 450, 4592, 46185, 462402, 4625478, 46258861, 462599818, 4626029362, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 0.4626... .
First differs from A056170 at n=128, 256, 384, 512, 640.... - R. J. Mathar, Jan 18 2024

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.5.

Cf. A001222, A000688.
Cf. A048109, A059956, A271971.
Cf. A369162, A369163, A369164.

Table[PrimeOmega[FiniteAbelianGroupCount[n]], {n, 1, 100}]

a(n) = bigomega(vecprod(apply(numbpart, factor(n)[, 2])));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))), where c = Sum_{k>=1} d(k) * A001222(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

A369168 Numbers k such that A000005(k) = A000688(k).

Original entry on oeis.org

1, 16, 81, 625, 1296, 2401, 10000, 14641, 23040, 28561, 32256, 38400, 38416, 50625, 50688, 59904, 75264, 78336, 83521, 87552, 89600, 105984, 125440, 130321, 133632, 140800, 142848, 166400, 170496, 185856, 188928, 194481, 198144, 216576, 217600, 234256, 243200

Offset: 1

Amiram Eldar, Jan 15 2024

nonn

The asymptotic density of this sequence is 0 (Ivić, 1983).

If k is a term, then every number with the same prime signature (A124832) as k is a term. The least term of each prime signature is given in A369169.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137.

Cf. A000005, A000688, A124832.
Subsequence of A369170.
A369169 is a subsequence.

Select[Range[250000], DivisorSigma[0, #] == FiniteAbelianGroupCount[#] &]

is(n) = {my(e = factor(n)[,2]); vecprod(apply(x -> x+1, e)) == vecprod(apply(numbpart, e));}

x * log(log(x))/log(x) << N(x) << x / log(x)^(1-eps) for every 0 < eps < 1, where N(x) is the number of terms not exceeding x (Ivić, 1983).

A369169 Terms k of A025487 such that A000005(k) = A000688(k).

Original entry on oeis.org

1, 16, 1296, 23040, 810000, 7257600, 16934400, 283852800, 1437004800, 1944810000, 13970880000, 30735936000, 232475443200, 852409958400, 1765360396800, 3269185920000, 7192209024000, 8029628006400, 28473963210000, 97893956160000, 181803061440000, 1086822696960000

Offset: 1

Amiram Eldar, Jan 15 2024

nonn

Since both A000005(k) and A000688(k) depend only on the prime signature of k (A124832), if k is a term of this sequence then every number m such that A046523(m) = k is a term of A369168.
From David A. Corneth, Jan 15 2024: (Start)
16 | a(n) for n > 1.
This sequence contains A002110(n)^4. (End)

			16 is in the sequence as 16 has 5 divisors (1, 2, 4, 8, 16) and 5 factorizations into prime powers (16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2).

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..377
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137.

Cf. A000005, A000688, A002110, A046523, A124832.

Intersection of A025487 and A369168.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; Select[lps, DivisorSigma[0, #] == FiniteAbelianGroupCount[#] &]

A379359 Numerators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 2, 3, 7, 9, 11, 13, 41, 22, 25, 28, 59, 65, 71, 77, 391, 421, 218, 233, 481, 511, 541, 571, 581, 298, 313, 106, 217, 227, 237, 247, 1739, 1809, 1879, 1949, 3933, 4073, 4213, 4353, 13199, 13619, 14039, 14459, 14669, 14879, 15299, 15719, 15803, 16013, 16223, 16643

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 2, 3, 7/2, 9/2, 11/2, 13/2, 41/6, 22/3, 25/3, 28/3, 59/6, ...

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 13-16, Theorem 1.3.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See section 5.1, Abelian group enumeration constants, p. 274.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000688, A063966, A084911, A370897, A379360 (denominators), A379361.

Numerator[Accumulate[Table[1/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / f(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A000688(k)).

a(n)/A379360(n) = D * n + O(sqrt(n/log(n))), where D = A084911.

A379360 Denominators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 5, 10, 10, 10, 10, 70, 70, 70, 70, 140, 140, 140, 140, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 140, 140, 140, 140, 140, 140

Offset: 1

Amiram Eldar, Dec 21 2024

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 13-16, Theorem 1.3.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See section 5.1, Abelian group enumeration constants, p. 274.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000688, A063966, A370897, A379359 (numerators), A379362.

Denominator[Accumulate[Table[1/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / f(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A000688(k)).

