cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A147616 Numbers k = concat(a,b) such that sigma(k) = sigma(a) * sigma(b), where sigma = A000203.

Original entry on oeis.org

38, 58, 66, 87, 118, 178, 205, 217, 275, 295, 298, 395, 451, 478, 492, 517, 538, 575, 660, 718, 766, 775, 838, 839, 870, 898, 1018, 1138, 1175, 1195, 1318, 1671, 1678, 1775, 1795, 1975, 2050, 2163, 2170, 2295, 2395, 2518, 2578, 2638, 2665, 2750, 2818
Offset: 1

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Author

M. F. Hasler, Nov 08 2008

Keywords

Comments

Concat(a,b) means decimal concatenation of a and b, i.e., a*10^[log_10(b)+1] + b, since we don't allow leading zeros in b. (However, allowing leading zeros in b would not give any additional term up to at least 10^6.)
This sequence has been suggested by David Wilson on the SeqFan mailing list, Nov 08 2008.
A possible variant would be to allow decomposition of k into an arbitrary number (>1) of substrings. If one requires decomposition of k into each of its digits, this yields A098771.

Examples

			a(1)=38 is in the sequence since sigma(38) = 60 = 4*15 = sigma(3)*sigma(8).

Links

Crossrefs

Cf. A000203, A098771, A147619 (analog for phi), A147624 (analog for omega), A147627 (analog for bigomega).

Programs

  • PARI
    is_A147616(n)={ local(p=1, s=sigma(n)); while( n>p*=10, n%p*10

Extensions

Precision about disallowed leading zeros, fix in PARI code, more cross references. - M. F. Hasler, Nov 09 2008

A147547 Smallest n-digit number m such that phi(10^n+1)=phi(m), gcd(10^n+1,m)=1 and 10 does not divide m, or zero if there is no such m.

Original entry on oeis.org

0, 0, 779, 9991, 90901, 990001, 9090901, 94139561, 681465373, 9898047311, 86925973487, 979104060601, 9080337988583, 95255589092561, 712493161316801, 9926748805307137, 90004044661864321, 989999011990088281, 9090909102763796801, 97910150575731744097, 713349371311332607153, 9789743000892702875281, 88299846937619669895601
Offset: 1

Views

Author

Farideh Firoozbakht, Nov 07 2008

Keywords

Comments

It is easily seen that if m is in the sequence, then phi(m.m)=phi(m)^2 where dot denotes concatenation. So the sequence b(n)=a(n).a(n) is a subsequence of A147619 and it seems that the nonzero terms of this sequence is an infinite subsequence of A147619. If 10^n+1 is prime (n must be of the form 2^k), then a(n)=0 because in this case there is no n-digit number m such that phi(10^n+1)=10^n=phi(m).

Examples

			phi(979104060601)=phi(10^12+1), gcd(10^12+1,979104060601)=1, 10 doesn't divide 979104060601 and 979104060601 is the smallest 12-digit number with these properties so a(12)=979104060601. Note that phi(979104060601.979104060601)=phi(979104060601)^2.

Links

Crossrefs

Cf. A147548, A147549, A147619.

Programs

  • Mathematica
    a[1]=a[2]=0;a[n_]:=(b=10^n+1;c=EulerPhi[b];For[m=c+1,!(Mod[m,10]>0&&GCD[m,b] ==1&&c==EulerPhi[m]),m++ ];m);Do[Print[a[n]],{n,12}]

Extensions

a(13) and a(14) from Max Alekseyev, Mar 12 2009
a(15)-a(20) from Hiroaki Yamanouchi, Aug 27 2014
a(21)-a(59) from Max Alekseyev, Sep 07 2014, Dec 27 2015

A147624 Numbers n = concat(a,b) such that omega(n) = omega(a) * omega(b), where omega = A001221.

Original entry on oeis.org

23, 25, 26, 27, 29, 32, 36, 37, 43, 46, 47, 49, 53, 56, 59, 62, 63, 65, 68, 69, 73, 76, 79, 83, 86, 89, 96, 97, 104, 108, 113, 116, 122, 123, 124, 129, 136, 137, 139, 142, 143, 144, 145, 147, 148, 152, 153, 155, 158, 159, 163, 166, 167, 169, 173, 176, 179, 183, 184
Offset: 1

Views

Author

M. F. Hasler, Nov 08 2008

Keywords

Comments

Concat(a,b) means decimal concatenation of a and b, i.e., a*10^[log[10](b)+1] + b, since we don't allow leading zeros in b.

Examples

			a(1)=23 is in the sequence since omega(23) = 1 = 1*1 = omega(2)*omega(3).
307 is not in this sequence although omega(307) = 1 = 1*1 = omega(3)*omega(07), since we don't allow leading zeros in the second part b.

Crossrefs

Cf. A001221, A147616 (analog for sigma), A147619 (analog for eulerphi), A147627 (analog for bigomega).

