A382469 -id:A382469 - cp's OEIS frontend

A383726 Square array read by ascending antidiagonals, where row n lists numbers m such that omega(m) = n and the largest prime factor of m equals the sum of its remaining distinct prime factors, where omega(m) = A001221(m).

30, 3135, 60, 3570, 6279, 70, 844305, 7140, 8855, 90, 1231230, 1218945, 8970, 9405, 120

Offset: 3

Paolo Xausa, May 07 2025

			Array begins:
  n\k|       1        2        3        4        5  ...
  -----------------------------------------------------
   3 |      30,      60,      70,      90,     120, ... = A365795
   4 |    3135,    6279,    8855,    9405,   10695, ... = A383728
   5 |    3570,    7140,    8970,   10626,   10710, ... = A383729
   6 |  844305, 1218945, 2496585, 2532915, 3024021, ...
   7 | 1231230, 2062830, 2181270, 2462460, 3327870, ...
  ...     |                                       \______ A383727 (main diagonal)
       A383725

Cf. A001221, A365795, A382469, A383725, A383727, A383728, A383729.

Module[{dmax = 5, a, m}, a = Table[m = Times @@ Prime[Range[n]] - 1; Table[While[Length[#] != n || Total[Most[#]] != Last[#] & [FactorInteger[++m][[All,1]]]]; m, dmax-n+3], {n, dmax+2, 3, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]

A383728 Numbers k such that omega(k) = 4 and the largest prime factor of k equals the sum of its remaining distinct prime factors, where omega(k) = A001221(k).

Original entry on oeis.org

3135, 6279, 8855, 9405, 10695, 11571, 15675, 16095, 17255, 17391, 18837, 20615, 20735, 26691, 28083, 28215, 31031, 32085, 34485, 34713, 36519, 41151, 41615, 43953, 44275, 45695, 46655, 47025, 47859, 48285, 48495, 50439, 52173, 53475, 54131, 56511, 56823, 57239, 59295, 59565

Offset: 1

Paolo Xausa, May 08 2025

nonn

			32085 is a term because it has 4 distinct prime factors (3, 5, 23 and 31) and the largest one is the sum of the others (3 + 5 + 23 = 31).

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

Row n = 4 of A383726.

Cf. A001221, A365795, A382469, A383725, A383726, A383729.

N:= 10^5: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N/(3*5*7),2)]):
V:= NULL:
for j from 1 while P[j]^3*(3*P[j]) < N do
  for k from j+1 while P[j]*P[k]^2*(P[j]+2*P[k]) < N do
    for l from k+1 while P[j]*P[k]*P[l] * (P[j]+P[k]+P[l]) <= N do
      p4:= P[j]+P[k]+P[l];
      if not isprime(p4) then next fi;
      for d1 from 1 while P[j]^d1 * P[k] * P[l] * p4 <= N do
       for d2 from 1 while P[j]^d1 * P[k]^d2 * P[l] * p4 <= N do
        for d3 from 1 while P[j]^d1 * P[k]^d2 * P[l]^d3  * p4 <= N do
         for d4 from 1 while  P[j]^d1 * P[k]^d2 * P[l]^d3 * p4^d4 <= N do
             V:= V,P[j]^d1 * P[k]^d2 * P[l]^d3 * p4^d4
od od od od od od od:
sort([V]); # Robert Israel, Jun 09 2025

A383728Q[k_] := Length[#] == 4 && Total[Most[#]] == Last[#] & [FactorInteger[k][[All, 1]]];
Select[Range[10^5], A383728Q]

A383729 Numbers k such that omega(k) = 5 and the largest prime factor of k equals the sum of its remaining distinct prime factors, where omega(k) = A001221(k).

Original entry on oeis.org

3570, 7140, 8970, 10626, 10710, 14280, 16530, 17850, 17940, 20706, 21252, 21420, 24738, 24882, 24990, 26910, 28560, 31878, 32130, 33060, 35700, 35880, 36890, 38130, 41412, 42504, 42840, 44330, 44850, 49476, 49590, 49764, 49938, 49980, 52170, 53550, 53820, 54834, 55986, 57120

Offset: 1

Paolo Xausa, May 08 2025

nonn

			10710 is a term because it has 5 distinct prime factors (2, 3, 5, 7 and 17) and the largest one is the sum of the others (2 + 3 + 5 + 7 = 17).

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

Row n = 5 of A383726.

Cf. A001221, A365795, A382469, A383725, A383726, A383728.