A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 3, 7, 5, 2, 5, 7, 13, 7, 13, 59, 89, 37, 52, 89, 119, 89, 119, 109, 62, 47, 52, 89, 119, 89, 119, 803, 1013, 803, 1013, 1921, 2341, 1921, 2341, 2201, 2621, 2201, 2621, 2411, 2621, 2201, 2621, 2537, 2747, 2537, 2957, 2747, 3167, 1009, 1149, 3307

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 0, 1, 1/2, 3/2, 1/2, 3/2, 7/6, 5/3, 2/3, 5/3, 7/6, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000041, A000688, A063966, A084911, A370897, A379359, A379362 (denominators).

Numerator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).

a(n)/A379362(n) ~ D * c * n, where D = A084911, c = 2/(1 + Sum_{k>=1} 1/(P(k)*2^k)) - 1 = 0.18634377034863729099..., and P(k) = A000041(k).

A379362 Denominators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 15, 30, 30, 30, 30, 210, 210, 210, 210, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 420, 420, 420, 420, 420, 420, 420

Offset: 1

Amiram Eldar, Dec 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000688, A063966, A370897, A379360, A379361 (numerators).

Denominator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).

A369166 Numbers k such that A000688(k) = A000688(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A358817 at n = 165.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 368, 3632, 36266, 362468, 3624664, 36246863, 362468411, 3624675258, ... . From these values the asymptotic density of this sequence, whose existence was proven by Erdős and Ivić (1987) (the constant c in the Formula section), can be empirically evaluated by 0.36246... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, pp. 475-476.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, 51, Number Theory, Budapest, 1987, pp. 45-91. See p. 60.

Cf. A000688, A358817.

Subsequences: A007674, A052213, A085651, A335328.

Select[Range[300], FiniteAbelianGroupCount[#] == FiniteAbelianGroupCount[#+1] &]

lista(kmax) = {my(c1 = 1, c2); for(k = 2, kmax, c2 = vecprod(apply(numbpart, factor(k)[, 2])); if(c1 == c2, print1(k-1, ", ")); c1 = c2);}

The number of terms not exceeding x, N(x) = c * x + O(x^(3/4) * log(x)^4), where c > 0 is a constant (Erdős and Ivić, 1987).

A369167 a(n) = A000688(n + A000688(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 7, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 3, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3

Offset: 1

Amiram Eldar, Jan 15 2024

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, page 478.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Haihong Fan and Wenguang Zhai, A Symmetric Form of the Mean Value Involving Non-Isomorphic Abelian Groups, Symmetry 2022, 14(9), 1755.
Haihong Fan and Wenguang Zhai, On some sums involving the counting function of nonisomorphic Abelian groups, Lithuanian Mathematical Journal, Vol. 63 (2023), pp. 166-180; arXiv preprint, arXiv:2204.02576 [math.NT], 2022.
Aleksandar Ivić, An asymptotic formula involving the enumerating function of finite abelian groups, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 3 (1992), pp. 61-66.

Cf. A000688, A369162.

Table[FiniteAbelianGroupCount[n + FiniteAbelianGroupCount[n]], {n, 1, 100}]

A000688(n) = vecprod(apply(numbpart, factor(n)[, 2]));
a(n) = A000688(n + A000688(n));

Sum_{k=1..n} a(k) = c * n + O(n^(k+eps)) for any eps > 0, where c > 0 is a constant and k = 11/12 (Ivić, 1992), 3/4 (Fan and Zhai, 2023), or 2/3 (Fan and Zhai, 2022).

A369170 Numbers k such that A000005(k) <= A000688(k).

Original entry on oeis.org

1, 16, 32, 64, 81, 128, 243, 256, 512, 576, 625, 729, 768, 1024, 1152, 1280, 1296, 1536, 1600, 1728, 1792, 2048, 2187, 2304, 2401, 2560, 2592, 2816, 2916, 3072, 3125, 3136, 3200, 3328, 3456, 3584, 3888, 4096, 4352, 4608, 4864, 5120, 5184, 5632, 5832, 5888, 6144

Offset: 1

Amiram Eldar, Jan 15 2024

The asymptotic density of this sequence is 0 (Ivić, 1983).

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137.

Cf. A000005, A000688.

Subsequences: A369168, A369169.

Select[Range[6000], DivisorSigma[0, #] <= FiniteAbelianGroupCount[#] &]

is(n) = {my(e = factor(n)[,2]); vecprod(apply(x -> x+1, e)) <= vecprod(apply(numbpart, e));}

The number of terms not exceeding x, N(x) << x / log(x)^(1-eps) for every 0 < eps < 1 (Ivić, 1983).

cp's OEIS Frontend

A369165 a(n) = A001222(A000688(n)).

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A369168 Numbers k such that A000005(k) = A000688(k).

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A369169 Terms k of A025487 such that A000005(k) = A000688(k).

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A379359 Numerators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

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A379360 Denominators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

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A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

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PARI

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A379362 Denominators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

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