Divisors Of Sum Of Previous Rows

[franktaw at netscape.net]

A120576 by rows: 2; 1; 3; 6; 4,12; 7,14,28; 11,77; 5,15,33,55,165; 
73,146,219,438; 9,18,657,1314; 
8,16,23,24,36,46,48,69,72,92,138,144,184,207,276,368,414,552,828,1104,165
6,3312; 1847,12929; 5541,9235,27705; 19,38,3694,35093,70186; 
487,974,1948,3896,7792,11201,22402,44804,89608,179216; 
139,278,556,1112,67693,135386,270772,541544; 
701,1402,2224,2804,5608,11216,97439,194878,389756,779512,1559024; 
22,44,253,506,1012,4549,9098,18196,50039,100078,104627,200156,209254,4185
08,1150897,2301794,4603588; 13776209; 27552418; 55104836; 110209672; 
220419344; 32,440838688; 
61,122,244,488,976,1952,451679,903358,1806716,3613432,7226864,14453728,27
552419,55104838,110209676,220419352,440838704,881677408

Smallest still missing is 10.

A120577 by rows: 3; 1; 2,4; 5,10; 25; 50; 20,100; 11,22,44,55,110,220; 
31,62,341,682; 29,58,899,1798; 79,158,2291,4582; 
37,74,148,316,2923,5846,11692; 8,4091,8182,16364,32728; 
7,21,4481,13443,31367,94101; 23,449,529,10327,237521; 
17,34,85,170,2861,5722,14305,28610,48637,97274,243185,486370; 
40,59,118,236,295,472,590,599,1180,1198,2360,2396,2995,4792,5990,11980,23
960,35341,70682,141364,176705,282728,353410,706820,1413640; 
46,115,230,20233,40466,101165,202330,465359,930718,2326795,4653590; 
9,53,159,477,28081,84243,252729,1488293,4464879,13394637; 
41,697,47501,807517,1947541,33108197; 13,5309207,69019691; 
103,206,695867,1391734,71674301,143348602; 
145,349,419,493,1745,2095,2465,5933,7123,10121,12151,29665,35615,50605,60
755,146231,172057,206567,731155,860285,1032835,2485927,4240699,12429635,2
1203495,72091883,360459415; 418369639,836739278; 2091848195

Still missing 6.

A120578 by rows: 4; 1,2; 7; 14; 28; 8,56; 
3,5,6,10,12,15,20,24,30,40,60,120; 31,93,155,465; 13,39,403,1209; 
17,169,221,2873; 21,293,879,2051,6153; 
25,50,311,622,1555,3110,7775,15550; 
37,43,74,86,148,172,259,301,518,602,1036,1204,1591,3182,6364,11137,22274,
44548; 4933,9866,19732,34531,69062,138124; 
42,84,14799,29598,59196,103593,207186,414372; 
9,11,18,22,23,26,27,33,46,54,63,66,69,77,78,91,99,117,126,138,143,154,161
,182,189,198,207,231,234,253,273,286,297,299,322,351,378,414,429,462,483,
506,546,594,598,621,693,702,759,819,858,897,966,1001,1242,1287,1386,1449,
1518,1638,1771,1794,2002,2079,2093,2277,2457,2574,2691,2898,3003,3289,354
2,3861,4158,4186,4347,4554,4914,5313,5382,6006,6279,6578,6831,7722,8073,8
694,9009,9867,10626,12558,13662,15939,16146,18018,18837,19734,23023,27027
,29601,31878,37674,46046,47817,54054,56511,59202,69069,88803,95634,113022
,138138,177606,207207,414414,621621,1243242; 2556907,5113814; 12784535; 
25569070; 10227628,51138140; 44,28125977,56251954,112503908; 
227,353,681,1059,2043,2497,2951,3177,3883,4589,6129,7491,8853,9531,11649,
13767,22473,26559,32461,34947,41301,50479,67419,79677,80131,97383,104841,
123903,151437,240393,292149,454311,721179,876447,881441,1041703,1362933,2
163537,2644323,3125109,7932969,9375327,11458733,23798907,28125981,3437619
9,103128597,309385791; 19,57,14943223,44829669,283921237,851763711; 
121,363,5639729,16919187,62037019,186111057,682407209,2047221627

Missing 16.

