zak08

Sigma(T(n))=T(m)
Sunday, September 26, 2010 7:30 PM
From:
<>

To:
"seqfaneu" <seqfan@seqfan.eu>
%N Sum of divisors of n-th triangular number is
m-th triangular number.
%F Sigma(T(n))=T(m)
%F A000203(A000217(n))=A000217(m)
%Y A000217 Triangular numbers:
%Y A000203 sigma(n) = sum of divisors of n.

The first, say, 1000 terms will be greatly appreciated
(private plz).

Thanks,
Zak

n,m
1,1
8,13
9,12
215,384
458,575
520,783
2232,4095
3251,4607
3634,4095
5349,6912
9489,12543
10051,13824
10463,16895
14072,21504
14705,20735
17463,27264
27812,40959
46552,68256
55889,76544
79614,104832
100055,175104
106941,130559
110682,146432
113839,180224
119098,129024
181690,202239
197223,316224
214600,328320
270570,372735
287585,395199
333291,512000
384463,532575
439206,512000
443115,732159
608563,787968
767496,1181439
1097448,1756160
1335300,2253824
1401829,1673216
1471870,1626624
1545794,2037503
1600173,1926144
1617333,1870847
1697586,2121984
1776076,2253824
1841379,2985983
1872262,2024703
2619859,3842559
3320063,5909760
3587786,4734975
3590569,4065984
3628856,6455295
3697866,4622335
4096453,4172544
4353683,6455295
4545736,6723584
4881166,4967424
5148307,6635519
5489297,6890688
5839422,6952959
6381878,8142848
6703433,8414783
6933073,7624448
7774688,13045760
8171639,15271424
9496063,13996800
9755711,17791487
10521801,13152255
10803634,11915775
11565552,18399744
11913956,18849024
17201718,20893184
18180439,26562815
18448336,25357760
18514175,33640704
19235999,37965824
21677350,24312959
21782493,27288575
23274555,36468224
24946264,39334400
25418006,31071039
30050970,39301119
30317125,36315135
34102705,43614207
34781974,39472640
36134516,53581824
37023301,40076288
37122528,60949503
41362253,49724415
41667961,44859392
42590138,49741824
43884297,53362880
50077313,70451199
58698839,99442944
60841449,85184000
70008381,85659167
75897865,87325695
81591984,153957375
86323894,103167999
89576459,156542463
98322338,118374399
10521801, 13152255
10803634, 11915775
11565552, 18399744
11913956, 18849024
17201718, 20893184
18180439, 26562815
18448336, 25357760
18514175, 33640704
19235999, 37965824
21677350, 24312959
21782493, 27288575
23274555, 36468224
24946264, 39334400
25418006, 31071039
30050970, 39301119
30317125, 36315135
34102705, 43614207
34781974, 39472640
36134516, 53581824
37023301, 40076288
37122528, 60949503
41362253, 49724415
41667961, 44859392
42590138, 49741824
43884297, 53362880
50077313, 70451199
58698839, 99442944
60841449, 85184000
70008381, 85659167
75897865, 87325695
81591984, 153957375
86323894, 103167999
89576459, 156542463
98322338, 118374399

Link

A083016

Sep. 26th, 2010 at 4:58 PM

Link

A058362: First 239 terms

Sep. 26th, 2010 at 2:49 AM

First 239 terms of A058362
For these I used 10804 terms of A059044 calculated by D. S. McNeil, Sep 24 2010

121174811,1128318991,2201579179,2715239543,2840465567,3510848161,
3688067693,3893783651,5089850089,5825680093,6649068043,6778294049,
7064865859,7912975891,8099786711,9010802341,9327115723,9491161423,
9544001791,10101930253,10523406343,13193702321,13343904191,13696222633,
13814960791,14270167963,14417987803,14785488167,15015589423,15107682347,
15190575311,15954967387,16355858549,16397510873,16860841447,16977684523,
18229994813,19198207247,19632877931,19939382171,19969756529,20340679067,
20624258069,20704857203,21606081947,21942140327,22655907689,22668740803,
22871565439,23078268971,24133414027,24285733271,25452376273,25516713623,
25533712129,25711162493,25845216427,26077317023,26096443039,27377538113,
27705831001,28433266787,28524113557,28589927473,29004438311,29176895687,
29551076389,29607445051,29697334487,30663283373,31075512701,31385634373,
31611804947,34482174547,34490811791,35613704947,35779139447,37010936671,
37211873519,37537789739,37562283943,37626325571,37660045607,38023061603,
38664114029,39079285681,39421901521,40367270071,40693734881,41077455547,
41705115391,43215328697,43655560489,43862523119,43878668143,44196120509,
45194847791,45321778427,46006843591,47023892827,47415377039,47428196251,
48235506811,48341742577,48732936191,49697365153,49899905491,50223008453,
51097407979,51260289019,51524014159,52362381983,52397429317,52564342679,
52781644099,53191868039,53513740807,53514693157,53902050617,53942296319,
54112157843,54186816133,55157908423,55182788201,55315106261,55542356011,
55850357411,55952708303,56441073229,56488396183,56826607763,58353520367,
58762211741,58807389181,59055426971,59323589047,59395508083,59518435643,
59699836723,60014315209,60319867099,60521788511,60571400801,63318867719,
63330976543,63399016501,63887535637,64107497393,64182969979,64346827561,
64852652993,64854182717,64930013381,65526355603,65607399181,65699099573,
65894084797,66266424457,66306357259,66562666571,67204937893,67466739251,
68256058873,68299107697,68771213513,68843678339,68914748723,69385686463,
70731413167,71071122047,71219836921,72622356823,72631417679,73505265283,
74002875977,75159871607,75190971179,75349139483,75638624077,75840843493,
76298094581,76909714933,77028613573,77063732671,77086359709,77220840517,
77660567707,77895447373,78417854237,78712829197,79071760567,79256189231,
80480768293,80793486881,80806055437,81120743153,81692882707,81886974017,
82149125473,82548317959,82762243489,83174507341,83904447041,84366808759,
84380624729,84697452163,84849503741,85744299349,86530670893,86921347697,
87111161279,87334358141,87797406517,87887156989,87969590711,88596346981,
89051153627,89482269403,89485651873,89952008813,91349177831,92157377839,
92428472537,92892934969,93460293649,93608096521,93886329259,94381622351,
95057227099,96563070053,97230135221,97294450381,97471441699,98264395861,
98350128403,98823150043,99097428047,99682312669,99981755323

Link

A059044: First 220 terms

Sep. 25th, 2010 at 3:35 PM

9843019,37772429,53868649,71427757,78364549,79080577,98150021,99591433,
104436889,106457509,111267419,121174811,121174841,168236119,199450099,
203908891,207068803,216618187,230952859,234058871,235524781,253412317,
263651161,268843033,294485363,296239787,297267973,299215331,318764189,
328232783,330905759,338176547,340348241,379879889,409838123,410949373,
424153669,454138211,456157853,494985941,512618927,514395457,557073967,
558582131,567204943,575091337,604948577,628222247,631209119,654377861,
659513467,679277111,682389173,697349699,697385119,702524363,711819929,
726076381,733664731,741844751,757216723,771959533,792582977,798224431,
807145033,814134043,830839783,839733647,840093613,849658721,856316771,
868170431,868965269,870053291,915176959,916042927,933640801,955221299,
959579051,960370597,962889883,966774503,975528691,979840201,981445481,
1010241773,1014361399,1045224287,1045699013,1055050691,1099040821,
1115921627,1121366899,1124024983,1124297563,1126721747,1128318991,
1128319021,1130357321,1146726809,1152015869,1187933567,1205389357,
1206788459,1209421957,1229089283,1239391339,1261934711,1263863561,
1274109713,1277468749,1360079767,1379228407,1381863611,1428580883,
1429749787,1441274839,1460977261,1496897131,1517056417,1526969413,
1536010627,1552068179,1578448447,1583146259,1583795047,1648018019,
1680402509,1680661987,1700689841,1702089713,1715452559,1730431523,
1739854547,1742237683,1743979729,1750795699,1751434913,1754686117,
1764185959,1785372437,1862719889,1880094869,1888318513,1895596537,
1905645499,1905697273,1924852687,1946640637,1956128563,1978917791,
1986229261,1995526999,1997019763,2014899163,2023809911,2031121229,
2051056543,2063279633,2068336393,2072088559,2081350903,2088114907,
2093949799,2120719267,2127895417,2151972181,2157202877,2170572497,
2201579179,2201579209,2206030241,2224631161,2229227401,2277055657,
2291460131,2343405283,2347937869,2367489833,2381232641,2383754419,
2385014713,2391374603,2393786441,2413458233,2414276867,2417113729,
2422151531,2422399919,2424431041,2430042493,2449222267,2451346627,
2466699503,2487010283,2489649551,2489751233,2512825361,2522045677,
2523949733,2526698827,2567685787,2572071259,2587772609,2588592143,
2603502073,2607085147,2619847969,2638252633,2656011607,2659870663,
2664410963,2669503393,2705581223,2712592943,2715239543,2715239573,
2726811047,2769191861,2770053071

Link

A175783 Primes p such that the sum of next p primes is prime.

Sep. 4th, 2010 at 5:19 AM

A175783
{3,5,11,19,37,41,113,149,163,281,421,541,547,607,661,691,739,823,827,941,991,1013,1019,
1093,1249,1453,1549,1553,1583,1619,1733,1747,1951,2069,2153,2239,2399,2549,2609,2663,
2687,2741,2777,2803,2833,2857,2861,2879,2969,3121,3299,3329,3331,3461,3491,3631,3643,
3691,3701,3739,3847,3907,3919,3989,4019,4049,4289,4583,4621,4639,4703,4733,4787,4789,
4967,5039,5437,5503,5639,5683,5689,5737,5743,5827,5851,5897,5953,5987,6047,6053,6133,
6301,6469,6473,6673,6827,6869,7019,7069,7177,7211,7253,7649,7823,7883}
Primes p such that the sum of next p primes is prime.
Primes p such that sum(prime(p+k), k=1..p) is prime.
a(1)=3 because sum s of 3 primes after 3, s=5+7+11=23 is prime
a(2)=5 because sum s of 5 primes after 5, s=7+11+13+17+19=67 is prime
Corresponding sums in A175784
S=Reap[Do[p=Prime[n];If[PrimeQ[s=Sum[Prime[n+k],{k,p}]],Sow[{p,s}]],{n,2,1000}]][[2,1]]
{p,s}
{{3,23},{5,67},{11,353},{19,1187},{37,4691},{41,5843},{113,51749},{149,90677},{163,110069},
{281,350983},{421,819913},{541,1387021},{547,1420039},{607,1773413},{661,2131859},{691,2337761},
{739,2684989},{823,3365581},{827,3402913},{941,4462811},{991,4974943},{1013,5208389},{1019,5274673},
{1093,6124501},{1249,8099023},{1453,11115259},{1549,12715649},{1553,12790949},{1583,13327723},
{1619,13986559},{1733,16111567},{1747,16387883},{1951,20626787},{2069,23326249},{2153,25399483},
{2239,27505129},{2399,31859521},{2549,36101623},{2609,37865083},{2663,39533201},{2687,40308791},
{2741,42109709},{2777,43261411},{2803,44159821},{2833,45107371},{2857,45939617},{2861,46088209},
{2879,46663489},{2969,49767761},{3121,55200209},{3299,61862291},{3329,63120719},{3331,63224671},
{3461,68434697},{3491,69693307},{3631,75778621},{3643,76322759},{3691,78411527},{3701,78885341},
{3739,80604773},{3847,85476977},{3907,88278271},{3919,88905689},{3989,92196427},{4019,93707897},
{4049,95151491},{4289,107430019},{4583,123226241},{4621,125351207},{4639,126369367},{4703,130110373},{4733,131873717},{4787,134924627},{4789,135079577},{4967,145745263},{5039,150330589},{5437,175979107},{5503,180546967},{5639,189836783},{5683,193114267},{5689,193558403},{5737,196956073},{5743,197464711},{5827,203507063},{5851,205337267},{5897,208830359},{5953,212905481},{5987,215333959},{6047,219811549},{6053,220287259},{6133,226472471},{6301,239687281},{6469,253215569},{6473,253579663},{6673,270022009},{6827,283238831},{6869,286980283},{7019,300374119},{7069,304806989},{7177,314358673},{7211,317486647},{7253,321531317},{7649,359146979},{7823,376293607},{7883,382453153}}