[QUOTE=CRGreathouse]When you submit that sequence, you may wish to include its relative density in the primes (you can probably work this out, or else find my post where I give it) and cross-references (at least to A180362).
[/QUOTE]

I made a crossref to A180362. I sent it. It's A175768.

I assume that when you say "no restrictions" you mean "k > 1".

[QUOTE=3.14159;228037]Made sequence A175768 for the sequence with no restriction on k, meaning the 4n + 1 primes are a subset of the sequence, and my guess on how much of the sequence are 4n + 1 primes = 85 to 99 percent.[/QUOTE]

It's 94.7028...%. I prefer to express this as a relative density of 0.527967... in the primes.

[QUOTE=3.14159;228037]I have the members of this sequence up to 4k.[/QUOTE]

To 10,000 it's just the 4n+1 primes plus 530 exceptions:
[code]163,271,379,487,811,919,1459,1567,1783,1999,2539,2647,2971,3079,3187,3511,3727,3943,4051,4159,4483,4591,5023,5347,5563,5779,6211,6427,6967,7507,7723,8263,8803,9127,9343,9883,10099,10531,10639,11071,11287,11503,11719,11827,12043,12583,12799,12907,13339,13879,14419,14851,15391,15607,15823,16363,16903,17011,17443,17551,17659,18199,18307,18523,19387,19603,19819,19927,20143,20359,20899,21871,22303,23059,23167,23599,24247,24571,25111,25219,25759,25867,26083,26407,26731,26839,26947,27271,27487,27919,28027,28351,29863,30187,30403,30727,31051,31159,31267,31699,32563,32779,32887,33211,33427,33751,33967,34183,34939,35803,35911,36343,36451,36559,37423,37747,37963,38287,38611,39043,39367,39799,40123,40231,40771,40879,41203,41851,41959,42283,42391,42499,43579,44119,44983,45307,45523,45631,46171,46279,46819,47143,47251,47791,48871,49411,49627,49843,50383,50599,50707,50923,51031,51679,51787,52543,53299,53407,53623,53731,54163,54919,55243,55351,56431,57727,57943,58699,59023,59239,59671,59779,59887,60103,60427,60859,61291,61507,61723,62047,63127,63559,63667,64747,65071,65179,65287,65719,65827,67231,67339,67447,67987,68311,68743,69067,69499,69931,70039,70687,71011,71119,71443,71551,71983,72091,72307,72739,73063,73387,73819,74143,75223,75979,76303,76519,77167,77383,77491,78031,78139,78571,78787,79111,79867,80191,80407,81163,81703,81919,82351,82567,82891,83431,85159,86131,86239,87103,87211,87427,87643,87751,89371,90019,90127,91099,91423,91639,92179,92503,93151,94447,94771,95203,95311,95419,95527,95959,96823,96931,97039,97579,97687,98011,98227,98443,99523,100279,100927,101359,101467,102547,102763,102871,103087,103843,103951,104059,104383,104491,104707,105031,106219,106543,106759,106867,107839,108271,108379,109567,109891,110323,110431,110647,110863,111187,112807,113023,113131,113779,114319,114643,114859,114967,115183,115399,115831,116047,116371,116803,116911,117127,117883,117991,118423,118747,118751,119179,119503,119611,119827,120691,120907,121123,121447,122203,122527,122743,123499,123931,124147,124363,124471,125119,125551,125659,126199,126307,126631,126739,127711,127819,128467,128683,129223,129439,129763,130087,130303,130411,130843,131059,131251,131707,132247,132679,133327,133543,134191,134731,134839,134947,135271,136027,136351,136999,138079,138403,138511,138727,139267,139483,139591,140779,141319,142183,142939,143263,144451,144667,144883,145207,145423,145531,145963,146719,147151,147583,147799,148123,148339,148663,149419,150067,150607,151471,151579,151687,151903,152443,152767,153523,153739,154279,154387,154927,155251,156007,156979,157303,157411,157519,157627,157951,158923,159463,159571,159787,160651,161407,161731,161839,161947,162703,163027,163243,163351,163567,164431,165079,165511,166807,167023,167779,167887,168211,168643,169399,169831,170047,170263,170371,171559,172423,173827,174259,174367,174583,174799,174907,175447,175663,176419,177283,177823,178039,178903,179119,179659,180307,180847,181063,181387,181603,181711,181927,182467,182899,183439,183763,183871,183979,184087,184627,184843,185167,185491,185599,185707,185923,186247,186679,187003,187111,187219,187651,188299,188407,189271,190027,190243,190783,190891,192187,193051,193751,194239,194671,195103,195319,195427,195751,195967,196291,196831,197371,197479,197803,198127,199207,201151,201907,202231,202339,202879,202987,203311,203419,204067,204931,206251,206551,207199,207307,207523,207847,208279,208387,208927,209359[/code]

To 1e5 there are 5313 exceptions, but b-files are usually limited to 10,000 terms.

[QUOTE=Charles]It's 94.7028...%. I prefer to express this as a relative density of 0.527967... in the primes.
[/QUOTE]

Excellent. Slightly less than 95% of them, or about 18 parts in 19 are 4n + 1 primes. How did you manage to work out the percentage?

