A325223 -id:A325223 - cp's OEIS frontend

A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 3, 3, 6, 3, 7, 2, 4, 4, 8, 3, 4, 5, 4, 3, 9, 3, 10, 4, 5, 6, 5, 4, 11, 7, 6, 3, 12, 4, 13, 4, 4, 8, 14, 4, 6, 4, 7, 5, 15, 5, 6, 3, 8, 9, 16, 4, 17, 10, 5, 5, 7, 5, 18, 6, 9, 5, 19, 5, 20, 11, 5, 7, 7, 6, 21, 4, 6, 12

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is smaller, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - min(4,5) = 4.

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

The number of times k appears in the sequence is A325232(k).

Cf. A001222, A052126, A056239, A061395, A064989, A065770, A112798, A174090, A257990, A263297, A325134, A325169, A325223, A325225, A325227.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Total[primeMS[n]]-Min[Length[primeMS[n]],Max[primeMS[n]]]],{n,100}]

a(n) = A056239(n) - min(A001222(n), A061395(n)) = A056239(n) - A325225(n).

A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

Original entry on oeis.org

6, 10, 14, 15, 18, 21, 22, 26, 33, 34, 35, 38, 39, 46, 50, 51, 54, 55, 57, 58, 62, 65, 69, 74, 75, 77, 82, 85, 86, 87, 91, 93, 94, 95, 98, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 147, 155, 158, 159, 161, 162, 166, 177, 178

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   54: {1,2,2,2}
   55: {3,5}
   57: {2,8}
   58: {1,10}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's in A325226.

Cf. A056239, A093641, A112798, A174090, A257541, A307517, A325223, A325224, A325225, A325231.

filter:= proc(n) local F;
   F:= sort(ifactors(n)[2],(a,b)-> a[1]Robert Israel, Apr 14 2019

Select[Range[100],PrimeOmega[#/Power@@FactorInteger[#][[-1]]]==1&]

from sympy import factorint
A325230_list = [n for n, m in ((n, factorint(n)) for n in range(2,10**6)) if len(m) == 2 and m[min(m)] == 1] # Chai Wah Wu, Apr 16 2019

A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

Original entry on oeis.org

6, 10, 12, 14, 22, 24, 26, 34, 38, 46, 48, 58, 62, 74, 82, 86, 94, 96, 106, 118, 122, 134, 142, 146, 158, 166, 178, 192, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 384, 386, 394, 398, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Also numbers n such that the sum of prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n is 1. 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, and their sum is A056239.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   48: {1,1,1,1,2}
   58: {1,10}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
   96: {1,1,1,1,1,2}
  106: {1,16}
  118: {1,17}

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Positions of 1's in A325223.

Cf. A001222, A056239, A060687, A061395, A093641, A112798, A174090, A257541, A265283, A325224, A325225, A325227, A325232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[primeMS[#]]-Max[Length[primeMS[#]],Max[primeMS[#]]]==1&]

from sympy import isprime
A325231_list = [n for n in range(6,10**6) if ((not n % 2) and isprime(n//2)) or (bin(n)[2:4] == '11' and bin(n).count('1') == 2)] # Chai Wah Wu, Apr 16 2019

cp's OEIS Frontend

A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

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A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

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A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python