cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A377869 Numbers k such that A276085(k) has no divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 18, 19, 22, 23, 24, 26, 27, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 50, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 70, 71, 72, 73, 74, 75, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109
Offset: 1

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Author

Antti Karttunen, Nov 10 2024

Keywords

Comments

Numbers k for which A276085(k) is in A048103, i.e., in the range of A276086.
Numbers k such that A276086(A276085(A276085(k))) is equal to A276085(k).
This is a subsequence of A369003 (numbers k for which A276085(k) is not a multiple of 4), from which it differs for the first time at n=122, where a(122) = 175, as A369003(122) = 174 is not included in this sequence.
From Antti Karttunen, Nov 17 2024: (Start)
More generally, this is equal to setwise difference A000027 \ (A369002 U A377872 U A377878 U ...).
Even semiprimes (A100484) is a subsequence, but the odd semiprimes (A046315) are all in the complement (A377873), because they are included in A369002.
For k=1..6, there are 8, 70, 656, 6531, 64773, 645301 terms <= 10^k. Question: What is the asymptotic density of this sequence, if it has one?
(End)

Examples

			A276085(11) = 210 = 2*3*5*7, which has no divisor of the form p^p, therefore 11 is included in this sequence.
A276085(15) = 8 = 2^2 * 2, which has a divisor of the form p^p, therefore 15 is NOT included in this sequence.
A276085(25) = 12 = 2^2 * 3, which has a divisor of the form p^p, therefore 25 is NOT included.
A276085(34) = 30031 = A002110(1-1)+A002110(7-1) (as 34 = 2*17 = prime(1)*prime(7)), and because 30031 = 59*509 (an odd semiprime), 34 is included.
A276085(60) = 10 = 2*5, which has no divisors of the form p^p, therefore 60 is included.
A276085(102) = 30033 = 3^2 * 47 * 71, which has no p^p divisors, therefore 102 is included.
A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which thus has a divisor of the form p^p, and therefore 174 is NOT included in this sequence.

Links

Crossrefs

Cf. A046315, A048103, A276085, A276086, A377868 (characteristic function), A377873 (complement).
Setwise difference A369003 \ A377875.
Cf. A000040, A100484, A377871, A377989 (subsequences).
Cf. A369002, A377872, A377878.

Programs

A377872 Numbers k for which A276085(k) is a multiple of 27, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

1, 55, 95, 115, 155, 174, 187, 203, 232, 265, 282, 297, 323, 325, 329, 335, 376, 391, 396, 438, 462, 474, 511, 513, 515, 527, 528, 539, 553, 584, 606, 616, 621, 632, 649, 654, 678, 684, 704, 707, 745, 763, 791, 798, 808, 828, 837, 872, 901, 904, 906, 912, 913, 931, 966, 978, 1002, 1057, 1064, 1073, 1074, 1075, 1104, 1105
Offset: 1

Views

Author

Antti Karttunen, Nov 10 2024

Keywords

Comments

A multiplicative semigroup; if m and n are in the sequence then so is m*n.
From Antti Karttunen, Nov 17 2024: (Start)
Question: What is the asymptotic density of this sequence? There are 1, 3, 56, 484, 4899, 50034, 508254 terms <= 10^k, for k=1..7. See also questions in A377869 and in A377878.
If 3*x is a term, then 4*x is also a term, and vice versa.
Contains no even semiprimes (A100484), semiprimes of the form 3*prime (A001748), nor terms of the form 4*prime (A001749).
(End)

Links

Crossrefs

Cf. A276085, A377869, A377876, A377878.
Subsequence of A339746, and of A377873.
Cf. also A369007, A377875.

Programs

  • PARI
    isA377872(n) = { my(m=27, f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= Mod(prime(i),m); i++); s += f[k, 2]*pr); (0==lift(s)); };

Formula

{k such that Sum e*A377876(A000720(p)-1) == 0 (mod 27), when k = Product(p^e)}.

