A276936 -id:A276936 - cp's OEIS frontend

A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.

1, 3, 4, 5, 7, 8, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the integer partitions counted by A276429.
The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.68974964705635552968... - Amiram Eldar, Jan 09 2021

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
  11: {5}
  12: {1,1,2}
  13: {6}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A056239, A087153, A112798, A124010, A276078, A276429.
Cf. A324525, A324571, A325127, A325128, A325130, A325131.
Complement of A276936.

q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while not q(k) do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 28 2019

Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k!=PrimePi[p]]&]

A276937 Numbers m with at least one prime factor prime(k) such that prime(k)^k is a divisor of m, but with no factor prime(j) such that prime(j)^(j+1) divides m.

Original entry on oeis.org

2, 6, 9, 10, 14, 18, 22, 26, 30, 34, 38, 42, 45, 46, 50, 58, 62, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 99, 102, 106, 110, 114, 117, 118, 122, 125, 126, 130, 134, 138, 142, 146, 150, 153, 154, 158, 166, 170, 171, 174, 178, 182, 186, 190, 194, 198, 202, 206, 207, 210, 214, 218, 222, 225, 226, 230, 234, 238, 242, 246, 250

Offset: 1

Antti Karttunen, Sep 24 2016

Numbers m for which A276077(m) = 0 and A276935(m) > 0.

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) - Product_{k>=1} (1 - 1/prime(k)^(k-1)) = 0.2803209124521781114031... . - Amiram Eldar, Sep 30 2023

			14 = 2*7 = prime(1)^1 * prime(4)^1 is a member as the first prime factor (2) satisfies the first condition, and neither prime factor violates the second condition.
36 = 4*9 = prime(1)^2 * prime(2)^2 is NOT a member because prime(1)^2 does not satisfy the second condition.
45 = 5*9 = prime(3)^1 * prime(2)^2 is a member as the latter prime factor satisfies the first condition, and neither prime factor violates the second condition.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Intersection of A276078 and A276936.
Topmost row of A276941 (leftmost column in A276942).
Cf. A276935, A276077.

p[n_]:=FactorInteger[n][[All,1]];f[n_]:=PrimePi/@p[n];
yQ[n_]:=Select[n/(Prime[f[n]]^f[n]),IntegerQ]!={};
nQ[n_]:=Select[n/(Prime[f[n]]^(f[n]+1)),IntegerQ]=={};
Select[Range[2,250],yQ[#]&&nQ[#]&] (* Ivan N. Ianakiev, Sep 28 2016 *)

is(n) = {my(f = factor(n), c = 0, k); for (i=1, #f~, k = primepi(f[i, 1]); if(f[i, 2] > k, return(0), if( f[i, 2] == k, c++))); c > 0;} \\ Amiram Eldar, Sep 30 2023

A276935 Number of distinct prime factors prime(k) of n such that prime(k)^k, but not prime(k)^(k+1) is a divisor of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 24 2016

			For n = 12 = 2*2*3 = prime(1)^2 * prime(2)^1, neither of the prime factors satisfies the condition, thus a(12) = 0.
For n = 18 = 2*3*3 = prime(1)^1 * prime(2)^2, both prime factors satisfy the condition, thus a(18) = 1+1 = 2.
For n = 750 = 2*3*5*5*5 = prime(1)^1 * prime(2)^1 * prime(3)^3, only the prime factors 2 and 5 satisfy the condition, thus a(750) = 1+1 = 2.

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

Cf. A276077, A276936.

f[p_, e_] := If[PrimePi[p] == e, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 30 2023 *)

a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i,1]) == f[i,2]);} \\ Amiram Eldar, Sep 30 2023

a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) = A055396(n)], where [] is Iverson bracket, giving 1 as its result when the stated equivalence is true and 0 otherwise.
From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = 1 if e = primepi(p), and 0 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (1/prime(k)^k - 1/prime(k)^(k+1)) = 0.33083690651252383414... . (End)

A276938 Second row of A276941: a(n) = A003961(A276937(n)).

Original entry on oeis.org

3, 15, 25, 21, 33, 75, 39, 51, 105, 57, 69, 165, 175, 87, 147, 93, 111, 275, 195, 231, 123, 255, 129, 141, 525, 159, 363, 325, 285, 177, 273, 345, 425, 183, 201, 343, 825, 357, 213, 435, 219, 237, 735, 475, 429, 249, 267, 399, 575, 465, 291, 561, 555, 483, 303, 975, 309, 321, 725

Offset: 1

Antti Karttunen, Sep 24 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..5000

Row 2 of A276941 (column 2 of A276942).
Cf. A003961, A276937.
Subsequence of A276078, but no common terms with A276936.

(define (A276938 n) (A003961 (A276937 n)))

a(n) = A003961(A276937(n)).

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A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.

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A276937 Numbers m with at least one prime factor prime(k) such that prime(k)^k is a divisor of m, but with no factor prime(j) such that prime(j)^(j+1) divides m.

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A276935 Number of distinct prime factors prime(k) of n such that prime(k)^k, but not prime(k)^(k+1) is a divisor of n.

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A276938 Second row of A276941: a(n) = A003961(A276937(n)).

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