A333634 -id:A333634 - cp's OEIS frontend

A348037 a(n) = n / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 5, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Oct 19 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Differs from A003557 at the positions given by A347960.

Cf. A003959, A003968, A333634, A348038, A348039, A348499 (positions of 1's).

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := n / GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348037(n) = (n/gcd(n, A003968(n)));

a(n) = n / A348036(n) = n / gcd(n, A003968(n)).

a(n) = A003557(n) / A348039(n).

A327877 Numbers having an odd number of non-unitary prime factors.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168

Offset: 1

Amiram Eldar, Sep 28 2019

nonn

Differs from A190641(n) from n = 310 (900, the least number with 3 non-unitary prime factors, is in this sequence but not in A190641).

The asymptotic density of the numbers in this sequence is 0.338682... = 1 - A065493.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Transactions of the American Mathematical Society, Vol. 112, No. 2, (1964), pp. 214-227.
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107, No. 1 (1933), pp. 188-232.
I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330.

Cf. A056170, A065493, A190641, A333634 (complement).

A056170[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Select[Range[200], OddQ[A056170[#]] &] (* after Jean-François Alcover at A056170 *)

\\ here b(n) is A056170(n)
b(n)={my(f=factor(n)[, 2]); sum(i=1, #f, f[i]>1)}
{ select(k->b(k)%2, [1..200]) } \\ Andrew Howroyd, Sep 28 2019

A065493 Decimal expansion of the Feller-Tornier constant (1 + A065474)/2.

Original entry on oeis.org

6, 6, 1, 3, 1, 7, 0, 4, 9, 4, 6, 9, 6, 2, 2, 3, 3, 5, 2, 8, 9, 7, 6, 5, 8, 4, 6, 2, 7, 4, 1, 1, 8, 5, 3, 3, 2, 8, 5, 4, 7, 5, 2, 8, 9, 8, 3, 2, 9, 1, 6, 3, 5, 4, 9, 8, 0, 9, 0, 5, 6, 2, 6, 2, 2, 6, 6, 2, 5, 0, 3, 1, 7, 4, 3, 1, 2, 2, 3, 0, 4, 9, 4, 2, 2, 6, 1, 7, 4, 0, 7, 8, 4, 2, 8, 1, 8, 7

Offset: 0

N. J. A. Sloane, Nov 19 2001

The asymptotic density of numbers with an even number of non-unitary prime divisors (A333634). - Amiram Eldar, May 23 2020

Named after the Croatian-American mathematician William Feller (1906-1970) and the German mathematician Erhard Tornier (1894-1982). - Amiram Eldar, Jun 16 2021

			0.661317049469622335289765846274...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4.1, p. 106.

Jayadev S. Athreya, Cristian Cobeli, and Alexandru Zaharescu, Visibility phenomena in hypercubes, arXiv:2204.03147 [math.NT], 2022.
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107 (1933), pp. 188-232.
Mizan R. Khan and Riaz R. Khan, To count clean triangles we count on imph(n), arXiv:2012.11081 [math.CO], 2020. Mentions this constant.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Feller-Tornier Constant.
Eric Weisstein's World of Mathematics, Prime Products.
Wikipedia, Feller-Tornier constant.

Cf. A065474, A078080, A333634.

digits = 98; r[n_] := -2^n; 1/2 + (1/2) Exp[NSum[r[n]*(PrimeZetaP[2*n]/n), {n, 1, Infinity}, NSumTerms -> 1000, WorkingPrecision -> 2 digits ]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

(1 + prodeulerrat(1 - 2/p^2))/2 \\ Amiram Eldar, Mar 17 2021

A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is a subsequence of A333634. See comments in A347892.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Union of A005117 and A347892 (terms that are not squarefree).
Subsequence of A333634.
Positions of ones in A348037.
Cf. A003968, A348030, A348036, A347960.

f[p_, e_] := p*(p + 1)^(e - 1); s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[120], Divisible[s[#], #] &] (* Amiram Eldar, Oct 29 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
isA348499(n) = !(A003968(n)%n);