A377871 Numbers k such that neither k nor A276085(k) has divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 41, 42, 43, 45, 46, 47, 50, 53, 58, 59, 61, 62, 63, 66, 67, 70, 71, 73, 74, 75, 78, 79, 82, 83, 86, 89, 90, 94, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 113, 114, 117, 118, 122, 125, 126, 127, 130, 131, 134, 137, 138, 139, 142
Offset: 1

Views

Author

Antti Karttunen, Nov 10 2024

Keywords

Comments

Range of A276087, where A276087(n) = A276086(A276086(n)) [the twofold application of the primorial base exp-function].
A276087(0) = 2, and for n >= 0, A276087(A143293(n)) = A000040(n+2), therefore all primes are included.
From Antti Karttunen, Nov 17 2024: (Start)
Even semiprimes > 4 form a subsequence, because A006862 (Euclid numbers) is a subsequence of A048103. Note that A276087(A376416(n)) = A276086(A006862(n)) = A100484(1+n). On the other hand, none of the odd semiprimes, A046315, occur here, because they are all included in A369002, and thus in A377873. Similarly, A276092 after its initial 1 is a subsequence, because A057588 (Kummer numbers) is also a subsequence of A048103.
For k=1..6, there are 6, 52, 486, 4775, 46982, 467372 terms <= 10^k. Question: Does this sequence have an asymptotic density?
(End)

Examples

			A276087(A002110(10)) = A276086(A276086(A002110(10))) = A276086(A000040(10+1)) = A276086(31) = 14, therefore 14 is included in this sequence.

Links

Crossrefs

Intersection of A048103 and A377869.
Sequence A276087 sorted into ascending order.
Cf. A377870 (characteristic function).
Subsequences: A000040, A100484 (after its initial 4), A276092 (after its initial 1).
Cf. A006862, A143293, A276086, A376416.

Programs

A377875 Numbers k for which A276085(k) is not a multiple of 4 and has at least one divisor of the form p^p, with p an odd prime, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

174, 232, 282, 325, 376, 438, 462, 474, 539, 584, 606, 616, 632, 654, 678, 798, 808, 872, 904, 906, 931, 966, 978, 1002, 1064, 1074, 1075, 1105, 1127, 1182, 1208, 1288, 1302, 1304, 1336, 1398, 1432, 1506, 1519, 1576, 1626, 1662, 1736, 1755, 1842, 1864, 1866, 2008, 2168, 2216, 2226, 2340, 2425, 2442, 2456, 2488, 2514
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2024

Keywords

Comments

Among the initial 15275 terms, in 110 cases A276085(k) is a multiple of 5^5, and for no cases a multiple of 7^7.

Examples

			A276085(55) = 216 = 2^3 * 3^3, which although it has a divisor of the form p^p, with p an odd prime, it is also a multiple of 4, and therefore 55 is NOT included in this sequence.
A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which has a divisor of the form p^p, with p an odd prime, thus 174 is included in this sequence.
A276085(4823) = 614889782588493750 =  2 * 3 * 5^5 * 13 * 2522624749081, thus 4823 is included.
A276085(1104299) = 11231250 = 2 * 3 * 5^5 * 599, thus 1104299 is included.

Links

Crossrefs

Setwise difference A369003 \ A377869.
Setwise difference A377873 \ A369002.
Cf. A083345, A276085, A377868, A377872, A377874.

Programs

  • PARI
    A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); };
A377868(n) = if(isprime(n), 1, my(x=A276085(n),pp); forprime(p=2,, pp = p^p; if(!(x%pp), return(0)); if(pp > x, return(1))));
isA377875(n) = ((A083345(n)%2) && !A377868(n));

Formula

{k such that A377868(k) < A377874(k)}.

A377878 Numbers k for which A276085(k) is a multiple of 3125, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

1, 4823, 8267, 9553, 15623, 15833, 15929, 20633, 23393, 28417, 33079, 34027, 36941, 37129, 37939, 42599, 43249, 44431, 47291, 49374, 60097, 65832, 66323, 69287, 69749, 70613, 74063, 74281, 74333, 74999, 77231, 83881, 86191, 86551, 87776, 88727, 99683, 106481, 108673, 111366, 113922, 115729, 118517, 124841, 126054, 129337
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2024

Keywords

Comments

A multiplicative semigroup; if m and n are in the sequence then so is m*n.
Question: Does this sequence have asymptotic density? See also questions in A377872 and A377869.

Links

Crossrefs

Cf. A000720, A276085, A377869, A377877.
Subsequence of A373140, and of A377873.
Cf. also A377872.

Programs

  • PARI
    isA377878(n) = { my(m=5^5, f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= Mod(prime(i),m); i++); s += f[k, 2]*pr); (0==lift(s)); };

Formula

{k such that Sum e*A377877(A000720(p)-1) == 0 (mod 5^5), when k = Product(p^e)}.
Showing 1-5 of 5 results.

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