A326151 -id:A326151 - cp's OEIS frontend

A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

2, 30, 154, 190, 390, 442, 506, 658, 714, 874, 1110, 1118, 1254, 1330, 1430, 1786, 1794, 1798, 1958, 2310, 2414, 2442, 2470, 2730, 2958, 3034, 3066, 3266, 3390, 3534, 3710, 3770, 3874, 3914, 4042, 4466, 4526, 4758, 4930, 5106, 5434, 5474, 5642, 6090, 6106

Offset: 1

Gus Wiseman, Jan 20 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
   154: {1,4,5}
   190: {1,3,8}
   390: {1,2,3,6}
   442: {1,6,7}
   506: {1,5,9}
   658: {1,4,15}
   714: {1,2,4,7}
   874: {1,8,9}
  1110: {1,2,3,12}

Even squarefree case of A326149.
For nonprime instead of even we have A326158.
Squarefree case of A379319.
Even case of A379844.
Partitions of this type are counted by A380221, see A379733, A379735.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A112798, A175508, A324850, A324851, A326150, A326151, A326153/A326154, A326156, A326157.

Select[Range[2,1000],EvenQ[#]&&SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]

A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

Original entry on oeis.org

49, 63, 65, 81, 125, 150, 154, 165, 169, 190, 198, 259, 273, 333, 351, 361, 364, 385, 390, 435, 442, 468, 481, 490, 495, 506, 525, 561, 580, 595, 609, 630, 658, 675, 700, 714, 741, 765, 781, 783, 810, 840, 841, 846, 874, 900, 918, 925, 931, 935, 952, 988

Offset: 1

Gus Wiseman, Jan 23 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The terms together with their prime indices begin:
   49: {4,4}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
  125: {3,3,3}
  150: {1,2,3,3}
  154: {1,4,5}
  165: {2,3,5}
  169: {6,6}
  190: {1,3,8}
  198: {1,2,2,5}
  259: {4,12}
  273: {2,4,6}
  333: {2,2,12}
  351: {2,2,2,6}
  361: {8,8}
  364: {1,1,4,6}
For example, 198 has prime indices {1,2,2,5}, and 20/10 is an integer > 1, so 198 is in the sequence.

The fraction A003963(n)/A056239(n) reduces to A326153(n)/A326154(n).
The non-proper version is A326149, superset of A326150.
Also a superset of A326151.
The squarefree case is A326158 without first term.
Partitions of this type are counted by A380219.
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379735, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A028422, A036844, A112798, A301988, A319000, A324850, A324851, A326156, A379319, A379844.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Divisible[Times@@prix[#],Total[prix[#]]]&&!SameQ[Times@@prix[#],Total[prix[#]]]&]

A326157 Squarefree numbers whose product of prime indices is twice their sum of prime indices.

Original entry on oeis.org

65, 154, 190

Offset: 1

Gus Wiseman, Sep 12 2019

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.

This sequence is finite. Proof: k = p_1*p_2*...*p_t is a term iff q_1*q_2*...*q_t = 2*(q_1 + q_2 + ... + q_t), where q_i = pi(p_i) and q_1 < q_2 < ... < q_t. If t = 2, then 1/2 = 1/q_1 + 1/q_2. Thus q_1 <= 3, we have k = prime(3)*prime(6) = 65. If t = 3, then 1/2 = 1/(q_1*q_2) + 1/(q_1*q_3) + 1/(q_2*q_3). Thus q_1*q_2 <= 5, we have k = prime(1)*prime(4)*prime(5) = 154 or k = prime(1)*prime(3)*prime(8) = 190. If t > 3, then 1/2 = Sum_{i=1..t} q_i/(q_1*q_2*...*q_t) < Sum_{i=1..t} i/t! < 1/2, a contradiction. - Jinyuan Wang, Jun 27 2020

			The sequence of terms together with their prime indices starts:
   65: {3,6}
  154: {1,4,5}
  190: {1,3,8}

Intersection of A005117 and A326151.
Product of prime indices is A003963.
Sum of prime indices is A056239.
Cf. A000720, A001222, A069016, A112798, A301987, A325041, A325042, A326152.

q:= n-> (l-> andmap(i-> i[2]=1, l) and (h-> mul(i, i=h)=2*add(i,
        i=h))(map(i-> numtheory[pi](i[1]), l)))(ifactors(n)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, Sep 12 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],SquareFreeQ[#]&&SameQ[Times@@primeMS[#],2*Plus@@primeMS[#]]&]

A003963(a(n)) = 2 * A056239(a(n)).

A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

Original entry on oeis.org

2, 1, 21, 25, 39, 35, 57, 55, 49, 65, 75, 77, 129, 95, 91, 105, 183, 119, 125, 143, 133, 185, 147, 161, 169, 195, 175, 209, 339, 217, 255, 253, 259, 305, 247, 285, 273, 245, 301, 299, 345, 323, 325, 357, 371, 435, 669, 391, 361, 403, 399, 473, 343, 469, 481

Offset: 0

Gus Wiseman, Jan 21 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The least number whose prime indices satisfy (product) - (sum) = 3 is 25 (prime indices {3,3}), so a(3) = 25.

Position of first appearance of n in A325036.
For sum instead of difference we have A379682, firsts of A379681.
A000040 lists the primes, differences A001223.
A003963 multiplies together prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
The subtraction A325036 takes the following values:
- zero: A301987, counted by A001055 (strict A045778).
- negative: A325037, counted by A114324, see A318029
- positive: A325038, counted by A096276 shifted right
- negative one: A325041, counted by A028422
- one: A325042, counted by A001055 shifted right
- nonnegative: A325044, counted by A096276
- nonpositive: A379721, counted by A319005
Cf. A000720, A178503, A175508, A318950, A319000, A325034, A326151, A326153/A326154, A379319, A379722.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pp=Table[Total[prix[n]]-Times@@prix[n],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[pp,-i][[1,1]],{i,0,mnrm[-DeleteCases[pp,0|_?Positive]]}]

Satisfies A003963(a(n)) - A056239(a(n)) = n.

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A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

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A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

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A326157 Squarefree numbers whose product of prime indices is twice their sum of prime indices.

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A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

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