A120579 (I presume) by rows: 5; 1; 2,3,6; 17; 34; 4,68; 
7,10,14,20,28,35,70,140; 8,16,29,58,116,232,464; 19,73,1387; 1433,2866; 
7165; 14330; 5732,28660; 11,22,44,15763,31526,63052; 
38,55,83,95,110,166,190,209,415,418,830,913,1045,1577,1826,2090,3154,4565
,7885,9130,15770,17347,34694,86735,173470; 
12,15,24,30,40,41,60,82,109,120,123,164,205,218,246,327,328,410,436,492,5
45,615,654,820,872,984,1090,1230,1308,1635,1640,2180,2460,2616,3270,4360,
4469,4920,6540,8938,13080,13407,17876,22345,26814,35752,44690,53628,67035
,89380,107256,134070,178760,268140,536280; 
23,133,161,437,719,3059,5033,13661,16537,95627,115759,314203,2199421; 
2482097,4964194; 12410485; 24820970; 9928388,49641940; 
27303067,54606134,109212268; 121,300333737; 
137,605,685,1507,7247,7535,16577,36235,79717,82885,398585,876887,992839,4
384435,4964195,10921229,54606145,120133519,600667595; 
46,1171,2342,25969,26933,51938,53866,597287,1194574,30409699,60819398,699
423077,1398846154

Missing 9.

These probably are all permutations (for starting value > 1, of 
course).  I don't see any approach to a proof.

Franklin T. Adams-Watters


-----Original Message-----
From: Leroy Quet <qq-quet at mindspring.com>

I just submitted these sequences:

>%S A120576 2,1,3,6,4,12,7,14,28,11,77
>%N A120576 Irregular array where the nth row are the divisors, not
>occurring earlier in the sequence, of the sum of the terms in all 
previous
>rows. a(1)=2.
>%C A120576 Is this sequence a permutation of the positive integers?
>%e A120576 Array begins:
>2
>1
>3
>6
>4,12
>7,14,28
>Now these terms add up to 77. So row 7 is the divisors of 77 which 
don't
>occur earlier in the sequence. 1 and 7 occur in earlier rows, so row 7 
is
>(11,77).
>%Y A120576 A120577,A120578,A120579
>%O A120576 1
>%K A120576 ,more,nonn,

>%S A120577 3,1,2,4,50,10,25,50,20,100
>%N A120577 Irregular array where the nth row are the divisors, not
>occurring earlier in the sequence, of the sum of the terms in all 
previous
>rows. a(1)=3.
>%C A120577 Is this sequence a permutation of the positive integers?
>%e A120577 Array begins:
>3
>1
>2,4
>5,10
>25
>50
>Now these terms add up to 100. So row 7 is the divisors of 100 which 
don't
>occur earlier in the sequence. 1,2,4,5,10,25, and 50 occur in earlier
>rows, so row 7 is (20,100).
>%Y A120577 A120576,A120578,A120579
>%O A120577 1
>%K A120577 ,more,nonn,

>%S A120578 4,1,2,7,14,28,8,56,3,5,6,10,12,15,20,24,30,40,60,120
>%N A120578 Irregular array where the nth row are the divisors, not
>occurring earlier in the sequence, of the sum of the terms in all 
previous
>rows. a(1)=4.
>%C A120578 Is this sequence a permutation of the positive integers?
>%e A120578 Array begins:
>4
>1,2
>7
>14
>28
>8,56
>Now these terms add up to 120. So row 7 is the divisors of 120 which 
don't
>occur earlier in the sequence. 1,2,4, and 8 occur in earlier rows, so 
row
>7 is (3,5,6,10,12,15,20,24,30,40,60,120).
>%Y A120578 A120576,A120577,A120579
>%O A120578 1
>%K A120578 ,more,nonn,

(I also submitted the beginning of the sequence for a(1) = 5, but you 
get
the idea.)

Question: Are the sequences generated this way each a permutation of 
the
positive integers for a(1) = any integer >= 2?

If some a(1)'s lead to permutations and some do not, what would the
sequence be of the a(1)'s which do (or don't) lead to permutations?