[QUOTE=Charles]To 10,000 it's just the 4n+1 primes plus 530 exceptions:
[/QUOTE]

How do you sort them in numerical order, anyway?

Also: I assume you listed the set of primes that were not 4n + 1s.

[QUOTE=3.14159;228048]I assume you listed the set of primes that were not 4n + 1s.[/QUOTE]

That is what I wrote, yes.

[QUOTE=3.14159;228048]Excellent. Slightly less than 95% of them, or about 18 parts in 19 are 4n + 1 primes. How did you manage to work out the percentage?[/QUOTE]

Clearly all primes of the form k * 4^4 + 1 are also of the form k1 * 2^2 + 1, and in general so are all primes of the form k * (2a)^(2a) + 1. Generalizing further, we need only consider primes; numbers of the form k * 25^25 + 1 are also of the form k1 * 5^5 + 1.

This avoids the general use of inclusion-exclusion (which requires a great number of terms) since powers of distinct primes are coprime. So you have a simple, rapidly-converging infinite product over primes. The first dozen primes are sufficient to calculate 60 decimal places.

[QUOTE=Charles]Clearly all primes of the form k * 4^4 + 1 are also of the form k1 * 2^2 + 1, and in general so are all primes of the form k * (2a)^(2a) + 1. Generalizing further, we need only consider primes; numbers of the form k * 25^25 + 1 are also of the form k1 * 5^5 + 1.
[/QUOTE]

I think you meant, k * (2[sup]a[/sup]) ^ (2[sup]a[/sup]) + 1.

[QUOTE=Charles]This avoids the general use of inclusion-exclusion (which requires a great number of terms) since powers of distinct primes are coprime. So you have a simple, rapidly-converging infinite product over primes. The first dozen primes are sufficient to calculate 60 decimal places.
[/QUOTE]

The series can be expressed as.. ?

[QUOTE=3.14159;228055]I think you meant, k * (2[sup]a[/sup]) ^ (2[sup]a[/sup]) + 1.[/QUOTE]

No. If that was all I knew I couldn't avoid terms like 6^6 which would cause a combinatorial explosion in the calculation.

[QUOTE=3.14159;228055]The series can be expressed as.. ?[/QUOTE]

See if you can come up with it! 1 in 4 are 1 mod 4, 1 in 27 are 1 mod 27, so since 4 and 27 are relatively prime, 1/4 + 1/27 - 1/(4 * 27) are 1 mod 4 or 1 mod 27.

[QUOTE=Charles]No. If that was all I knew I couldn't avoid terms like 6^6 which would cause a combinatorial explosion in the calculation.
[/QUOTE]

Oh, I see my error. Pardon.
[QUOTE=Charles]See if you can come up with it! 1 in 4 are 1 mod 4, 1 in 27 are 1 mod 27, so since 4 and 27 are relatively prime, 1/4 + 1/27 - 1/(4 * 27) are 1 mod 4 or 1 mod 27.
[/QUOTE]

1/4 + 1/27 + 1/3125 + 1/823543 + 1/285311670611 + 1/302875106592253 + 1/827240261886336764177 + 1/1978419655660313589123979 + 1/20880467999847912034355032910567 + .... - 1/(4 * 27 * 3125 * 823543 * 285311670611 * 302875106592253 * 827240261886336764177 * 1978419655660313589123979 * 20880467999847912034355032910567 * ....) ??

That will be too large, since it's only removing one congruence class mod H (the huge product). You removed 1 mod H, but you also need to remove 1 + 4*27, 1 + 2*4*27, ... mod H.

I suggest looking at it as a product rather than a sum, it's easier to express.

[QUOTE=Charles]I suggest looking at it as a product rather than a sum, it's easier to express.
[/QUOTE]

Speaking of the sums.. The sums converge to this: [URL="http://www.research.att.com/~njas/sequences/A094289"]A094289[/URL]

Here are the first 300 digits: [code]0.287358251306224179736418045878932206955908802685881709299499368947089329278688939770209124280029090055929603180132199757677833174625274203928613500682866624372279071764951496386358568820464783694988950221338310990369641738444509170337274489547045606825482008978904241753401587644678759939089840746020[/code]

When extended to all integers:

[code]1.29128599706266354040728259059560054149861936827452231731000244513694453876523445555881704112942970898499507092481543054841048741928486419757916355594791369649697415687802079972917794827300902564923055072096663812846701205368574597870300127789412928825355177022238337531934574925996777964830084954911[/code]

Factoring each for a bit: Former: 2 ^ 2 * 5 * 7 * 62039 * 4685224417 * c283

Latter: 43 * 199 * 353 * 285101 * 546233 * 72659134783 * 74772299267 * c260

[QUOTE=3.14159;228063]Speaking of the sums.. The sums converge to this: [URL="http://www.research.att.com/~njas/sequences/A094289"]A094289[/URL] [/QUOTE]

Right. That's too big, though -- it double-counts 4 residues mod 108, for example, making it too large by at least 4/108 = 3.7...%.