Programs

  • PARI
    is_A147624(n)={ local(p=1, o=omega(n)); while( n>p*=10, n%p*10
    A147624(n) & print1(n","))

A147627 Numbers n = concat(a,b) such that bigomega(n) = bigomega(a) * bigomega(b), where bigomega = A001222.

Original entry on oeis.org

23, 26, 28, 34, 37, 39, 53, 62, 65, 73, 74, 78, 93, 95, 104, 113, 119, 125, 134, 137, 138, 142, 143, 145, 155, 156, 173, 182, 193, 194, 197, 207, 211, 212, 213, 214, 215, 217, 221, 223, 226, 229, 230, 233, 235, 238, 241, 242, 244, 245, 249, 253, 260, 262, 265
Offset: 1

Views

Author

M. F. Hasler, Nov 08 2008

Keywords

Comments

Concat(a,b) means decimal concatenation of a and b, i.e., a*10^[log[10](b)+1] + b, since we do not allow leading zeros in b.

Examples

			a(4)=34 is in the sequence since bigomega(34) = 2 = 1*2 = bigomega(3)*bigomega(4).
206 is not in this sequence although bigomega(206) = 2 = 1*2 = bigomega(2)*bigomega(06), we do not allow leading zeros in the second part b.

Crossrefs

Cf. A001222, A147616 (analog for sigma), A147619 (analog for eulerphi), A147624 (analog for omega).

Programs

  • PARI
    is_A147627(n)={ local(p=1, o=bigomega(n)); while( n>p*=10, n%p*10
    A147627(n) & print1(n","))

A147548 a(n) is the largest n-digit number m such that phi(10^n+1)=phi(m), gcd(10^n+1,m)=1 & 10 doesn't divide m and zero if there is no such m.

Original entry on oeis.org

0, 0, 925, 9991, 95969, 995681, 9595969, 99820697, 894463345, 9992684743, 97451082703, 999896409437, 9935266565443, 99974409884813, 999999115863815, 9999446015088757, 99942773726308253, 999999997876532621, 9220779220779220841, 99999797970236297071
Offset: 1

Views

Author

Farideh Firoozbakht, Nov 07 2008

Keywords

Comments

It is easily seen that if m is in the sequence then phi(m.m)=phi(m)^2 where dot means concatenation. So the sequence b(n)=a(n).a(n) is a subsequence of A147619 and it seems that the nenzero terms of this sequence is an infinite subsequence of the sequence A147619. If 10^n+1 is prime (n must be of the form 2^k) then a(n)=0 because in this case there is no n-digit number m such that phi(10^n+1)=10^n=phi(m).

Examples

			phi(894463345)=phi(10^9+1), gcd(10^9+1,894463345)=1, 10 doesn't divide 894463345 and 894463345 is the largesst 9-digit number number with these properties so a(9)=894463345. Note that phi(894463345.894463345)=phi(894463345)^2,

Crossrefs

Cf. A147547, A147549, A147619.

Programs

  • Mathematica
    a[n_]:=(b=10^n+1;c=EulerPhi[b];If[PrimeQ[b],0,For[m=0,!(Mod[m,10]>0&&GCD [10^n-m,b]==1&&c==EulerPhi[10^n-m]),m++ ];10^n-m]);Do[Print[a[n]], {n,9}]

Extensions

a(10)-a(14) from Max Alekseyev, Mar 12 2009
a(15)-a(20) from Hiroaki Yamanouchi, Aug 27 2014

A147549 a(n) is the number of n-digit numbers m such that phi(m)=phi(10^n+1), gcd(10^n+1,m)=1 and 10 doesn't divide m.

Original entry on oeis.org

0, 0, 3, 1, 3, 4, 11, 17, 116, 25, 222, 1806, 54, 223, 302422, 213, 35, 320146, 8, 1403
Offset: 1

Views

Author

Farideh Firoozbakht, Nov 12 2008

Keywords

Comments

If 10^n+1 is prime (n must be of the form 2^k) then a(n)=0 because in this case there is no n-digit number m such that phi(10^n+1) = 10^n = phi(m). I defined this sequence and sequences A147547 and A147548 to answer a question (Nov 06 2008) from M. F. Hasler about the infiniteness of the "primitive" elements (those that aren't multiples of 10) of sequence A147619.

Crossrefs

Cf. A147547, A147548.

Programs

  • Mathematica
    a[n_]:=(b=10^n+1;c=EulerPhi[b];e=b-2; If[PrimeQ[b],0,Length[Select[Range[ c+1,e],Mod[ #,10]>0 && GCD[ #,b]==1 && EulerPhi[b]==EulerPhi[ # ]&]]]); Do[Print[a[n]],{n,9}]

Extensions

a(10)-a(14) from Max Alekseyev, Mar 12 2009
a(15)-a(20) from Hiroaki Yamanouchi, Aug 27 2014
Showing 1-6 of 6 results.

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