A197987 -id:A197987 - cp's OEIS frontend

A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Aug 18 2016

nonn

Numbers not divisible by p^(1+A000720(p)) for any prime p, where A000720(p) gives the index of prime p: 1 for 2, 2 for 3, 3 for 5, and so on.
Also Heinz numbers of integer partitions where the multiplicity of i does not exceed i for any i (A052335). Differs from A048103 in lacking {625, 1250, 1875, 3750, 4375, 5625, 6875, 8125, 8750, ...}. - Gus Wiseman, Mar 09 2019
Asymptotic density is Product_{i>=1} 1-prime(i)^(-1-i) = 0.72102334... - Amiram Eldar, Oct 20 2020

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros in A276077.
Complement: A276079.
Sequence A276076 sorted into ascending order.
Cf. A000040, A000720.
Subsequence of A048103 from which it differs for the first time at n=451, where a(451) = 626, while A048103(451) = 625, a value missing from here.
Cf. A052335, A056239, A062457, A087153, A109298, A112798, A115584, A117144, A118914, A124010, A197987, A276078.
Cf. A324524, A324525, A324571, A324587, A324588.

Select[Range@ 121, Or[# == 1, AllTrue[FactorInteger[#], PrimePi[#1] >= #2 & @@ # &]] &] (* Michael De Vlieger, Jun 24 2017 *)

isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > primepi(f[k, 1]), return(0))); return (1); \\ Michel Marcus, Jun 24 2017

is(n) = {my(t=1);forprime(p = 2, , t++; pp = p^t; if(n%pp==0, return(0)); if(pp > n, return(1)))} \\ David A. Corneth, Jun 24 2017

upto(n) = {my(v = vector(n,i,1), t=1, res=List()); forprime(p=2, , t++; pp = p^t; if(pp>n, break); for(i=1, n\pp, v[pp*i] = 0)); for(i=1, n, if(v[i]==1, listput(res, i))); res} \\ David A. Corneth, Jun 24 2017

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return all(f[i] <= primepi(i) for i in f)
print([n for n in range(1, 151) if ok(n)]) # Indranil Ghosh, Jun 24 2017

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293

Offset: 0

Omar E. Pol, Sep 09 2018

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.

The partial sums of column k give the column k of A319076.

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

PARI
```
T(n, k) = prime(k)^n;
```

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

cp's OEIS Frontend

A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula