PARI's commands

[3.14159]

Originally Posted by 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).

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

[CRGreathouse]

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

Originally Posted by 3.14159

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.

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

Originally Posted by 3.14159

I have the members of this sequence up to 4k.

To 10,000 it's just the 4n+1 primes plus 530 exceptions:

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

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

[3.14159]

Originally Posted by Charles

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

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?

Originally Posted by Charles

To 10,000 it's just the 4n+1 primes plus 530 exceptions:

How do you sort them in numerical order, anyway?

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

[CRGreathouse]

Originally Posted by 3.14159

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

That is what I wrote, yes.

Originally Posted by 3.14159

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?

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.

[3.14159]

Originally Posted by 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.

I think you meant, k * (2^a) ^ (2^a) + 1.

Originally Posted by 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.

The series can be expressed as.. ?

[CRGreathouse]

Originally Posted by 3.14159

I think you meant, k * (2^a) ^ (2^a) + 1.

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

Originally Posted by 3.14159

The series can be expressed as.. ?

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.

[3.14159]

Originally Posted by 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.

Oh, I see my error. Pardon.

Originally Posted by 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.

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 * ....) ??

[CRGreathouse]

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.

[3.14159]

Originally Posted by Charles

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

Speaking of the sums.. The sums converge to this: A094289

Here are the first 300 digits:

0.287358251306224179736418045878932206955908802685881709299499368947089329278688939770209124280029090055929603180132199757677833174625274203928613500682866624372279071764951496386358568820464783694988950221338310990369641738444509170337274489547045606825482008978904241753401587644678759939089840746020

When extended to all integers:

1.29128599706266354040728259059560054149861936827452231731000244513694453876523445555881704112942970898499507092481543054841048741928486419757916355594791369649697415687802079972917794827300902564923055072096663812846701205368574597870300127789412928825355177022238337531934574925996777964830084954911

[3.14159]

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

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

[CRGreathouse]

Originally Posted by 3.14159

Speaking of the sums.. The sums converge to this: A094289

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...%.