A335428 Prime exponent of the first Fermi-Dirac factor (number of form p^(2^k), A050376) reached, when starting from n and iterating with A334870, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jun 26 2020

nonn

			For n=27, when iterating with A334870, we obtain the path 27 -> 18 -> 9, with that 9 being the first prime power encountered that has an exponent of the form 2^k, as 9 = 3^2, thus a(27) = 2. See the binary tree A334860 or A334866 for how such paths go.
For n=900, when iterating with A334870 we obtain the path 900 -> 30 -> 15 -> 10 -> 5, and 5^1 is finally a prime power with an exponent that is two's power, thus a(900) = 1. Note that 900 is the first such position of 1 in this sequence that is not listed in A333634.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A209229, A302777, A334860, A334866, A334870, A334872.

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A335428(n) = { while(n>1 && !A302777(n), n = A334870(n)); isprimepower(n); };

\\ Faster, A209229 and A302777 like in above:
A335428(n) = if(1==n,0, while(!A302777(n), if(issquarefree(n), return(1)); if(issquare(n), n = sqrtint(n), n /= core(n))); isprimepower(n));

A348961 Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105

Offset: 1

Amiram Eldar, Nov 05 2021

nonn

First differs from A005117 at n = 24, from A333634 and A348499 at n = 47, and from A336223 at n = 63.
Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 1, and that an e-perfect number (A054979) is a term of this sequence if and only if at least one of the exponents in its prime factorization is not a perfect square.
Since all the e-perfect numbers are products of a primitive e-perfect number (A054980) and a coprime squarefree number, and all the known primitive e-perfect numbers have a nonsquare exponent in their prime factorizations, there is no known e-perfect number that is not in this sequence.

			3 is a term since esigma(3) = 3, 3 * d_e(3) = 3 * 1, so esigma(3) | 3 * d_e(3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348962 are subsequences.

Cf. A049419, A051377, A322791, A333634, A336223, A348499, A348964.

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^# &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

Numbers k such that A000188(k) is a term of A028260.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then A000188(k) = 1, 1 has 0 prime divisors, and 0 is even.
A number k is a term if and only if its powerful part, A057521(k), is a term.
The asymptotic density of this sequence is 7/10 (Cohen, 1964).
The corresponding sequence of numbers k such that the square root of the largest square dividing k has an even number of distinct prime divisors, i.e., numbers k such that A000188(k) is a term of A030231, is A333634.

			2 is a term since the largest square dividing 2 is 1, sqrt(1) = 1, 1 has 0 prime divisors, and 0 is even.
16 is a term since the largest square dividing 16 is 16, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Cf. A000188, A001222, A005117, A028260, A030231, A057521, A333634.

f[p_, e_] := p^Floor[e/2]; Select[Range[100], EvenQ[PrimeOmega[Times @@ (f @@@ FactorInteger[#])]] &]

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

First differs from A333634 at n = 47.
Terms k of A335275 such that A000188(k) is a term of A030231.
Numbers whose powerful part (A057521) is a square term of A030231.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) = 1 has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (Product_{p prime} (1 - 1/(p^2*(p+1))) + Product_{p prime} (1 - (2*p+1)/(p^2*(p+1))))/2 = (0.881513... + 0.394391...)/2 = 0.637952807730728551636349961980617856650450613867264... (Cohen, 1964; the first product is A065465).

			36 is a term since the largest square dividing 36 is 36, which is a unitary divisor, sqrt(36) = 6, 6 = 2 * 3 has 2 distinct prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A333634 and A335275.

Cf. A000188, A001221, A005117, A030231, A057521, A065465.

seqQ[n_] := EvenQ @ Length[(e = Select[FactorInteger[n][[;; , 2]], # > 1 &])] && AllTrue[e, EvenQ[#] &]; Select[Range[100], seqQ]

cp's OEIS Frontend

A348037 a(n) = n / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327877 Numbers having an odd number of non-unitary prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A065493 Decimal expansion of the Feller-Tornier constant (1 + A065474)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A335428 Prime exponent of the first Fermi-Dirac factor (number of form p^(2^k), A050376) reached, when starting from n and iterating with A334870, with a(1) = 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

A348961 Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

Views

Author

Keywords

Comments

Examples

Links