A321650 -id:A321650 - cp's OEIS frontend

A345959 Numbers whose prime indices have alternating sum -1.

6, 15, 24, 35, 54, 60, 77, 96, 135, 140, 143, 150, 216, 221, 240, 294, 308, 315, 323, 375, 384, 437, 486, 540, 560, 572, 600, 667, 693, 726, 735, 864, 875, 884, 899, 960, 1014, 1147, 1176, 1215, 1232, 1260, 1287, 1292, 1350, 1500, 1517, 1536, 1715, 1734, 1748

Offset: 1

Gus Wiseman, Jul 12 2021

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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with even Omega (A001222) and exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

			The initial terms and their prime indices:
    6: {1,2}
   15: {2,3}
   24: {1,1,1,2}
   35: {3,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   77: {4,5}
   96: {1,1,1,1,1,2}
  135: {2,2,2,3}
  140: {1,1,3,4}
  143: {5,6}
  150: {1,2,3,3}
  216: {1,1,1,2,2,2}
  221: {6,7}
  240: {1,1,1,1,2,3}

These multisets are counted by A000070.
The k = 0 version is A000290, counted by A000041.
The k = 1 version is A001105.
The k > 0 version is A026424.
These are the positions of -1's in A316524.
The k = 2 version is A345960.
The k = -2 version is A345962.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A001791/A345910/A345912 count/rank compositions with alternating sum -1.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534/A325535 count separable/inseparable partitions.
A344607 counts partitions with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, A345961.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==-1&]

A345960 Numbers whose prime indices have alternating sum 2.

Original entry on oeis.org

3, 12, 27, 30, 48, 70, 75, 108, 120, 147, 154, 192, 243, 270, 280, 286, 300, 363, 432, 442, 480, 507, 588, 616, 630, 646, 675, 750, 768, 867, 874, 972, 1080, 1083, 1120, 1144, 1200, 1323, 1334, 1386, 1452, 1470, 1587, 1728, 1750, 1768, 1798, 1875, 1920, 2028

Offset: 1

Gus Wiseman, Jul 12 2021

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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with odd Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321650 and ranked by A122111.

			The initial terms and their prime indices:
    3: {2}
   12: {1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   48: {1,1,1,1,2}
   70: {1,3,4}
   75: {2,3,3}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  147: {2,4,4}
  154: {1,4,5}
  192: {1,1,1,1,1,1,2}
  243: {2,2,2,2,2}
  270: {1,2,2,2,3}
  280: {1,1,1,3,4}
  286: {1,5,6}
  300: {1,1,2,3,3}

These partitions are counted by A000097.
The k = 0 version is A000290, counted by A000041.
The k = 1 version is A001105 (reverse: A345958).
The k > 0 version is A026424.
These multisets are counted by A120452.
These are the positions of 2's in A316524 (reverse: A344616).
The k = -1 version is A345959.
The version for reversed alternating sum is A345961.
The k = -2 version is A345962.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A002054/A345924/A345923 count/rank compositions with alternating sum -2.
A056239 adds up prime indices, row sums of A112798.
A088218/A345925/A345922 count/rank compositions with alternating sum 2.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
Cf. A000037, A000070, A028260, A035363, A239830, A341446, A344609, A344610, A344651, A344741, A345197.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==2&]

A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2

Offset: 2

Matthew Vandermast, Apr 28 2014

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)

Row lengths are A051903(n); row sums are A001222(n).
Cf. A217171.
These partitions are ranked by A238745.
For prime indices A296150 instead of exponents we get A321649, rev A321650.
A000700 counts self-conjugate partitions, ranked by A088902.
A003963 gives product of prime indices, conjugate A329382.
A008480 gives number of permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221.
A352486-A352490 are sets related to the fixed points of A122111.
Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Row n is identical to row A124859(n) of table A212171.

A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 2, 2, 4, 2, 3, 2, 6, 4, 2, 2, 4, 2, 2, 4, 8, 2, 5, 2, 2, 3, 2, 2, 5, 4, 2, 6, 2, 2, 4, 2, 10, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 6, 2, 2, 6, 4, 4, 2, 2, 2, 7, 4, 2, 2, 2, 2, 4, 2, 2, 4, 12, 3, 2, 2, 2, 2, 4, 2, 7, 2, 2, 6, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 2

Offset: 1

Gus Wiseman, May 25 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.

Twice the row media of A321649 or A321650.
For mean instead of twice median we have A326839/A326840.
This is the conjugate version of A360005.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation in terms of Heinz numbers.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A352491 gives n minus Heinz number of conjugate.
A363220 counts partitions with same median as conjugate.
Cf. A046682, A321648, A325040, A326567/A326568, A326848, A362617, A362618, A363223, A363224.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]

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A345959 Numbers whose prime indices have alternating sum -1.

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A345960 Numbers whose prime indices have alternating sum 2.

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A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

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A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

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