N:= 10^5: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N/(2*3*5*7),2)]):
V:= NULL:
i:= 1:
for j from i+1 while P[i]*P[j]^3*(P[i]+3*P[j]) < N do
  for k from j+1 while P[i]*P[j]*P[k]^2*(P[i]+P[j]+2*P[k]) < N do
    for l from k+1 while P[i]*P[j]*P[k]*P[l] * (P[i]+P[j]+P[k]+P[l]) <= N do
      p5:= P[i]+P[j]+P[k]+P[l];
      if not isprime(p5) then next fi;
      for d1 from 1 while P[i]^d1 * P[j] * P[k] * P[l] * p5 <= N do
       for d2 from 1 while P[i]^d1 * P[j]^d2 * P[k] * P[l] * p5 <= N do
        for d3 from 1 while P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l] * p5 <= N do
         for d4 from 1 while P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l]^d4 * p5 <= N do
           for d5 from 1 while  P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l]^d4 * p5^d5 <= N do
             V:= V,P[i]^d1 * P[j]^d2 * P[k]^d3 * P[l]^d4 * p5^d5
od od od od od od od od:
sort([V]); # Robert Israel, Jun 09 2025

A383729Q[k_] := Length[#] == 5 && Total[Most[#]] == Last[#] & [FactorInteger[k][[All, 1]]];
Select[Range[10^5], A383729Q]

A383725 a(n) is the least number k such that omega(k) = n and the largest prime factor of k equals the sum of its remaining prime factors, where omega(k) = A001221(k).

Original entry on oeis.org

30, 3135, 3570, 844305, 1231230, 463798335, 1089218130, 410825520105, 905980145070, 818186519485335, 1461885412557570, 2023416377587710105, 3676255934199278430, 6175645531427513476335, 14590719651042312667890, 29263451149172039260325865, 67794672364404337821058590

Offset: 3

Paolo Xausa, May 07 2025

nonn

a(n) and n have opposite parity. - David A. Corneth, May 08 2025

			a(3) = 30 is the smallest number having 3 distinct prime factors (namely 2, 3, and 5) such that the largest one is the sum of the others (2 + 3 = 5).
a(4) = 3135 is the smallest number having 4 distinct prime factors (namely 3, 5, 11 and 19) such that the largest one is the sum of the others (3 + 5 + 11 = 19).

Cf. A001221, A002110, A068873, A102330, A365795, A382469, A383728, A383729.

First column of A383726.

isok(k, n) = my(f=factor(k)); (omega(f)==n) && (vecsum(f[,1]) == 2*vecmax(f[,1]));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 08 2025

a(n) = (Product_{i=1..n-1} A102330(n-1,i))*A068873(n-1).
a(n) = A383726(n,1).
a(n) >= A002110(n).

A382468 a(n) = (largest prime factor of n) minus (its remaining distinct prime factors).

Original entry on oeis.org

2, 3, 2, 5, 1, 7, 2, 3, 3, 11, 1, 13, 5, 2, 2, 17, 1, 19, 3, 4, 9, 23, 1, 5, 11, 3, 5, 29, 0, 31, 2, 8, 15, 2, 1, 37, 17, 10, 3, 41, 2, 43, 9, 2, 21, 47, 1, 7, 3, 14, 11, 53, 1, 6, 5, 16, 27, 59, 0, 61, 29, 4, 2, 8, 6, 67, 15, 20, 0, 71, 1, 73, 35, 2, 17, 4, 8, 79, 3, 3

Offset: 2

Paolo Xausa, Mar 31 2025

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A212665 (positions of negative terms), A215142 (positions of ones), A382469 (positions of zeros).

Cf. A006530, A008472, A027748.

A382468 := proc(n)
    2*A006530(n)-A008472(n) ;
end proc:
seq( A382468(n),n=1..50) ; # R. J. Mathar, Apr 01 2025

A382468[n_] := Last[#] - Total[Most[#]] & [FactorInteger[n][[All, 1]]];
Array[A382468, 100, 2]

a(p) = p, for p prime.

a(n) = 2*A006530(n) - A008472(n).

A383677 Irregular triangle read by rows: T(n,k), 2 <= n , 3 <= k <= largest k such that A067175(k) <= n , is the smallest n-digit number m such that omega(m) = A001221(m) = k, and its largest prime factor equals the sum of its remaining prime factors. or -1 if no such number exists.

Original entry on oeis.org

30, 120, -1, 1080, 3135, 3570, 10336, 10695, 10626, -1, 100672, 102695, 103730, 844305, -1, 1003520, 1005039, 1003450, 1218945, 1231230, -1, 10017286, 10000295, 10003390, 10064145, 10314150, -1, 100216924, 100019275, 100017216, 100367745, 100327920, 463798335, -1

Offset: 2

Jean-Marc Rebert, May 11 2025

For n ranging from 2 to 20, the corresponding maximum values of k are as follows: [3, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16].

			T(4,3) = 1080 is the smallest 4-digit number having 3 distinct prime factors (namely 2, 3, and 5) such that the largest one is the sum of the others (2 + 3 = 5).
T(5,4) = 10695 is the smallest 5-digit number having 4 distinct prime factors (namely 3, 5, 23 and 31) such that the largest one is the sum of the others (3 + 5 + 23 = 31).
Triangle begins:
       30;
      120,      -1;
     1080,    3135,    3570;
    10336,   10695,   10626,      -1;
   100672,  102695,  103730,  844305,      -1;
  1003520, 1005039, 1003450, 1218945, 1231230, -1;
  ...

Cf. A001221, A002110, A067175, A365795, A382469, A383725, A383726, A383728, A383729.

A383858 Irregular triangle read by rows: T(n,k) (n >= 4, 4 <= k <= A384502(n)) is the smallest n-digit number m with k distinct prime factors, such that these factors can be divided into two subsets of at least two elements each, both summing to the same value. If no such number exists, T(n,k) = -1.

Original entry on oeis.org

2145, 2310, 10725, 10374, 101065, 100050, 255255, 510510, 1005993, 1000350, 1036035, 1009470, 10006081, 10000130, 10012065, 10004610, 100010225, 100001300, 100001195, 100009910, 111546435, 223092870, 1000083889, 1000008758, 1000001751, 1000005270, 1002569295, 1001110110

Offset: 4

Jean-Marc Rebert, May 12 2025

The maximum values of k for each row n>=1 are respectively 0, 0, 0, 5, 5, 7, 7, ...
The corresponding sums are:
3+13 = 5+11, 3+11 = 2+5+7, 3+13 = 5+11, 3+19 = 2+7+13, 5+41 = 17+29, 2+29 = 3+5+23, 11+17 = 3+5+7+13, 5+7+17 = 2+3+11+13, 3+53 = 19+37, 2+19 = 3+5+13, 3+5+23 = 7+11+13, 5+7+23 = 2+3+11+19, 17+163 = 23+157, 2+383 = 5+7+373, 3+17+71 = 5+7+79, 2+7+71 = 3+5+11+61, 5+109 = 7+107, 2+383 = 5+7+373, 37+53 = 5+7+31+47, 5+11+103 = 2+7+13+97, 7+19+23 = 3+5+11+13+17, 3+5+19+23 = 2+7+11+13+17, ...

			T(4,4) = 2145 = 3*5*11*13 is the smallest four-digit number with four distinct prime factors (3, 5, 11, and 13), where the prime factors can be partitioned into two subsets of at least two elements each, both summing to the same value: 3+13 = 5+11.
T(5,4) = 2310 = 2*3*5*7*11 is the smallest five-digit number with four distinct prime factors (2, 3, 5, 7 and 11), where the prime factors can be partitioned into two subsets of at least two elements each, both summing to the same value: 3+11 = 2+5+7.
The lower triangle begins at T(4,4):
[     2145,      2310];
[    10725,     10374];
[   101065,    100050,    255255,    510510];
[  1005993,   1000350,   1036035,   1009470];
[ 10006081,  10000130,  10012065,  10004610];
[100010225, 100001300, 100001195, 100009910, 111546435, 223092870]; ...

Cf. A001221, A365795, A382469, A383677, A383725, A383726, A383728, A383729, A384502.

cp's OEIS Frontend

A383726 Square array read by ascending antidiagonals, where row n lists numbers m such that omega(m) = n and the largest prime factor of m equals the sum of its remaining distinct prime factors, where omega(m) = A001221(m).

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A383728 Numbers k such that omega(k) = 4 and the largest prime factor of k equals the sum of its remaining distinct prime factors, where omega(k) = A001221(k).

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A383729 Numbers k such that omega(k) = 5 and the largest prime factor of k equals the sum of its remaining distinct prime factors, where omega(k) = A001221(k).

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A383725 a(n) is the least number k such that omega(k) = n and the largest prime factor of k equals the sum of its remaining prime factors, where omega(k) = A001221(k).

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A382468 a(n) = (largest prime factor of n) minus (its remaining distinct prime factors).

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A383677 Irregular triangle read by rows: T(n,k), 2 <= n , 3 <= k <= largest k such that A067175(k) <= n , is the smallest n-digit number m such that omega(m) = A001221(m) = k, and its largest prime factor equals the sum of its remaining prime factors. or -1 if no such number exists.

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